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@@ -136,3 +136,8 @@ aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf filter=lfs diff=lfs merge=lfs -tex
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3tFKT4oBgHgl3EQf8i6_/content/tmp_files/2301.11950v1.pdf.txt

@@ -0,0 +1,1654 @@
+TUM-EFT 173/22
+Strong decays of T +
+cc at NLO in an effective field theory
+Lin Dai,1, ∗ Sean Fleming,2, † Reed Hodges,3, ‡ and Thomas Mehen3, §
+1Physik Department, Technische Universit¨at M¨unchen, 85748 Garching, Germany
+2Department of Physics and Astronomy,
+University of Arizona, Tucson, Arizona 85721, USA
+3Department of Physics, Duke University,
+Durham, North Carolina 27708, USA
+Abstract
+The T +
+cc exotic meson, discovered by the LHCb collaboration in 2021, can be interpreted as a
+molecular state of D(∗)0 and D(∗)+ mesons. We compute next-leading order (NLO) contributions to
+the strong decay of T +
+cc in an effective field theory for D mesons and pions, considering contributions
+from one-pion exchange and final state rescattering. Corrections to the total width, as well as the
+differential distribution in the invariant mass of the final state D meson pair are computed. The
+results remain in good agreement with LHCb experimental results when the NLO contributions
+are added. The leading uncertainties in the calculation come from terms which depend on the
+scattering length and effective range in D meson scattering.
+∗Electronic address: lin.dai@tum.de
+†Electronic address: spf@email.arizona.edu
+‡Electronic address: reed.hodges@duke.edu
+§Electronic address: mehen@phy.duke.edu
+1
+arXiv:2301.11950v1  [hep-ph]  27 Jan 2023
+
+I.
+INTRODUCTION
+The LHCb collaboration has observed a narrow resonance, the exotic tetraquark T +
+cc,
+in the final state D0D0π+ [1–5]. The resonance is close to both the D∗0D+ and D∗+D0
+thresholds. When using a unitarized Breit-Wigner profile appropriate for a coupled channel
+problem, LHCb finds the difference between the resonance mass and the D∗+D0 threshold,
+δm, and the decay width, Γ, to be: [5]
+δm = −360 ± 40+4
+−0 keV ,
+Γ = 48 ± 2+0
+−14 keV .
+(1)
+The D∗0D+ threshold is 1.7 MeV above the resonance. The closeness of the resonance to
+the two thresholds suggests the possibility that T +
+cc has a molecular nature.
+After the announcement of the discovery of T +
+cc, many theory papers attempted to under-
+stand various aspects of the exotic meson [6–26]. Several papers tried to predict its decay
+width and differential decay width, with considerable success [6, 7, 10, 13, 14, 20, 21]. In one
+of these papers [6], we wrote down an effective field theory for T +
+cc considering it a molecular
+state of two D mesons treated nonrelativistically, and computed leading-order strong and
+electromagnetic decays. Special attention was paid to the coupled channel nature of the
+problem. We found a decay width of 52 keV when the tetraquark is in an isospin-0 state,
+using a value of δm = −273 keV, which arises from using a relativistic P-wave two-body
+Breit-Wigner function with a Blatt-Weisskopf form factor. This was in good agreement with
+the LHCb experiment. The predicted differential spectra as a function of the invariant mass
+of the final state charm meson pair were also in good agreement with the binned experimen-
+tal data. In this paper we investigate how these conclusions are affected by next-to-leading
+order (NLO) strong decays.
+The effective theory we will use is similar to XEFT for the χc1(3872) [27–41].
+Refs.
+[27, 42, 43] have considered NLO XEFT diagrams for χc1(3872) decays. One-pion exchange
+was found to have a negligible contribution to the decay width [27, 43], while final state
+rescattering led to uncertainty in the decay rate of +50%
+−30% when the binding energy of the
+χc1(3872) is 0.2 MeV [43]. The differential spectrum dΓ[χc1(3872) → D0 ¯D0π0]/dEπ was
+found to have a curve whose peak location and overall shape are insensitive to NLO correc-
+tions; only the normalization is affected [43]. The sharply peaked nature of the differential
+2
+
+spectrum can inform about the molecular nature of the χc1(3872): since it is a function of
+the virtual D∗0 propagator (p2
+D + γ2)−1, where γ is the binding momentum, as the binding
+energy goes to zero the distribution becomes sharply peaked as pD → 0.
+By analogy with this earlier work on χc1(3872), in this paper we compute NLO contri-
+butions to the decay of T +
+cc to find the uncertainties due to one-loop one-pion exchange and
+final state rescattering diagrams. We calculate the uncertainty in the decay width, as well
+as in the shape, peak location, and normalization of differential spectra. The calculation is
+complicated by the presence of a coupled channel, which is not present for χc1(3872). We
+find the decay width including NLO corrections to be 47+53%
+−25% keV, which is consistent with
+XEFT [43]. We also discuss the physical significance of several of the parameters in the
+effective theory, and their effect on the decay width.
+In Section II we write down the effective Lagrangian to NLO. The required Feynman
+diagrams and their amplitudes, along with the explicit formulae for the partial widths are
+shown in Section III. Plots of the differential distribution are shown in Section IV, followed
+by concluding remarks in Section V.
+II.
+EFFECTIVE LAGRANGIAN
+The leading-order effective Lagrangian for strong decays of T +
+cc is [6]
+LLO = H∗i†
+�
+i∂0 +
+∇2
+2mH∗ − δ∗
+�
+H∗i + H†
+�
+i∂0 + ∇2
+2mH
+− δ
+�
+H
++ g
+fπ
+H†∂iπH∗i + H.c.
+−C(0)
+0 (H∗Tτ2H)†(H∗Tτ2H) − C(1)
+0 (H∗Tτ2τaH)†(H∗Tτ2τaH) .
+(2)
+Here H and H∗ are isodoublets of the pseudoscalar and vector charm meson fields, respec-
+tively, and π is the usual matrix of pion fields. The diagonal matrices δ and δ∗ contain the
+residual masses, which are the difference between the mass of the charm meson D(∗)i, where
+i = 0, +, and that of the D0. The coupling g = 0.54 is the heavy hadron chiral perturbation
+theory (HHχPT) axial coupling [44–46] and fπ = 130 MeV is the pion decay constant. The
+terms on the last two lines are contact interactions mediating D∗D scattering, where C(n)
+0
+mediates S-wave scattering in the isospin-n channel, and τa are Pauli matrices acting in
+isospin space.
+3
+
+Several new classes of terms appear at NLO in the effective theory. There are new contact
+interactions involving two derivatives:
+LC2 = C(0)
+2
+4 (H∗Tτ2H)†(H∗Tτ2
+←→
+∇ 2H) + C(1)
+2
+4 (H∗Tτ2τaH)†(H∗Tτ2τa
+←→
+∇ 2H) .
+(3)
+These interactions occur in XEFT and are proportional to the effective range [27]. We can
+also write down Dπ interaction terms by constructing isospin invariants out of the fields.
+LCπ = C(1/2)
+π
+(πH)†(πH) + C(3/2)
+π
+�
+vaH − 1
+3τaπH
+�†�
+vaH − 1
+3τaπH
+�
+.
+(4)
+Here v =
+�
+π1 π2 π0
+�T
+/
+√
+2 is a vector of pion fields, with π± ≡ (π1 ∓ iπ2)/
+√
+2, such that
+vaτa = π. C(1/2)
+π
+and C(3/2)
+π
+mediate scattering in the isospin-1/2 and isospin-3/2 channels,
+respectively. The interactions which are relevant to our calculation are:
+LCπ → C(1)
+π D0†π0†D+π− − C(1)
+π D+†π0†D0π+ + H.c.
++C(2)
+π D0†π0†D0π0 + C(2)
+π D+†π0†D+π0
++C(3)
+π D0†π+†D0π+ ,
+(5)
+where the couplings C(1)
+π , C(2)
+π , and C(3)
+π
+are particular linear combinations of C(1/2)
+π
+and C(3/2)
+π
+as governed by Eq. (4). These interactions can be matched onto the chiral Lagrangian [47].
+The values we use for these Cπ couplings are computed from lattice data; see Appendix C
+for details.
+We can write down D∗D → DDπ interactions by using the same strategy of constructing
+isospin invariants out of the fields. That would lead to:
+LB1 = B(I=0)
+1
+εαβ(H∗
+αHβ)†(Hτ2τiH∇vi)
++B(I=1)
+1
+(H∗τ2τkH)†(εijkHτ2τiH∇vj) + H.c. .
+(6)
+However, we need isospin-breaking terms in order to fully renormalize the theory at NLO,
+so ultimately we have four unique B1 couplings, one for each possible channel. Written in
+terms of the charm meson fields, the interactions become:
+LB1 → B(1)
+1 (D+D∗0)†(D+D0∇π0) + B(2)
+1 (D0D∗+)†(D+D0∇π0)
++B(3)
+1
+2 (D0D∗+)†(D0D0∇π+) + B(4)
+1
+2 (D+D∗0)†(D0D0∇π+) .
+(7)
+4
+
+Relations between the B(i)
+1
+implied by Eq. (6) are given in the Appendix. We can construct
+DD contact terms out of the isospin invariants. There are only interactions in the isospin-1
+channel,
+LC0D = C(1)
+0D(Hτ2τaH)†(Hτ2τaH)
+→ C(1)
+0D
+2 (D0D0)†(D0D0) + C(1)
+0D(D+D0)†(D+D0) ,
+(8)
+where in the second line we have restricted to terms that are relevant to our calculation.
+The authors in Ref. [43] chose to vary their C(1)
+0D coupling, which described D ¯D scattering
+as opposed to DD, over a range of [−1, 1] fm2. We test several different values for it within
+that range. Lastly, we need a kinetic term for the pions; in contrast to XEFT, we treat them
+relativistically,
+Lπ = tr(∂µπ†∂µπ − m2
+ππ†π) .
+(9)
+The full NLO Lagrangian is then LNLO = LC2 + LCπ + LB1 + LC0D + Lπ.
+III.
+FORMULAE FOR DECAY WIDTHS
+Writing down the decay width for the T +
+cc at NLO requires care due to the coupled channel
+nature of the problem. We define a two-point correlation function matrix ˆG as
+ˆG =
+�
+d4x e−iEt ⟨0|T[X(x)XT(0)]|0⟩ = iΣ(1 + CΣ)−1 ,
+(10)
+where the interpolating field is
+X =
+�
+�
+�
+D0D∗+
+D+D∗0
+�
+�
+� .
+(11)
+The right-hand side of Eq. (10) arises from expressing ˆG to all orders as an infinite sum
+of the C0-irreducible two-point function Σ, in a manner similar to that in Appendix A of
+Ref. [48], but here C0 and Σ are matrices due to the presence of a coupled channel. −iΣ is
+given by the sum of D∗D self-energy diagrams in Fig. 1. Its diagonal elements correspond
+to those two-point diagrams which do not swap channels, and the off-diagonal elements to
+those which do swap channels. We can then project out the isospin-0 and isospin-1 channels,
+5
+
+−iΣ
+=
++
++
+C2
++
++
+C0D
++
+Cπ
++
+B1
+FIG. 1: Some of the D∗D self-energy diagrams contributing to −iΣ. Bold solid lines represent D∗
+mesons, regular solid lines represent D mesons, and dashed lines represent pions. The first row is
+LO, the second row is NLO, and the third and fourth rows are NNLO. There are also other NNLO
+diagrams not shown which are C0-reducible combinations of the NLO diagrams.
+and tune the parameters of the two-point correlators so that there is a pole corresponding
+to the location of the T +
+cc bound state. Near the vicinity of the pole, the Green’s function
+can be written as
+G0/1 =
+�
+�
+�
+1
+∓1
+�
+�
+�
+T
+ˆG
+�
+�
+�
+1
+∓1
+�
+�
+� ≈ 1
+2
+iZ0/1
+E + ET +
+iΓ0/1
+2
+,
+(12)
+where Γ0/1 is the decay width and the residue Z0/1 is the wave function renormalization. We
+find for the decay width in the isospin-0 channel
+ΓNLO
+0
+≈ −ΓLO Re Σ′NLO
+0
+(−ET)
+Re tr Σ′LO(−ET) + 2 Im ΣNLO
+0
+(−ET)
+Re tr Σ′LO(−ET) ,
+(13)
+where Σ0 ≡ Σ11 + Σ22 − Σ12 − Σ21 is a particular combination of the elements of the Σ
+matrix appropriate for isospin-0. The first term of Eq. (13) is a correction to the LO decay
+width from NLO D∗D self-energy corrections, i.e., diagrams on the second row of Fig. 1.
+The second term of Eq. (13) consists of NLO decay diagrams, from various cuts of diagrams
+6
+
+on the third and fourth rows of Fig. 1. Note that Im ΣNLO is from Σ diagrams of one
+higher order than in Re ΣNLO because the LO self-energy graph has no imaginary part
+below threshold. The derivatives of Σ are with respect to E and evaluated at E = −ET.
+For a more detailed derivation of Eq. (13) refer to Appendix A.
+Three diagrams in Fig. 1 contribute to Re Σ to NLO. They are the LO self-energy dia-
+gram (−iΣ1), the one-pion exchange diagram (−iΣ2), and the C2 contact diagram (−iΣ3).
+They are evaluated in the power divergence subtraction (PDS) scheme [49]. This scheme
+corresponds to using MS to handle logarithmic divergences as well as subtracting poles in
+d = 3 to keep track of linear divergences. A 1/ϵ pole appears in Σ2, but the dependence
+on the renormalization scale drops out when the derivative with respect to E is taken. We
+neglect terms in the propagators that go as p4/m2
+H or (δm)p2/mH, where δm is of the order
+of the pion mass, compared to p2. In Σ2 and Σ3 we use a Fourier transform to evaluate
+the integrals over three-momentum, using a procedure outlined in Ref. [50]. We define a
+reduced mass µ(m1, m2) ≡ m1m2/(m1 + m2) and the binding momenta are defined to be
+γ2(m1, m2) = 2µ(m1, m2)(m1 + m2 − mT). The expressions for the self energy diagrams are:
+−iΣ1(m, m∗) = −iµ(m, m∗)
+2π
+[ΛPDS − γ(m, m∗)] ,
+(14)
+−iΣ2(m1, m∗
+1, m2, m∗
+2, mπ, g1, g2) = −4ig1g2
+3
+µ(m1, m∗
+1)µ(m2, m∗
+2)
+×
+�
+1
+16π2[ΛPDS − γ(m1, m∗
+1)][ΛPDS − γ(m2, m∗
+2)]
++(m∗
+2 − m1)2 − m2
+π
+(8π)2
+�1
+ϵ + 2
+−4 log
+�
+γ(m1, m∗
+1) + γ(m2, m∗
+2)
+−i(m∗
+2 − m1)2 + im2
+π
+�
+− 4 log µ
+��
+,
+(15)
+−iΣ3(m1, m∗
+1, m2, m∗
+2, C2) = − i
+4π2C2[γ2(m1, m∗
+1) + γ2(m2, m∗
+2)]µ(m1, m∗
+1)
+×µ(m2, m∗
+2)[ΛPDS − γ(m1, m∗
+1)][ΛPDS − γ(m2, m∗
+2)] . (16)
+To be consistent with the implementation of the PDS scheme in the decay diagrams (see
+Appendix B), for the double integral in Σ2 we have used rotational symmetry to replace
+7
+
+p
+m
+(a)
+g1
+p
+pπ
+g2
+g3
+m∗
+1
+mext
+mπ
+m
+m∗
+2
+(b)
+pπ
+p
+Cπ
+mπ
+m
+(c)
+C2
+m∗
+1
+m
+p
+m∗
+2
+mext
+(d)
+B1
+m
+(e)
+C0D
+m∗
+m1
+(f)
+FIG. 2: Feynman diagrams at LO and NLO contributing to the decay of T +
+cc. We label the vertices
+and lines whose naming might be ambiguous. These diagrams arise from cuts of the diagrams on
+the third and fourth lines of Fig. 1.
+the tensor structure in the numerator with δij/3 and not δij/(d − 1).
+This choice does
+not affect the derivative of Σ2 as it only changes the constant terms which drop out upon
+differentiation with respect to E.
+The decay diagrams that contribute to 2 Im ΣNLO
+0
+(−ET) are shown in Fig. 2. By the
+optical theorem the square of these diagrams are given by the sum over the cuts of the
+NNLO diagrams in Fig. 1. If there is only one pion/charm meson vertex in a diagram, its
+coupling is labeled gπ. If there are more than one such vertex, the couplings are numbered
+gi. Depending on the type of pion and charm meson, these couplings will be either g/fπ or
+±g/(
+√
+2fπ).The expressions are written in terms of the basis integrals given in Appendix B.
+These basis integrals depend on parameters b, c1, and c2, the definitions for c1 and c2 are
+provided where appropriate, b = 1 unless otherwise specified, and the momentum arguments
+for the integrals are p unless otherwise specified.
+8
+
+iA(2a)(p, m, m∗, gπ) = 2igπϵT · pπµ(m, m∗)
+p2 + γ2(m, m∗)
+.
+(17)
+iA(2b)(p, m, mext, mπ, m∗
+1, m∗
+2, g1, g2, g3) = 4iµ(m, m∗
+1)µ(mext, m∗
+2)g1g2g3
+p2 + γ2(mext, m∗
+2)
+×
+�
+ϵT · p pπ · p
+�
+I(2)
+0
+− 2I(1) + I
+�
++ϵT · pπp2I(2)
+1
+�
+,
+(18)
+c1 = γ2(m, m∗
+1) ,
+c2 = p2 − (mT − m − mext)2 + m2
+π .
+iA(2c)(m, mext, mπ, m∗, gπ, Cπ) = 2iµ(m, m∗)gπCπϵT · p[I(1) − I] ,
+(19)
+c1 = γ2(m, m∗) ,
+c2 = p2 − (mT − m − mext)2 + m2
+π .
+iA(2d)(m, mext, m∗
+1, m∗
+2, gπ, C2) = 1
+πiC2gπϵT · pπµ(m, m∗
+1)µ(mext, m∗
+2)
+× p2 − γ2(m, m∗
+1)
+p2 + γ2(mext, m∗
+2)[γ(m, m∗
+1) − ΛPDS] .
+(20)
+iA(2e)(m, m∗, B1) = −iB1
+2π ϵT · pπµ(m, m∗)[γ(m, m∗) − ΛPDS] .
+(21)
+iA(2f)(m1, m2, m∗, p0
+π, gπ, C0D) = 4iµ(m1, m2)µ(m2, m∗)gπC0DϵT · pπI(pπ) , (22)
+c1 = γ2(m2, m∗) ,
+c2 = −2µ(m1, m2)
+�
+mT − m1 − m2 − p0
+π − p2
+π
+2m1
+�
+,
+b = µ(m1, m2)
+m1
+.
+Following Eq. (13) and using the amplitudes defined above, the decay widths for the two
+strong decays of T +
+cc are
+9
+
+dΓNLO
+0
+(T +
+cc → D+D0π0)
+dp2
+0dp2
++
+=
+2
+Re tr Σ′LO(−ET)Re
+�
+A(2a)(p+, m+, m∗
+0, −g/
+√
+2fπ)
+×
+�
+A(2b)(p0, m+, m0, mπ0, m∗
+0, m∗
++, −g/
+√
+2fπ, g/
+√
+2fπ, g/
+√
+2fπ)
++A(2b)(p+, m+, m+, mπ−, m∗
+0, m∗
+0, g/fπ, g/fπ, −g/
+√
+2fπ)
+−A(2b)(p0, m0, m0, mπ+, m∗
++, m∗
++, g/fπ, g/fπ, g/
+√
+2fπ)
+−A(2b)(p+, m0, m+, mπ0, m∗
++, m∗
+0, g/
+√
+2fπ, −g/
+√
+2fπ, −g/
+√
+2fπ)
++A(2c)(p0, m+, m0, mπ0, m∗
+0, −g/
+√
+2fπ, C(2)
+π )
+−A(2c)(p0, m0, m0, mπ+, m∗
++, g/fπ, C(1)
+π )
++A(2f)(m0, m+, m∗
+0, −g/
+√
+2fπ, C(1)
+0D)
+−A(2f)(m+, m0, m∗
++, g/
+√
+2fπ, C(1)
+0D)
+�∗
++ (D0 ↔ D+, π+ ↔ π−)
+�
+−
+1
+Re tr Σ′LO(−ET)
+�
+[β1(p2
++ + γ2
++) + β2]
+���A(2a)(p+, m+, m∗
+0, −g/
+√
+2fπ)
+��2
+−A(2a)(p0, m0, m∗
++, g/
+√
+2fπ)A∗
+(2a)(p+, m+, m∗
+0, −g/
+√
+2fπ)
+�
++[β3(p2
+0 + γ2
+0) + β4]
+���A(2a)(p0, m0, m∗
++, g/
+√
+2fπ)
+��2
+−A(2a)(p+, m+, m∗
+0, −g/
+√
+2fπ)A∗
+(2a)(p0, m0, m∗
++, g/
+√
+2fπ)
+��
+−dΓLO
+0 (T +
+cc → D+D0π0)
+dp2
+0dp2
++
+Re Σ′NLO
+0
+Re tr Σ′LO
+����
+C2→0,E=−ET
+(23)
+10
+
+dΓNLO
+0
+(T +
+cc → D0D0π+)
+dp2
+1dp2
+2
+=
+1
+Re tr Σ′LO(−ET)Re
+�
+A(2a)(p2, m0, m∗
++, g/fπ)
+×
+�
+A(2b)(p1, m0, m0, mπ+, m∗
++, m∗
++, g/fπ, g/fπ, g/fπ)
++A(2b)(p2, m0, m0, mπ+, m∗
++, m∗
++, g/fπ, g/fπ, g/fπ)
+−A(2b)(p1, m+, m0, mπ0, m∗
+0, m∗
++, −g/
+√
+2fπ, g/
+√
+2fπ, g/fπ)
+−A(2b)(p2, m+, m0, mπ0, m∗
+0, m∗
++, −g/
+√
+2fπ, g/
+√
+2fπ, g/fπ)
++A(2c)(p1, m0, m0, mπ+, m∗
++, g/fπ, C(3)
+π )
+−A(2c)(p1, m+, m0, mπ0, m∗
+0, −g/
+√
+2fπ, C(1)
+π )
++A(2f)(m0, m0, m∗
++, g/fπ, C(1)
+0D/2)
+�∗
++ (p1 ↔ p2)
+−
+�2gµ0
+fπ
+�2p2
+π
+3 β5
+�
+1
+p2
+1 + γ2
+0
++
+1
+p2
+2 + γ2
+0
+��
+−dΓLO
+0 (T +
+cc → D0D0π+)
+dp2
+1dp2
+2
+�
+β4 + Re Σ′NLO
+0
+Re tr Σ′LO
+����
+C2→0,E=−ET
+�
+(24)
+In the previous formulae we have used subscripts on µ and γ to indicate which charm
+meson is a pseudoscalar in that particular channel, e.g., µ0 = µ(m0, m∗
++). The combinations
+of self-energy diagrams that we need are Re tr Σ′LO(−ET) and Re Σ′NLO
+0
+(−ET, C2 → 0). In
+terms of the functions defined above, these are given by:
+Re tr Σ′LO = Re Σ′
+1(m0, m∗
++) + Re Σ′
+1(m+, m∗
+0) ,
+Re Σ′NLO
+0
+|C2→0 = Re
+�
+Σ′
+2(m+, m∗
+0, m+, m∗
+0, mπ+, g/fπ, g/fπ)
++Σ′
+2(m0, m∗
++, m0, m∗
++, mπ+, g/fπ, g/fπ)
++Σ′
+2(m+, m∗
+0, m0, m∗
++, mπ0, −g/
+√
+2fπ, g/
+√
+2fπ)
++Σ′
+2(m0, m∗
++, m+, m∗
+0, mπ0, g/
+√
+2fπ, −g/
+√
+2fπ)
+�
+(25)
+The expressions for βi are given in Appendix C. The terms dependent on A(2b) and Re Σ′
+2
+have linear divergences that must cancel against each other. They cancel exactly in the limit
+µ0 = µ+. We make that approximation in those terms only to ensure the cancellation; it
+is a reasonable approximation as µ0/µ+ ≈ 0.99948. See Appendix B for more discussion of
+these linear divergences.
+11
+
+3730
+3732
+3734
+3736
+3738
+0
+20
+40
+60
+80
+100
+FIG. 3: A plot of the differential decay width as a function of the invariant mass of the final state
+D meson pair. Solid lines represent the LO calculation; the dashed lines represent the addition
+of non-analytic and NLO self-energy corrections. Overlaid is the binned experimental data from
+LHCb, with the background subtracted.
+IV.
+DIFFERENTIAL DECAY DISTRIBUTIONS AND PARTIAL WIDTHS
+Once we have formulae for the T +
+cc → DDπ partial widths, we can numerically integrate
+over part of three-body phase space in Mathematica and plot the differential distribution
+dΓ/dmDD. It is insightful to compare our predicted curves to the LHCb experimental data
+for the total yield. This will inform us about the effect and importance of the different
+interactions in the effective theory. We normalize our distributions by performing a least-
+squares fit of the LO distribution to the data, and using the same normalization factor for
+the NLO distributions. The Cπ decay diagrams, individually and as a whole, contribute
+negligibly to the distributions. The parameters β1, β3, and β5 also have a small impact on
+the distributions over the range in which we vary them. We therefore do not show plots
+varying these parameters individually.
+The contributions from the non-C2-dependent NLO self-energy corrections (i.e. the first
+12
+
+3730
+3732
+3734
+3736
+3738
+0
+20
+40
+60
+80
+100
+FIG. 4: A plot of the differential decay width as a function of the invariant mass of the final state
+D meson pair. Solid lines represent the LO calculation; The dashed and dotted lines represent
+two different ranges for C0D.
+Overlaid is the binned experimental data from LHCb, with the
+background subtracted.
+diagram on the second line of Fig. 1), as well as the contributions from Fig. 2b, serve to
+increase the partial widths by a small but noticeable amount (Fig. 3). The effect of the C0D,
+β2, and β4 terms on the distributions can be significant. In the following we will investigate
+their impact by setting all other contributions to dΓNLO/dmDD to zero and varying them
+individually.
+The C0D interaction has a sizeable contribution to the partial widths, as evidenced in
+Fig. 4, where we plot the differential distributions and vary this coupling in two possible
+ranges: C0D ∈ [−1, 1] fm2 and ∈ [−0.25, 0.25] fm2. Its effect on the neutral pion decay is
+twice as large as on the charged pion decay, because the coupling of charged pions to D
+mesons is bigger by a factor of
+√
+2. Clearly the differential distributions are sensitive to the
+coupling’s magnitude. If C0D is +1 fm2 the peak of theD+D0 mass distribution is too high,
+and if it is −1 fm2 three higher data points are underpredicted. It would be interesting to
+13
+
+3730
+3732
+3734
+3736
+3738
+0
+50
+100
+150
+FIG. 5: A plot of the differential decay width as a function of the invariant mass of the final state
+D meson pair. Solid lines represent the LO calculation. The dashed and dotted lines represent
+two different values of β2 and β4. Overlaid is the binned experimental data from LHCb, with the
+background subtracted.
+do a more careful analysis of the constraints this data puts on C0D but that is beyond the
+scope of this paper. C0D is directly proportional to the I = 1 D meson scattering length,
+so more precise knowledge of C0D from lattice simulations or experiments would allow us to
+sharpen our predictions for T +
+cc.
+We can glean the significance of β2 and β4 by taking the isospin limit m0 = m+. In
+Appendix C we see that in this limit:
+β2 = β4 = −γr0 ,
+(26)
+where γ is the binding momentum and r0 is the effective range in the I = 0 channel. The
+effective range is positive and we expect γr0 < 1. In Fig. 5, we plot the distribution with all
+other NLO interactions turned off, and for two values of β2 = β4 ≡ β: −0.1 and −0.59, along
+with the LO curve (β = 0). We get γr0 = 0.59 if we use the largest binding momentum
+(γ+) and r0 = 1/(100 MeV).
+For nucleons, r0 ≈ 1/(100 MeV); since charm mesons are
+14
+
+LO result NLO lower bound NLO upper bound
+Γ[T +
+cc → D0D0π+]
+28
+21
+44
+Γ[T +
+cc → D+D0π0]
+13
+7.8
+21
+Γstrong[T +
+cc]
+41
+29
+66
+Γstrong[T +
+cc] + ΓLO
+EM[T +
+cc]
+47
+35
+72
+TABLE I: Partial and total widths in units of keV at LO and NLO.
+considerably more compact objects one might expect the effective range for charm mesons
+to be smaller. We can see that the distribution is highly sensitive to the choice of β. A
+β of −0.59 greatly increases the differential distribution, and is in much poorer agreement
+with the experimental data. This suggests that the effective range for T +
+cc is smaller than
+for nucleons.
+Clearly the partial widths and their differential distributions can vary substantially de-
+pending on the choice of parameters in the effective field theory. However, the availability
+of experimental data for the decays presents the possibility of performing fits of the dis-
+tributions to the data to obtain estimates for these parameters. This could improve the
+predictive power of the effective theory. We save such a careful statistical analysis for a
+future publication.
+We can use these plots that show the effect of a subset of the NLO contributions to inform
+which ranges for the parameters to use when estimating the total NLO contribution to the
+differential distribution (Fig. 6). The upper and lower bounds in the figure reflect varying
+C0D from −1 fm2 to 0.25 fm2. The parameters β1, β3, and β5 are varied from −1/(100 MeV)2
+to +1/(100 MeV)2. The parameters β2 and β4, which reduce to −γr0 in the isospin limit,
+are varied between 0 and −0.26. The latter value corresponds to a binding momentum for
+the D∗+D0 channel, γ0, and r0 = 1/(100 MeV). While the uncertainty in the total width
+of the T +
+cc can be significant depending on the values of the NLO couplings, the qualitative
+aspects of the plots of the differential decay widths in Fig. 6 are consistent between LO and
+NLO. The overall shape and location of the peaks are unchanged by pion exchange and final
+state rescattering.
+When integrating over the full phase space to get the partial widths, we use the same
+15
+
+3730
+3732
+3734
+3736
+3738
+0
+20
+40
+60
+80
+100
+120
+140
+FIG. 6: A plot of the differential decay width as a function of the invariant mass of the final state
+D meson pair. Solid lines represent LO calculation; the dashed lines represent the lower and upper
+bounds of the NLO corrections. Here, we vary −1 fm2 ≤ C0D ≤ 0.25 fm2 and −0.26 ≤ β2/4 ≤ 0.
+Overlaid is the binned experimental data from LHCb, with the background subtracted.
+ranges for the parameters as in Fig. 6. The partial widths are given in Table I. Note that
+the LO numbers differ from those in our original paper [6] because here we use the binding
+energy from the unitarized Breit-Wigner fit, whereas in Ref. [6] we used the value from the
+P-wave two-body Breit Wigner fit with a Blatt-Weisskopf form factor. This has the effect
+of slightly increasing the prediction for the width compared to the initial paper, bringing
+it closer to the experimental value. When adding the LO electromagnetic decay width of
+6.1 keV (which is only slightly affected by the different binding energy) the total LO width
+predicted by our effective theory is 47 keV which is already in excellent agreement with the
+LHCb experimental value of 48 keV. Adding in the NLO contribution to the strong decay
+widths, the total width of the T +
+cc can range from 35 keV to 72 keV. So we can establish
+an uncertainty in the width due to NLO strong decays of Γ[T +
+cc] = 47+53%
+−25% keV. This is
+comparable to the uncertainty from similar operators contributing to the decay of χc1(3872)
+16
+
+3730
+3732
+3734
+3736
+3738
+0
+2.×10-7
+4.×10-7
+6.×10-7
+8.×10-7
+1.×10-6
+1.2×10-6
+1.4×10-6
+FIG. 7: Comparing our LO differential decay width to one where the D∗ propagators are taken to
+be constant. The curves are fixed to have the same normalization. Note the lack of a sharp peak
+in the constant propagator curves.
+in XEFT [43].
+We did not consider NLO corrections to the electromagnetic decay, because the LO
+electromagnetic decay was already a small contribution to the total width. In particular,
+the differential distribution for the electromagnetic decay was negligible compared to the
+strong decays’ distributions.
+To illustrate why these differential decay width plots are good tests of the molecular
+nature of the T +
+cc, in Fig. 7 we can compare the LO differential curves to those which would
+arise if we replaced the virtual D∗ propagators with a constant. The latter do not have
+sharp peaks and thus would be in poor agreement with the experimental data.
+V.
+CONCLUSIONS
+In this paper we have determined the effects of NLO strong decays on the total width
+and differential decay width of the exotic meson T +
+cc. We considered pion exchange and
+final state rescattering diagrams, from similar operators to those in XEFT for the χc1(3872)
+17
+
+[43]. We arrive at similar conclusions as Ref. [43]. The differential decay width plots have
+shapes and peaks that are relatively unchanged by the NLO effects, but the total width has
+significant uncertainty: Γ[T +
+cc] = 47+53%
+−25% keV. The central value (the LO result) is in good
+agreement with data.
+We varied the parameters in the NLO calculation to get a sense of the uncertainty in
+the predictions and determine which parameters in the NLO calculation give the biggest
+corrections. Nonanalytic corrections for pion loops are not important. The parameter C0D,
+which is proportional to the I = 1 D meson scattering length, and β2 and β4, which in the
+isospin limit are equal and proportional to the I = 0 D meson effective ranges, significantly
+affect the decay width and normalization of the differential distribution. It would be inter-
+esting to fit the NLO differential curves to the experimental data and obtain bounds on the
+undetermined couplings, thereby learning more about these physical quantities. Alterna-
+tively, one might hope to get information about these parameters from lattice simulations or
+other experiments. Any improvement in our understanding of these parameters in D meson
+scattering would increase the predictive power of the effective field theory.
+Acknowledgments - L. D. is supported by the Alexander von Humboldt Foundation.
+S. F. is supported by the U.S. Department of Energy, Office of Science, Office of Nuclear
+Physics, under award number DE-FG02-04ER41338. T. M. and R. H. are supported by
+the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under grant
+Contract Numbers DE-FG02-05ER41367.
+Appendix A: Coupled channel decay width
+The full expression for the isospin-0 two-point correlator is
+−iG0 = 1
+2
+−Σ0 − 4C(1)
+0 det Σ
+1 + C(0)
+0 Σ0 + C(1)
+0 Σ1 + 4C(0)
+0 C(1)
+0 det Σ
+,
+(A1)
+where Σ0/1 ≡ Σ11 + Σ22 ∓ Σ12 ∓ Σ21 are the isospin-0 and isopsin-1 combinations of the
+elements of Σ. Since we expect T +
+cc to be an isospin-0 state we treat C(1)
+0
+perturbatively and
+expand to NLO in C(1)
+0 .
+−iG0 ≈ 1
+2
+−Σ0
+1 + C(0)
+0 Σ0
++ 1
+2
+C(1)
+0 (ΣLO
+11 − ΣLO
+22 )2
+(1 + C(0)
+0 Σ0)2
+.
+(A2)
+18
+
+We see that the real numerator of the C(1)
+0
+term is the residue of a double pole at 1+C(0)
+0 Σ0 =
+0. That can be interpreted physically as a small shift in the location of the bound state,
+which can be seen from expanding the right-hand side of Eq. (12) about ENLO
+T
+= ET −ELO
+T .
+But since we are already tuning ET to be the location of the T +
+cc bound state, we can set
+C(1)
+0
+to zero to remove the double pole from the amplitude.
+−iG0 → 1
+2
+−Σ0
+1 + C(0)
+0 Σ0
+.
+(A3)
+At this stage the problem is identical to the single-channel problem in XEFT [27], with the
+single-channel two-point function replaced by our isospin-0 combination of coupled-channel
+two-point functions. The wave function renormalization and decay width are therefore:
+Z0 =
+1
+�
+C(0)
+0
+�2Re Σ′
+0(−ET)
+,
+Γ0 = 2 Im Σ0(−ET)
+Re Σ′
+0(−ET) .
+(A4)
+Σ0 has LO contributions from the diagonal elements, and NLO contributions from all ele-
+ments. After expanding in the NLO terms we find our corrections to the LO decay width.
+Γ0 ≈ ΓLO
+�
+1 − Re Σ′NLO
+0
+(−ET)
+Re tr Σ′LO(−ET)
+�
++ 2 Im ΣNLO
+0
+(−ET)
+Re tr Σ′LO(−ET) .
+(A5)
+Appendix B: Basis integrals and the PDS scheme
+The most basic integral that arises when evaluating the one-loop diagrams in the PDS
+scheme is:
+�ΛPDS
+2
+�4−d �
+dd−1l
+(2π)d−1
+1
+l2 + c − iϵ = 1
+4π(ΛPDS −
+√
+c − iϵ) .
+(B1)
+This result is obtained by subtracting the pole in d = 3 with a counterterm, then evaluating
+the result in d = 4, yielding a linear divergence in ΛPDS.
+The scalar integral I(p) is finite in d = 3 and d = 4, so no PDS counterterm is needed.
+I(p) =
+�
+dd−1l
+(2π)d−1
+1
+l2 + c1 − iϵ
+1
+l2 − 2bl · p + c2 − iϵ
+=
+1
+8π
+1
+�
+b2p2
+�
+tan−1
+� c2 − c1
+2
+�
+b2p2c1
+�
++ tan−1
+�
+2b2p2 + c1 − c2
+2
+�
+b2p2(c2 − b2p2)
+��
+.
+(B2)
+19
+
+The linear tensor integral I(1)(p) can be solved using algebraic manipulation of the nu-
+merator, which yields two integrals of the form of Eq. (B1) that have opposite sign for the
+divergence, and so I(1)(p) is UV finite.
+piI(1)(p) =
+�
+dd−1l
+(2π)d−1li
+1
+l2 + c1 − iϵ
+1
+l2 − 2bl · p + c2 − iϵ ,
+→ p2I(1)(p) =
+1
+2b
+� 1
+4π
+√
+c1 − iϵ − 1
+4π
+�
+c2 − b2p2 − iϵ + (c2 − c1)I(p)
+�
+.
+(B3)
+The quadratic tensor integrals I(2) require care when implementing the PDS scheme. The
+linear divergences which arise in the decay width can only cancel if the subtraction scheme
+is implemented correctly. After using Feynman parameters to combine the propagators and
+obtain an integrand like liljf(l2), the correct procedure is to replace lilj → δij/3 immediately,
+and not with δij/(d − 1). The latter would cancel the factor of d − 1 that arises when
+evaluating the loop momentum integral, and this results in the incorrect coefficient for the
+PDS subtraction scale ΛPDS. Additionally, algebraic manipulation of the numerator of I(2)
+to reduce it to integrals of the form of I(1) and I leads to yet another incorrect coefficient.
+This is the method used to obtain the expressions in the appendix of Ref. [43]; as such, the
+formulae for the decay width in that paper are only correct if ΛPDS = 0 and d = 4.
+Using the correct procedure for the basis integrals gives the following results:
+pipjI(2)
+0 (p) + δijp2I(2)
+1 (p) =
+�
+dd−1l
+(2π)d−1lilj
+1
+l2 + c1 − iϵ
+1
+l2 − 2bl · p + c2 − iϵ ,
+I(2)
+0 (p) = b2
+8π
+� 1
+0
+dx
+x2
+�
+∆(x)
+,
+(B4)
+→ p2I(2)
+1 (p) =
+1
+8π
+�2
+3ΛPDS −
+� 1
+0
+dx
+�
+∆(x)
+�
+,
+(B5)
+for ∆(x) = −b2p2x2 + (c2 −c1)x+c1 −iϵ. One can be reassured that this implementation of
+the PDS scheme is correct because the same relative weight of the ΛPDS and
+� 1
+0 dx
+�
+∆(x)
+terms is obtained when using a hard cutoff. That does not occur when using lilj → δij/(d−1)
+or algebraic manipulation of the numerator. Furthermore, unless the relative weight of the
+two terms in I(2)
+1
+is 2/3, the linear divergences that appear in ΓNLO
+0
+as A(2b) and Re Σ′
+2 do
+not cancel in the isospin limit, as they do in XEFT. For the T +
+cc, they cancel when µ0 = µ+,
+an approximation we make in the cutoff-dependent terms to ensure cancellation.
+With algebraic manipulation of the integrand in Eq. (B4) and integration by parts in
+20
+
+Eq. (B5), we can rewrite these expressions in terms of I and I(1).
+p2I(2)
+0
+= − 1
+16π
+�
+c2 − b2p2 − iϵ + c1
+2 I(p) + 3
+4
+c2 − c1
+b
+I(1)(p) ,
+(B6)
+p2I(2)
+1
+= ΛPDS
+12π −
+1
+16π
+�
+c2 − b2p2 − iϵ − c1
+2 I(p) − 1
+4
+c2 − c1
+b
+I(1)(p) .
+(B7)
+Appendix C: Cπ couplings and βi expressions
+In the isospin |I, mI⟩ basis, we use the phase convention
+|π+⟩ = − |1, 1⟩ ,
+|π0⟩ = |1, 0⟩ ,
+|D+⟩ =
+����
+1
+2, 1
+2
+�
+,
+|D0⟩ =
+����
+1
+2, −1
+2
+�
+.
+(C1)
+Then the Clebsch-Gordan decomposition of the Dπ pairs is
+|D0π0⟩ =
+�
+2
+3
+����
+3
+2, −1
+2
+�
++ 1
+√
+3
+����
+1
+2, −1
+2
+�
+,
+|D+π0⟩ =
+�
+2
+3
+����
+3
+2, 1
+2
+�
++ 1
+√
+3
+����
+1
+2, 1
+2
+�
+,
+|D0π+⟩ = −
+�
+2
+3
+����
+1
+2, 1
+2
+�
+− 1
+√
+3
+����
+3
+2, 1
+2
+�
+.
+(C2)
+From this we can deduce
+aD0π0 = aD+π0 = 2
+3a3/2
+Dπ + 1
+3a1/2
+Dπ ,
+aD0π+ = 1
+3a3/2
+Dπ + 2
+3a1/2
+Dπ .
+(C3)
+These scattering lengths are calculated on the lattice in Ref. [51] to be a1/2
+Dπ = 0.37+0.03
+−0.02 fm
+and a3/2
+Dπ = −(0.100±0.002) fm. The matching from tree level scattering tells us that, for the
+diagonal couplings C(2)
+π
+and C(3)
+π , we can use Cπ = 4π(1+mπ/mD)aDπ, with the appropriate
+masses and scattering lengths for each process. We can then use those two values to solve
+for C(1/2)
+π
+and C(3/2)
+π
+and obtain C(1)
+π . We get
+C(1)
+π
+= −3.0+0.32
+−0.40 fm ,
+C(2)
+π
+= −0.76+0.14
+−0.09 fm ,
+C(3)
+π
+= 2.9+0.3
+−0.2 fm .
+(C4)
+The expressions for the βi are given below. The subscripts on the γ and µ variables indicate
+the pseudoscalar charm meson is in that channel, e.g.
+γ+ = γ(m+, m∗
+0) is the binding
+momentum in the channel with the D+ meson.
+21
+
+β1 = (ΛPDS − γ+)
+�
+fπ
+√
+2πgB(1)
+1
++ 1
+πC(+)
+2
+µ+ − 1
+πC(−)
+2
+µ0
+ΛPDS − γ0
+ΛPDS − γ+
+�
+,
+(C5)
+β2 =
+� 1
+πC(+)
+2
+µ+(−2γ2
++)(ΛPDS − γ+) − 1
+πC(−)
+2
+µ0(−γ2
+0 − γ2
++)(ΛPDS − γ0)
++2π
+�µ2
+0
+γ0
++ µ2
++
+γ+
+�−1�
+− 1
+π2C(+)
+2
+µ3
++(γ+ − ΛPDS)(2γ+ − ΛPDS)
+− 1
+π2C(+)
+2
+µ3
+0(γ0 − ΛPDS)(2γ0 − ΛPDS)
+−C(−)
+2
+(γ2
++ + γ2
+0)µ+µ0
+2π
+�µ+
+γ0
+(ΛPDS − γ0) + µ0
+γ+
+(ΛPDS − γ+)
+�
++C(−)
+2
+µ+µ0(µ+ + µ0)
+π2
+(ΛPDS − γ+)(ΛPDS − γ0)
+��
+,
+(C6)
+β3 = (ΛPDS − γ0)
+�
+−
+fπ
+√
+2πgB(2)
+1
++ 1
+πC(+)
+2
+µ0 − 1
+πC(−)
+2
+µ+
+ΛPDS − γ+
+ΛPDS − γ0
+�
+,
+(C7)
+β4 =
+� 1
+πC(+)
+2
+µ0(−2γ2
+0)(ΛPDS − γ0) − 1
+πC(−)
+2
+µ+(−γ2
+0 − γ2
++)(ΛPDS − γ+)
++2π
+�µ2
+0
+γ0
++ µ2
++
+γ+
+�−1�
+− 1
+π2C(+)
+2
+µ3
++(γ+ − ΛPDS)(2γ+ − ΛPDS)
+− 1
+π2C(+)
+2
+µ3
+0(γ0 − ΛPDS)(2γ0 − ΛPDS)
+−C(−)
+2
+(γ2
++ + γ2
+0)µ+µ0
+2π
+�µ+
+γ0
+(ΛPDS − γ0) + µ0
+γ+
+(ΛPDS − γ+)
+�
++C(−)
+2
+µ+µ0(µ+ + µ0)
+π2
+(ΛPDS − γ+)(ΛPDS − γ0)
+��
+,
+(C8)
+β5 = 1
+πC(+)
+2
+µ0(ΛPDS − γ0) − 1
+πC(−)
+2
+µ+(ΛPDS − γ+)
++B(3)
+1 fπ
+4πg (γ0 − ΛPDS) − B(4)
+1 fπ
+4πg (γ+ − ΛPDS)µ+
+µ0
+.
+(C9)
+It is instructive to take the isospin limit of these β expressions and compare to XEFT.
+Referring to Eq. (6), we can write down the B1 couplings in this limit.
+B(1)
+1
+= −B(2)
+1
+= −
+√
+2B(I=0)
+1
+,
+B(3)
+1
+= 2(B(I=1)
+1
++ B(I=0)
+1
+) ,
+B(4)
+1
+= 2(B(I=1)
+1
+− B(I=0)
+1
+) .
+(C10)
+22
+
+Then taking µ+ = µ0 = µ, γ+ = γ0 = γ we find:
+β1 = β3 = β5 = 1
+π(γ − ΛPDS)
+�B(I=0)
+1
+fπ
+g
+− 2C(0)
+2 µ
+�
+,
+β2 = β4 = −4C(0)
+2 µγ
+π
+(γ − ΛPDS)2 .
+(C11)
+The isospin-1 couplings drop out, which is to be expected given that we have projected out
+the isospin-0 state and are here dropping isospin-breaking interactions. These expressions
+also match the dependence of the decay rate on C2 and B1 in XEFT [27]. Using Eq. (24)
+of [27] (and adjusting for a factor of 4 in the definition of C2 in that paper) we see that
+β2 = β4 = −γr0 in the isospin limit. It is an important check on our calculation that in the
+isospin limit the theory can be properly renormalized with isospin respecting counterterms.
+When isospin breaking in the masses and binding momentum is included, isospin breaking
+in the B1 operators needs to be included as we have done in this paper.
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+

5NFAT4oBgHgl3EQfFRzt/content/tmp_files/2301.08427v1.pdf.txt

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+arXiv:2301.08427v1  [cs.CL]  20 Jan 2023
+Arxiv preprint
+WHICH FEATURES ARE LEARNED BY CODEBERT:
+AN EMPIRICAL STUDY OF THE BERT-BASED SOURCE
+CODE REPRESENTATION LEARNING
+Lan Zhang∗, Chen Cao∗, Zhilong Wang∗ and Peng Liu
+The Pennsylvania State University
+State College, PA 16801, USA
+{lfz5092,cuc96,zzw169,pxl20}@psu.edu
+ABSTRACT
+The Bidirectional Encoder Representations from Transformers (BERT) were pro-
+posed in the natural language process (NLP) and shows promising results. Re-
+cently researchers applied the BERT to source-code representation learning and
+reported some good news on several downstream tasks. However, in this pa-
+per, we illustrated that current methods cannot effectively understand the logic of
+source codes. The representation of source code heavily relies on the programmer-
+defined variable and function names. We design and implement a set of experi-
+ments to demonstrate our conjecture and provide some insights for future works.
+1
+INTRODUCTION
+Deep learning has demonstrated its great learning ability in natural language processing (NLP).
+To deploy a natural language task, e.g. translation and text classification, researchers first pre-
+train a model to embed words into vectors using ELMo
+Sarzynska-Wawer et al. (2021), GPT
+Radford et al. (2018) and BERT Devlin et al. (2018). These pre-trained models are first learned on a
+large unsupervised text corpus and then fine-tuned on different downstream tasks. Those language-
+based techniques have been deployed to the source code to learn a program representation. Simi-
+lar to natural language, the program representation learned from the source code using pre-trained
+models can be applied for several sub-tasks for example program analysis. In 2020, Feng et al.
+proposed a pre-trained model called CodeBERT Feng et al. (2020) based on Bidirectional Encoder
+Representations from Transformers (BERT) that learns general-purpose representations to support
+downstream NL-PL applications such as natural language code search, code documentation genera-
+tion, etc. In 2021, Guo et al. proposed a new pre-trained model called GraphCodeBERT Guo et al.
+(2020), which improves the CodeBERT by enabling the model to capture more program semantic
+information, such as data flow.
+The difference between natural language and program language leads to an unintended consequence
+if these methods are directly employed to program language. In natural language, the meaning of a
+word is deterministic in a specific context, whereas in program language, a programmer can assign
+any string to a variable, method, or function as their name. In such a case, most strings in the code
+could be replaced by other words and may not have meaningful information. In this case, if a BERT
+model still heavily relies on the literal meaning of a variable/methods/function name, it may leave
+a pitfall when the assigned name does not literally contain any useful information or controversial
+meaning.
+Furthermore, limited words are used in natural language, while in the programming language, the
+number of words can be unlimited because a programmer can casually create a string to name
+a variable, no matter whether the created string is interpretable or not. Therefore, it is doubtful
+whether the word embedding adopted in natural language is still efficient in solving the program
+analysis tasks. If a model designer ignores the numerous difference between natural language and
+programing language and naively adopt methods from NLP, the designed model may suffer from the
+above limitations.
+∗equal contribution
+1
+
+Arxiv preprint
+In this paper, we aim to provide an explanation of these limitations of the BERT-based code rep-
+resentation learning techniques. Specifically, we want to understand what kind of features can be
+learned and cannot be learned by current pre-trained models.
+1
+template<typename It, typename Pred=std::less<typename std::iterator_traits<It>::
+value_type>>
+2
+inline void bubble_sort(It begin, It end, Pred pred=Pred()){
+3
+if ( std::distance( begin, end ) <= 1 ){ return; }
+4
+auto it_end
+= end;
+5
+bool finished
+= false;
+6
+while ( !finished ){
+7
+finished = true;
+8
+std::advance( it_end, -1 );
+9
+for (auto it = begin; it! = it_end; ++ it ){
+10
+auto next = detail::advance( it, 1 );
+11
+if (pred( * next, * it)){
+12
+std::swap( * it, * next);
+13
+finished = false;
+14
+}
+15
+}
+16
+}
+17
+}
+Code 1: A piece of code with meaningful variable/function names.
+1
+template<typename It, typename Fun2=std::less<typename std::iterator_traits<It>::
+value_type>>
+2
+inline void fun1(It var1, It var2, Pred fun2=Fun2()){
+3
+if ( std::distance( var1, var2 ) <= 1 ){ return; }
+4
+auto var3
+= var2;
+5
+bool var4
+= false;
+6
+while ( !var4 ){
+7
+var4 = true;
+8
+std::advance( var3, -1 );
+9
+for (auto var5 = var1; var5! = var3; ++ var5 ){
+10
+auto var6 = detail::advance( var5, 1 );
+11
+if (fun2( * var6, * var5)){
+12
+std::swap( * var5, * var6);
+13
+var4 = false;
+14
+}
+15
+}
+16
+}
+17
+}
+Code 2: A piece of code without meaningful variable/function names.
+Code 1 and Code 2 are two pieces of code that achieve the same logic – bubble sorting. The Code 1
+has well-named functions and variables whereas the Code 2 does not. If an analyst wants to know
+their purpose, through a quick glance, even a beginner can easily conclude that Code 1 is a bubble-
+sort function based on the literal meaning of the function name. However, it is much more chal-
+lenging for an analyst to understand the purpose of Code 2. Therefore, despite the exactly the same
+program logic that they have, Code 2 is much more difficult to analyze. We can draw the following
+conclusions from the analysis of these two code examples: 1) a source code can be understood in
+two ways: literal analysis, and logic analysis. 2) The literal analysis makes a conclusion based on
+the name of variables and functions, which is easier to analyze but is not always reliable. 3) The
+logic analysis requires a high-level understanding of the code, which is more reliable but hard to
+analyze.
+To understand whether the existing models learn the logic of the code, we identify two features in
+the source code: 1) literal feature. 2) logic feature. For instance, a logical expression is the logic
+feature, whereas the variable names in the expression are literal features. Then, we design a set of
+experiments that mask out different kinds of features in the training set and observe corresponding
+model performance. The result shows that the current models for source code representation learning
+still have limited ability to learn logic features.
+2
+
+Arxiv preprint
+2
+BACKGROUND
+2.1
+DEEP LEARNING FOR PROGRAM ANALYSIS
+Compared with traditional deep learning methods, researchers recognized several benefits of deep
+learning for the program analysis: First, deep learning involves less domain knowledge. Second, the
+representations learned by a DL model could be used for various downstream tasks. The applications
+of deep learning in program analysis can be grouped into two categories:
+Source code level deep learning. CodeBert and GraphCodeBERT Feng et al. (2020); Guo et al.
+(2020) are pre-trained models based on Transformer which learns code representations through self-
+supervised training tasks ( masked language modeling and structure-aware tasks) and a large-scale
+unlabeled corpus. Specifically, CodeBERT, which is pre-trained over 6 programming languages,
+is trained based on three tasks: masked language modeling, code structure edge predication, and
+representation alignment.
+Assembly code level deep learning. Previous research use DL to conduct various binary analysis
+tasks Chua et al. (2017); Shin et al. (2015); Li et al. (2021). The main focus of these works is to
+learn a good embedding from binary instructions or raw bytes, and then predict the label for a target
+task through a classification output layer.
+3
+INSIGHTS AND EXPERIMENTS
+A source code file of a program consists of a sequence of tokens. The tokens can be grouped into
+three categories: keywords, operators, and user-defined names.
+Keywords are reserved words that have special meanings and purposes and can only be used for
+specific purposes. For example, for, if, and break are widely known keywords used in many
+programming languages. A programming language usually only contains a limited number of key-
+words. For example, C programming language contains 32 keywords and Python3.7 contains 35
+keywords.
+Besides the keywords, a programming language needs to define a set of operators. For example,
+arithmetic operators (e.g., +, -, and *) and logical operators (e.g., and, or, and not) are two of
+most important categories. The keywords and operators are defined by a programming language. A
+programmer needs to define some tokens (i.e., names) to represent a variable, structure, function,
+method, class, and package. When programmers write a code snippet, they can randomly choose
+any string to name these elements. However, he/she has limited flexibility to choose the keywords
+and operators. Only some keywords (such as for and while), operators (such as ++, +1) are
+exchangeable.
+Currently, GraphCodeBert takes code pieces of functions or class methods as data samples. It to-
+kenizes keywords, operators, and user-defined names from the code pieces. Inside a function or a
+method, we can group the user-defined names into three categories: 1) variable name. 2) method
+name. 3) method invocation name. Program logic is not affected if we map these user-defined names
+with other strings in the same namespace. To evaluate whether the model learns the code semantics,
+we design 4 groups of experiments. For each group of experiments, we anonymize certain categories
+of user-defined names.
+1. In the first group of experiments, we anonymize the variable names. An example is the
+change from it end to var3 and finished to var4 between Code 1 and Code 2.
+2. In the second group of experiments, we anonymize the method names. An example is the
+change from bubble sort to fun1 between Code 1 and Code 2.
+3. In the third group of experiments, we anonymize the method/function invocation names.
+An example is the change from swap to fun2 between Code 1 and Code 2.
+4. The last group of experiments are a combination of the first three experiments, which
+anonymize all three kinds of user-defined names.
+Besides, we adopt two strategies to anonymize the name: The first strategy called “randomly-
+generated” randomly generates strings (e.g., “oe4yqk4cit2maq7t”) with any literal meaning. The
+3
+
+Arxiv preprint
+Table 1: Results on Code Search.
+Language
+Original
+Anonymizing
+w/o Variable
+w/o Method Def.
+w/o Method Inv.
+All
+Java
+70.36%
+Random
+67.73%
+60.89%
+69.84%
+17.42%
+Meaningful
+67.14%
+58.36%
+69.84%
+17.03%
+Python
+68.17%
+Random
+59.8%
+55.43%
+65.61%
+24.09%
+Meaningful
+59.78%
+55.65%
+65.61%
+23.73%
+Table 2: Results on Clone Detection.
+Language
+Original
+Anonymizing
+w/o Variable
+w/o Method Def.
+w/o Method Inv.
+All
+Java
+94.87%
+Random
+92.64%
+93.97%
+94.72%
+86.77%
+Meaningful
+92.52%
+94.27%
+93.67%
+84.76%
+second strategy called “meaningfully-generated” generates strings with a literal meaning. However
+the literal meaning does not reflect the intention of the variable/function/invocation. For example,
+this strategy could replace “bubble sort” with “aes encryption”.
+Based on the four types of name-set to replace and two replacing strategies, we eventually generated
+8 variants of the original dataset from Guo et al. (2020). Then, we retrain the existing models and
+evaluated their performance on the existing 2 downstream tasks: natural language code search, and
+clone detection.
+3.1
+EXPERIMENT RESULTS
+Figure 2 and Figure 1 show experiment results (accuracy) on the downstream task of code search
+and code clone detection, respectively. The second column shows the module performance reported
+by the original paper Guo et al. (2020). The fourth, fifth, and sixth columns show the module per-
+formance when we anonymize the variable name, method definition name, and method invocation
+name, respectively. The last column shows the model performance after we remove all three user-
+defined names.
+The results show that the anonymization of the variable names, method definition names, and method
+invocation names will result in a huge downgrade in model performance not matter we replace user-
+defined names with “randomly-generated” strings or a “meaningfully-generated” strings. Also, on
+average the dateset with meaningfully-generated strings shows worse result then the dataset with
+randomly-generatedstrings, which indicates that “meaningfully-generated”strings could misleading
+the models. An adversarial machine learning could be trained to further exploit the weakness of the
+CodeBert.
+Overall, our experiments proves that current source-code level representation learning methods still
+largely rely on the literal feature and ignore the logic feature. However, the literal feature is not
+always reliable as mentioned in section 1. The current mode still cannot effectively learn the hidden
+logic feature in the source code.
+3.2
+DISCUSSION
+Through a set of experiments and empirical analysis, this paper tries to explain the learning ability of
+current BERT-based source code representation learning schemes. The results show that CodeBERT
+and GraphCodeBERT are efficient to learn literal features but less efficient to learn logic features.
+The insights provided by this paper can help future researchers or users in two aspects: Firstly, Code-
+BERT and GraphCodeBERT, which open a new area for source analysis, are efficient methods for
+“well-named” source code. However, the user and researcher should expect a lower model perfor-
+mance if they want to apply them to analyze source code that does not provide enough information
+in a variable, method, and function names, e.g., the code generated from decompilation Katz et al.
+(2018) and code that does not follow standard code naming convention Butler et al. (2015).
+4
+
+Arxiv preprint
+Secondly, this paper indicates that models borrowed from NLP are not very suitable for code anal-
+ysis. The code analysis has some significant differences compared with NLP. Logical analysis is
+more important in many sophisticated program analysis tasks, such as vulnerability analysis, and
+patching generation. But it cannot be well performed by existing model designs. It is important to
+investigate how to improve the model’s ability for logical analysis in future research.
+REFERENCES
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+Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. Bert: Pre-training of deep
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+Zhangyin Feng, Daya Guo, Duyu Tang, Nan Duan, Xiaocheng Feng, Ming Gong, Linjun Shou, Bing
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+2015.
+5
+

5NFAT4oBgHgl3EQfFRzt/content/tmp_files/load_file.txt

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
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+page_content='CL]  20 Jan 2023  Arxiv preprint  WHICH FEATURES ARE LEARNED BY CODEBERT:  AN EMPIRICAL STUDY OF THE BERT-BASED SOURCE  CODE REPRESENTATION LEARNING  Lan Zhang∗, Chen Cao∗, Zhilong Wang∗ and Peng Liu  The Pennsylvania State University  State College, PA 16801, USA  {lfz5092,cuc96,zzw169,pxl20}@psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='edu  ABSTRACT  The Bidirectional Encoder Representations from Transformers (BERT) were pro-  posed in the natural language process (NLP) and shows promising results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Re-  cently researchers applied the BERT to source-code representation learning and  reported some good news on several downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' However, in this pa-  per, we illustrated that current methods cannot effectively understand the logic of  source codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' The representation of source code heavily relies on the programmer-  defined variable and function names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' We design and implement a set of experi-  ments to demonstrate our conjecture and provide some insights for future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  1  INTRODUCTION  Deep learning has demonstrated its great learning ability in natural language processing (NLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  To deploy a natural language task, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' translation and text classification, researchers first pre-  train a model to embed words into vectors using ELMo  Sarzynska-Wawer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' (2021), GPT  Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' (2018) and BERT Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' These pre-trained models are first learned on a  large unsupervised text corpus and then fine-tuned on different downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Those language-  based techniques have been deployed to the source code to learn a program representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Simi-  lar to natural language, the program representation learned from the source code using pre-trained  models can be applied for several sub-tasks for example program analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' In 2020, Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  proposed a pre-trained model called CodeBERT Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' (2020) based on Bidirectional Encoder  Representations from Transformers (BERT) that learns general-purpose representations to support  downstream NL-PL applications such as natural language code search, code documentation genera-  tion, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' In 2021, Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' proposed a new pre-trained model called GraphCodeBERT Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  (2020), which improves the CodeBERT by enabling the model to capture more program semantic  information, such as data flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  The difference between natural language and program language leads to an unintended consequence  if these methods are directly employed to program language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' In natural language, the meaning of a  word is deterministic in a specific context, whereas in program language, a programmer can assign  any string to a variable, method, or function as their name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' In such a case, most strings in the code  could be replaced by other words and may not have meaningful information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' In this case, if a BERT  model still heavily relies on the literal meaning of a variable/methods/function name, it may leave  a pitfall when the assigned name does not literally contain any useful information or controversial  meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  Furthermore, limited words are used in natural language, while in the programming language, the  number of words can be unlimited because a programmer can casually create a string to name  a variable, no matter whether the created string is interpretable or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Therefore, it is doubtful  whether the word embedding adopted in natural language is still efficient in solving the program  analysis tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' If a model designer ignores the numerous difference between natural language and  programing language and naively adopt methods from NLP, the designed model may suffer from the  above limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  ∗equal contribution  1  Arxiv preprint  In this paper, we aim to provide an explanation of these limitations of the BERT-based code rep-  resentation learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Specifically, we want to understand what kind of features can be  learned and cannot be learned by current pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  1  template<typename It, typename Pred=std::less<typename std::iterator_traits<It>::  value_type>>  2  inline void bubble_sort(It begin, It end, Pred pred=Pred()){  3  if ( std::distance( begin, end ) <= 1 ){ return;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' }  4  auto it_end  = end;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  5  bool finished  = false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  6  while ( !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='finished ){  7  finished = true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  8  std::advance( it_end, -1 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  9  for (auto it = begin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' it!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' = it_end;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' ++ it ){  10  auto next = detail::advance( it, 1 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  11  if (pred( * next, * it)){  12  std::swap( * it, * next);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  13  finished = false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  14  }  15  }  16  }  17  }  Code 1: A piece of code with meaningful variable/function names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  1  template<typename It, typename Fun2=std::less<typename std::iterator_traits<It>::  value_type>>  2  inline void fun1(It var1, It var2, Pred fun2=Fun2()){  3  if ( std::distance( var1, var2 ) <= 1 ){ return;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' }  4  auto var3  = var2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  5  bool var4  = false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  6  while ( !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='var4 ){  7  var4 = true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  8  std::advance( var3, -1 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  9  for (auto var5 = var1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' var5!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' = var3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' ++ var5 ){  10  auto var6 = detail::advance( var5, 1 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  11  if (fun2( * var6, * var5)){  12  std::swap( * var5, * var6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  13  var4 = false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  14  }  15  }  16  }  17  }  Code 2: A piece of code without meaningful variable/function names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  Code 1 and Code 2 are two pieces of code that achieve the same logic – bubble sorting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' The Code 1  has well-named functions and variables whereas the Code 2 does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' If an analyst wants to know  their purpose, through a quick glance, even a beginner can easily conclude that Code 1 is a bubble-  sort function based on the literal meaning of the function name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' However, it is much more chal-  lenging for an analyst to understand the purpose of Code 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Therefore, despite the exactly the same  program logic that they have, Code 2 is much more difficult to analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' We can draw the following  conclusions from the analysis of these two code examples: 1) a source code can be understood in  two ways: literal analysis, and logic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' 2) The literal analysis makes a conclusion based on  the name of variables and functions, which is easier to analyze but is not always reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' 3) The  logic analysis requires a high-level understanding of the code, which is more reliable but hard to  analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  To understand whether the existing models learn the logic of the code, we identify two features in  the source code: 1) literal feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' 2) logic feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' For instance, a logical expression is the logic  feature, whereas the variable names in the expression are literal features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Then, we design a set of  experiments that mask out different kinds of features in the training set and observe corresponding  model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' The result shows that the current models for source code representation learning  still have limited ability to learn logic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  2  Arxiv preprint  2  BACKGROUND  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='1  DEEP LEARNING FOR PROGRAM ANALYSIS  Compared with traditional deep learning methods, researchers recognized several benefits of deep  learning for the program analysis: First, deep learning involves less domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Second, the  representations learned by a DL model could be used for various downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' The applications  of deep learning in program analysis can be grouped into two categories:  Source code level deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' CodeBert and GraphCodeBERT Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  (2020) are pre-trained models based on Transformer which learns code representations through self-  supervised training tasks ( masked language modeling and structure-aware tasks) and a large-scale  unlabeled corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Specifically, CodeBERT, which is pre-trained over 6 programming languages,  is trained based on three tasks: masked language modeling, code structure edge predication, and  representation alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  Assembly code level deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Previous research use DL to conduct various binary analysis  tasks Chua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Shin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' The main focus of these works is to  learn a good embedding from binary instructions or raw bytes, and then predict the label for a target  task through a classification output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  3  INSIGHTS AND EXPERIMENTS  A source code file of a program consists of a sequence of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' The tokens can be grouped into  three categories: keywords, operators, and user-defined names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  Keywords are reserved words that have special meanings and purposes and can only be used for  specific purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' For example, for, if, and break are widely known keywords used in many  programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' A programming language usually only contains a limited number of key-  words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' For example, C programming language contains 32 keywords and Python3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='7 contains 35  keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  Besides the keywords, a programming language needs to define a set of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' For example,  arithmetic operators (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=', +, -, and *) and logical operators (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=', and, or, and not) are two of  most important categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' The keywords and operators are defined by a programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' A  programmer needs to define some tokens (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=', names) to represent a variable, structure, function,  method, class, and package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' When programmers write a code snippet, they can randomly choose  any string to name these elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' However, he/she has limited flexibility to choose the keywords  and operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Only some keywords (such as for and while), operators (such as ++, +1) are  exchangeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  Currently, GraphCodeBert takes code pieces of functions or class methods as data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' It to-  kenizes keywords, operators, and user-defined names from the code pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Inside a function or a  method, we can group the user-defined names into three categories: 1) variable name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' 2) method  name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' 3) method invocation name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Program logic is not affected if we map these user-defined names  with other strings in the same namespace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' To evaluate whether the model learns the code semantics,  we design 4 groups of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' For each group of experiments, we anonymize certain categories  of user-defined names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' In the first group of experiments, we anonymize the variable names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' An example is the  change from it end to var3 and finished to var4 between Code 1 and Code 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' In the second group of experiments, we anonymize the method names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' An example is the  change from bubble sort to fun1 between Code 1 and Code 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' In the third group of experiments, we anonymize the method/function invocation names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  An example is the change from swap to fun2 between Code 1 and Code 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' The last group of experiments are a combination of the first three experiments, which  anonymize all three kinds of user-defined names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  Besides, we adopt two strategies to anonymize the name: The first strategy called “randomly-  generated” randomly generates strings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=', “oe4yqk4cit2maq7t”) with any literal meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' The  3  Arxiv preprint  Table 1: Results on Code Search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  Language  Original  Anonymizing  w/o Variable  w/o Method Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  w/o Method Inv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  All  Java  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='36%  Random  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='73%  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='89%  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='84%  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='42%  Meaningful  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='14%  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='36%  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='84%  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='03%  Python  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='17%  Random  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='8%  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='43%  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='61%  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='09%  Meaningful  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='78%  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='65%  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='61%  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='73%  Table 2: Results on Clone Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  Language  Original  Anonymizing  w/o Variable  w/o Method Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  w/o Method Inv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  All  Java  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='87%  Random  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='64%  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='97%  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='72%  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='77%  Meaningful  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='52%  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='27%  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='67%  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='76%  second strategy called “meaningfully-generated” generates strings with a literal meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' However  the literal meaning does not reflect the intention of the variable/function/invocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' For example,  this strategy could replace “bubble sort” with “aes encryption”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  Based on the four types of name-set to replace and two replacing strategies, we eventually generated  8 variants of the original dataset from Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Then, we retrain the existing models and  evaluated their performance on the existing 2 downstream tasks: natural language code search, and  clone detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='1  EXPERIMENT RESULTS  Figure 2 and Figure 1 show experiment results (accuracy) on the downstream task of code search  and code clone detection, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' The second column shows the module performance reported  by the original paper Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' The fourth, fifth, and sixth columns show the module per-  formance when we anonymize the variable name, method definition name, and method invocation  name, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' The last column shows the model performance after we remove all three user-  defined names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  The results show that the anonymization of the variable names, method definition names, and method  invocation names will result in a huge downgrade in model performance not matter we replace user-  defined names with “randomly-generated” strings or a “meaningfully-generated” strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Also, on  average the dateset with meaningfully-generated strings shows worse result then the dataset with  randomly-generatedstrings, which indicates that “meaningfully-generated”strings could misleading  the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' An adversarial machine learning could be trained to further exploit the weakness of the  CodeBert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  Overall, our experiments proves that current source-code level representation learning methods still  largely rely on the literal feature and ignore the logic feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' However, the literal feature is not  always reliable as mentioned in section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' The current mode still cannot effectively learn the hidden  logic feature in the source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='2  DISCUSSION  Through a set of experiments and empirical analysis, this paper tries to explain the learning ability of  current BERT-based source code representation learning schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' The results show that CodeBERT  and GraphCodeBERT are efficient to learn literal features but less efficient to learn logic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  The insights provided by this paper can help future researchers or users in two aspects: Firstly, Code-  BERT and GraphCodeBERT, which open a new area for source analysis, are efficient methods for  “well-named” source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' However, the user and researcher should expect a lower model perfor-  mance if they want to apply them to analyze source code that does not provide enough information  in a variable, method, and function names, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=', the code generated from decompilation Katz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  (2018) and code that does not follow standard code naming convention Butler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  4  Arxiv preprint  Secondly, this paper indicates that models borrowed from NLP are not very suitable for code anal-  ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' The code analysis has some significant differences compared with NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Logical analysis is  more important in many sophisticated program analysis tasks, such as vulnerability analysis, and  patching generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' But it cannot be well performed by existing model designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' It is important to  investigate how to improve the model’s ability for logical analysis in future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  REFERENCES  Simon Butler, Michel Wermelinger, and Yijun Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Investigating naming convention adherence in  java references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' In 2015 IEEE International Conference on Software Maintenance and Evolution  (ICSME), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' 41–50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' IEEE, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  Zheng Leong Chua, Shiqi Shen, Prateek Saxena, and Zhenkai Liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Neural Nets Can Learn Func-  tion Type Signatures from Binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' In 26th USENIX Security Symposium (USENIX Security 17),  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' 99–116, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Bert: Pre-training of deep  bidirectional transformers for language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' arXiv preprint arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='04805, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  Zhangyin Feng, Daya Guo, Duyu Tang, Nan Duan, Xiaocheng Feng, Ming Gong, Linjun Shou, Bing  Qin, Ting Liu, Daxin Jiang, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Codebert: A pre-trained model for programming and natural  languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' arXiv preprint arXiv:2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='08155, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  Daya Guo, Shuo Ren, Shuai Lu, Zhangyin Feng, Duyu Tang, Shujie Liu, Long Zhou, Nan Duan,  Alexey Svyatkovskiy, Shengyu Fu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Graphcodebert: Pre-training code representations with  data flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' arXiv preprint arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='08366, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  Deborah S Katz, Jason Ruchti, and Eric Schulte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Using recurrent neural networks for decompilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  In 2018 IEEE 25th International Conference on Software Analysis, Evolution and Reengineering  (SANER), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' 346–356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' IEEE, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Qu, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Yin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' PalmTree: Learning an Assembly Language Model for Instruction Em-  bedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' In ACM CCS, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  Alec Radford, Karthik Narasimhan, Tim Salimans, and Ilya Sutskever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Improving language under-  standing by generative pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  Justyna Sarzynska-Wawer, Aleksander Wawer, Aleksandra Pawlak, Julia Szymanowska, Izabela  Stefaniak, Michal Jarkiewicz, and Lukasz Okruszek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Detecting formal thought disorder by deep  contextualized word representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Psychiatry Research, 304:114135, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  Eui Chul Richard Shin, Dawn Song, and Reza Moazzezi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' Recognizing functions in binaries with  neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' In 24th {USENIX} Security Symposium ({USENIX} Security 15), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content=' 611–626,  2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}
+page_content='  5' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'}

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+arXiv:2301.03077v1  [stat.ML]  8 Jan 2023
+Stochastic Langevin Monte Carlo for (weakly) log-concave
+posterior distributions.
+Marelys Crespo Navas1, S´ebastien Gadat2,3, Xavier Gendre1
+1 ISAE-SUPAERO, Universit´e de Toulouse
+2Toulouse School of Economics (CNRS UMR 5314), Universit´e Toulouse I Capitole
+3 Institut Universitaire de France
+January 10, 2023
+Abstract
+In this paper, we investigate a continuous time version of the Stochastic Langevin Monte Carlo
+method, introduced in [39], that incorporates a stochastic sampling step inside the traditional over-
+damped Langevin diffusion. This method is popular in machine learning for sampling posterior
+distribution. We will pay specific attention in our work to the computational cost in terms of
+n (the number of observations that produces the posterior distribution), and d (the dimension
+of the ambient space where the parameter of interest is living). We derive our analysis in the
+weakly convex framework, which is parameterized with the help of the Kurdyka-�Lojasiewicz (KL)
+inequality, that permits to handle a vanishing curvature settings, which is far less restrictive when
+compared to the simple strongly convex case. We establish that the final horizon of simulation
+to obtain an ε approximation (in terms of entropy) is of the order (d log(n)2)(1+r)2[log2(ε−1) +
+n2d2(1+r) log4(1+r)(n)] with a Poissonian subsampling of parameter
+�
+n(d log2(n))1+r�−1, where the
+parameter r is involved in the KL inequality and varies between 0 (strongly convex case) and 1
+(limiting Laplace situation).
+Keywords: Langevin Monte Carlo sampling; Log concave models; Weak convexity.
+AMS classifications: Primary 6265C05; secondary ; 62C10; 65C30; 60H3520.
+1
+1
+Markovian Stochastic Langevin Dynamics and main results
+1.1
+Introduction
+Motivations
+In the recent past years, a huge amount of methods have been developed in machine
+learning to handle large scale massive datasets with a large number n of observations (X1, . . . , Xn)
+embedded in a high dimensional space Rd. These methods generally involve either optimization of a
+data-dependent function (for frequentist learning) or sampling a data-dependent measure (for Bayesian
+learning with posterior distributions). In both approaches, a bottleneck lies on the size of n and d
+that usually generates numerical difficulties for the use of standard algorithms. We are interested
+in this paper in the simulation of a posterior distribution following a Bayesian point of view with a
+statistical model described by a collection of densities (pθ)θ∈Θ on X, where the parameter of interest
+θ⋆ belongs to Θ = Rd and where the (Xi)1≤i≤n are assumed to be i.i.d. observations in X distributed
+according to pθ⋆. A standard Bayesian approach consists in defining a prior distribution π0 on Θ and
+then sample the posterior distribution denoted by µn (which will be denoted by exp(−Uνn) below)
+using a numerical probabilistic approximation with the help of an over-damped Langevin diffusion:
+dθt = −∇Uνn(t)dt +
+√
+2dBt.
+1We are grateful to Patrick Cattiaux and Arnaud Guillin for helpful discussions and references on functional inequal-
+ities and especially on weak log Sobolev inequalities.
+1
+
+In this work, we manage to deal with an adaptation of the Langevin Monte Carlo (LMC) algorithm
+proposed in [39], that exploits some old ideas of stochastic algorithms introduced in [36]: instead of
+using the previous equation, the authors propose a modification of the diffusion that generates a noisy
+drift in the LMC due to a sampling strategy among the set of observations (Xi)1≤i≤n. Before we
+provide some details on the precise objects and algorithm necessary to properly define this method,
+we first give some literature insights related to it.
+State of the art
+Ergodicity and quantitative mixing properties of over-damped LMC and many
+other sampling algorithms is a popular subject of research initiated in the probabilistic works around,
+roughly speaking, two strategies. The first one relies on pathwise considerations and dynamical proper-
+ties of random dynamical system and is built with some coupling argument and Lyapunov controls. We
+refer to the seminal contributions [32, 27], that exploits the approach of the Doeblin coupling and total
+variation (TV) bounds. Many extensions may be derived from this Lyapunov approach and may lead
+to Wasserstein or L2 upper bounds, we refer to [8] and the references therein of the same authors for a
+description of the link between Lyapunov conditions and ergodicity. The second strategy derives from
+spectral properties of Markov operators and is related to famous functional inequalities (Poincar´e and
+Log-Sobolev among others). The general idea is to differentiate the distance along the time-evolution
+and apply a Gronwall Lemma to obtain a quantitative estimate of the long-time evolution of the semi-
+group. We refer to the seminal contributions of [26, 2], and to [3] for an almost exhaustive survey of
+all possible inequalities and consequences on the ergodicity of the Markov semi-groups. Finally, let us
+emphasize that some strong links exist between the spectral and the Lyapunov approaches, as pointed
+out by [9]. If functional inequalities are then strongly related to mixing properties and especially from
+a quantitative point of view, it is therefore necessary to develop a machinery that is able to assess these
+inequalities carefully, especially with a specific attention to our statistical setting of large n and d in the
+completely non-trivial situation where the target measure is log-concave but not strongly log-concave,
+which is a common feature of Bayesian posterior distributions.
+On the statistical side, the mixing properties of LMC has been largely investigated during the past
+decade, strongly motivated by machine learning methods such as Exponentially Weighted Aggregation
+introduced by [11], which involves sampling a non log-concave and heavy tailed posterior distribution.
+A first paper of Dalalyan [12] establishes the cost of LMC to obtain an ε TV bound in terms of d
+and ρ when the target measure is ρ strongly log-concave and proposes a penalized version of LMC to
+circumvent the lack of strong log-concavity when the target distribution is only log-concave. Since this
+pioneering paper, a huge impressive literature expanded. Among others, we refer to [16] that gives a
+careful study of discretized LMC, [14] for a kinetic version of LMC and [15] where the penalized LMC in
+non strongly-concave situation is studied in depth. Among all these papers, first, the lack of strong log-
+concavity is dealt with a modification of the initial LMC using a surrogate and asymptotically vanishing
+penalty. Second, these papers assume that a noiseless gradient of the log-posterior is available at each
+iteration of the algorithm, which may not be realistic, especially with large n.
+Stochastic LMC (SLMC below) has attracted the interest of several works: [39] introduced this
+method and described its efficiency from a numerical point of view in the particular case of Bayesian
+learning, which is exactly our framework. Some recent advances and related contributions may be also
+cited: [13] studies a noisy version of LMC and derives some non-asymptotic upper bounds (in terms of
+Wasserstein distance) of the sampling strategy in presence of a possibly biased noise for strongly log-
+concave posterior distribution. The recent contribution of [40] is also related to our work: the authors
+develop a machinery for the study of SLMC essentially based on the Poincar´e inequality but the way
+the lower bound on the spectral gap involved in the LMC is dealt with appears to be inappropriate. In
+particular, the diffusion involved in (Stochastic)-LMC is used at a very low-temperature, proportional
+to 1/n, which generates some important troubles in the size of the spectral gap in non strongly log-
+concave framework. In [35], the authors derives some close bounds to our framework for optimization
+purpose, and the authors identify the important dependency of the spectral gap denoted by λ∗ in
+their paper with the temperature level 1/β they introduced. They obtain some very highly pessimistic
+bounds in some general situations (see their discussion in [35][Section 4]), they conclude their discussion
+by the urgent need to find some non-trivial situations where some better lower bounds of λ∗ may be
+derived.
+2
+
+Indeed, the final remark of [35][Section 4]) is related to the well known metastability phenomenon:
+at a low temperature, the mixing rates of a lot of reversible and irreversible Markov semi-groups
+are strongly deteriorated by the low temperature settings, which is implicitly induced by a Bayesian
+posterior sampling problem with a large number n of observations. In a regime of variance noise
+of the order O(β−1), the first study of large deviation principle of invariant measures traces back
+to [18] where the authors establish the asymptotic of the spectral gap of the over-damped Langevin
+diffusion as exp(−Iβ) ( [18][Chapter 6]) where I is an explicit constant that depends on the potential
+of the Gibbs field. This result has been extended in depth by [26], which leads to the first precise
+analyses of the so-called simulated annealing method (see e.g. [24, 33]). These works, and more recent
+contributions with irreversible dynamical systems in a stochastic settings ([22, 19]) show that there
+is almost nothing to expect in metastable situations in terms of asymptotic behaviour of the spectral
+gap, and indirectly in terms of mixing rate. Hence, the only situation that may lead to reasonable
+results is an intermediary situation between the (almost) trivial strongly log-concave case and the
+metastable multi-welled case. This is the purpose of the weakly log-concave situation that is described
+by the family of Kurdyka-�Lojasiewicz inequalities [28, 30] used in optimization theory [5] that have
+shown to be efficient for stochastic optimization [20] or for sampling [21]. We also refer to the recent
+contributions [6] that derives some functional inequalities within an intermediary framework in which
+the curvature ρ is related to their keystone function α that controls the constants involved in the
+functional inequalities they are studying.
+Taking together the statistical considerations and limitations, we are motivated in this paper in
+the study of the continuous time Stochastic Langevin Monte Carlo procedure. This process will be
+described precisely in the next paragraph as well as the Kurdyka-�Lojasiewicz setup parametrized by a
+real value r, which varies between 0 (strongly convex case) and 1 (limiting Laplace asymptotic tail).
+We will show that the final horizon of simulation to obtain an ε approximation is of the order:
+(d log(n)2)(1+r)2[log2(ε−1) + n2d2(1+r) log4(1+r)(n)]
+with a Poissonian subsampling of parameter
+1
+n(d log2(n))1+r .
+The rest of the introduction consists in the definitions of the algorithm in Subsection 1.2, the way we
+assess the quality of our result with an entropy criterion in Subsection 1.3, as well as the quantitative
+weakly log-concave assumption in Subsection 1.4. We finally state our main result in Subsection 1.5.
+1.2
+Continuous time evolution
+Below, we briefly remind the continuous time SLMC algorithm for Bayesian learning, for which a
+discretized form has been introduced in [39]. For this purpose, we consider a statistical model that
+is built with the help of a function (x, θ) �−→ pθ(x) where θ ∈ Rd encodes the parameter of the
+statistical model and x the observation in a space denoted by X. We then assume that we have n i.i.d.
+observations denoted by (X1, . . . , Xn) distributed according to pθ. Given a prior distribution π0 on
+Rd, the posterior distribution µn is then defined as:
+µn(θ) ∝ π0(θ) ×
+n
+�
+i=1
+pθ(Xi).
+We introduce the log-parametrization that leads to the Gibbs form:
+Ux(θ) = −[log π0(θ) + n log pθ(x)],
+and we then observe that:
+µn(θ) ∝ exp
+�
+− 1
+n
+n
+�
+i=1
+UXi(θ)
+�
+= exp (−Uνn(θ)) ,
+where νn refers to the empirical distribution and Uνn the average value of UX(θ) when X ∼ νn:
+νn(x) = 1
+n
+n
+�
+i=1
+δXi(x)
+and
+Uνn(θ) = EX∼νn[UX(θ)].
+3
+
+The standard Langevin Monte Carlo approach relies on the ergodic behaviour of the stochastic differ-
+ential equation:
+dθt = −∇Uνn(θt)dt +
+√
+2dBt,
+(1)
+that possesses under some mild assumptions a unique invariant distribution µn.
+The SLMC algorithm takes benefit of both sampling with a S.D.E. and homogenization of the drift
+that may be written as an expectation on X that is sampled uniformly over the set of observations
+according to νn. The leading idea is to replace the expectation in Uνn that depends on the overall set
+of observations (X1, . . . , Xn) by a single unique observation that is randomized uniformly all along
+the evolution of the stochastic differential equation, and modified according to a Markov exponential
+clock. That being said, we can write an explicit formal definition of the algorithm as follows. We
+define
+�
+ξ(n)
+j
+�
+j≥1 an infinite sequence of exponential random variables of mean α−1
+n
+that will be fixed
+later on.
+We also consider a sequence
+�
+V (n)
+j
+�
+j≥0 of i.i.d. random variables uniformly distributed in {1, 2, . . ., n}.
+We then define the process (Xt)t≥0 as a jump process that takes its values in {1, 2, . . ., n} such that:
+Xt =
+
+
+
+
+
+
+
+
+
+XV (n)
+1
+,
+if
+0 ≤ t < ξ(n)
+1
+,
+XV (n)
+j
+,
+if
+j−1
+�
+k=1
+ξ(n)
+k
+≤ t <
+j�
+k=1
+ξ(n)
+k ,
+j > 1.
+(2)
+Informally, (Xt)t≥0 should be understood as follows: the process takes the value of one observation
+uniformly chosen from the n observations X1, . . . , Xn during exponential times with intensity αn. The
+stochastic Langevin over-damped diffusion we consider is then given by the joint evolution (θt, Xt)t≥0
+and that is defined by:
+dθt = −∇θUXt(θt)dt +
+√
+2dBt,
+t > 0,
+(3)
+where (Bt)t≥0 is a multivariate standard Brownian Motion.
+Algorithm 1: Stochastic Langevin over-damped
+Data: (X1, . . . , Xn) i.i.d. observations, n0 initial distribution, π0 prior distribution
+1 t0 = 0
+2 Generate θ0 according to n0
+3 for k = 0, 1, . . . do
+4
+Pick Xk uniformly in {X1, . . . , Xn}
+5
+Generate ξk according to an Exponential distribution with mean α−1
+n
+6
+tk+1 = tk + ξk
+7
+θtk+1 = θtk −
+� tk+1
+tk
+∇θUXk(θs)ds +
+√
+2Bξk
+8 end
+9 return lim
+k→∞ θtk
+1.3
+Entropic divergence
+To assess the long-time behaviour of the SLMC, we introduce several notations related to the pair
+(θt, Xt)t≥0. Below, we denote by λd the Lebesgue measure over Rd. The semi-group induced by L
+being elliptic on the θ coordinate, trivially irreducible and finitely supported on the x coordinate,
+makes the law of (θt, Xt) absolutely continuous with respect to the measure λd ⊗ νn as soon as t > 0.
+We introduce the notation of mt to refer to the joint density of (θt, Xt) at time t with respect
+to λd ⊗ νn. In the meantime, nt denotes the marginal distribution of θt and mt(·|θ) the conditional
+distribution of Xt given θt = θ. That is:
+Law(θt, Xt) = mt,
+nt(θ) =
+n
+�
+i=1
+mt(θ, Xi),
+mt(x|θ) = mt(θ, x)
+nt(θ) ,
+(4)
+4
+
+for θ ∈ Rd and x ∈ {X1, . . . , Xn}.
+To show that the SLMC algorithm recovers the correct asymptotic behaviour, i.e. that nt(θ) −→ µn
+when t −→ ∞, we consider the relative entropy (or Kullback-Leibler divergence) of nt with respect to
+µn that is well defined thanks to the ellipticity, and given by:
+Jt = Entµn
+� nt
+µn
+�
+=
+�
+Rd
+log
+� nt(θ)
+µn(θ)
+�
+dnt(θ).
+(5)
+Jt measures at any time t > 0 a divergence between the instantaneous law of the process at time t
+and the (presumably) invariant distribution µn of the process (θt, Xt). It would also be possible to
+measure this difference between the two distributions in terms of the L2 or the χ-square distance and
+to produce a theoretical analysis with the help of functional analysis but it would rely on stronger
+assumptions on the function Uνn.
+In the meantime, we also introduce a weighted L2 distance between the conditional distribution of
+Xt given θt = θ and the measure νn. This distance is denoted by It and is defined as:
+It =
+�
+Rd
+n
+�
+i=1
+�mt(Xi|θ)
+νn(Xi)
+− 1
+�2
+νn(Xi)dnt(θ).
+(6)
+This quantity measures the average closeness (w.r.t. θ) of the conditional law of x given θ at time t to
+νn.
+1.4
+Main assumptions
+Weak convexity
+We will study the SLMC into a weakly convex framework, i.e. when Uνn is assumed
+to be convex but not necessarily strongly convex. SLMC has recently received an important interest in
+the machine learning community and has been studied essentially in its explicit Euler discretized form
+in various situations where functional inequalities are involved. We refer to [38] (uniform Log-Sobolev
+inequality), to [35] (uniform Poincar´e inequality) where the authors develop a Wasserstein-2 analysis
+of the algorithm, and to [40] (uniform Poincar´e inequality). In these works, the functional inequalities
+play a crucial role to analyze the behaviour of SLMC and these inequalities are assumed, which is an
+important hypothesis. Importantly, Poincar´e or Log-Sobolev inequalities are not so innocent since they
+generally require convexity (see e.g. [4, 3]) to be reasonably dimension-dependent, and even strong
+convexity to be dimension free. Otherwise, the constant involved in these functional inequalities are
+exponentially degraded by the “temperature” (n−1(d log2β(n))−(1+r) in our case) and the dimension
+(d for us) as indicated in [26].
+In our work, we have chosen to parameterize this lack of strong convexity with the help of the
+Kurdyka-�Lojasiewicz inequality [28, 30], which is a standard tool in optimization to describe the tran-
+sition between convexity and strong convexity and makes the bounds more explicit. This assumption
+allows to observe how the entropy evolves according to the key exponent involved in the KL inequality.
+In particular, it makes possible to understand the influence of the lack of strong convexity that is more
+or less hidden in the uniform Poincar´e or Log-Sobolev inequalities that are assumed in the previous
+works. We introduce a parametric form of the KL inequalities following [20].
+For this purpose, for any V twice differentiable function, we denote the spectrum of the Hessian
+matrix of V as Sp(∇2V (θ)). Furthermore, if V is convex, we denote:
+λ∇2V (θ) = inf Sp(∇2V (θ)).
+Hypothesis Hr
+KL(c, L) We say that a function V : Rd → R satisfies a Hr
+KL(c, L)-condition if:
+a) V is a C2-function.
+b) V is a convex function and minθ∈RdV (θ) = V (θ∗) > 0.
+c) ∇V is L-Lipschitz.
+5
+
+d) There exist some constants 0 ≤ r < 1 and c > 0 such that:
+cV −r(θ) ≤ λ∇2V (θ)
+∀θ ∈ Rd.
+(7)
+Let us briefly comment this assumption.
+• In [21], a slightly different parametrization is used with the introduction of another exponent
+q related to λ∇2V (θ) = sup Sp(∇2V (θ)). The authors also assume the upper bound λ∇2V (θ) ≤
+˜cV −q(θ). Here, we have chosen to simplify this assumption and use a rough upper bound on the
+eigenvalues of the Hessian matrix given by the Lipschitz constant L, i.e. in the last inequality
+we simply use ˜c = L and q = 0.
+• We shall observe that if V (θ) = (1 + ∥θ∥2
+2)p with p ∈ [1/2, 1], then V satisfies Hr
+KL(c, L) with
+r = 1−p
+p
+and c = 2p(1 − 2(1 − p)), see Remark 7 of [21] for further details. In particular, the
+larger p, the smaller r, which translates into a better curvature of the potential function V .
+• When r = q, we recover a global standard KL inequality (see [20, 5]) and when r = 1 it
+corresponds to the limiting Laplace case.
+• The case r = 0 is of course associated to the strongly convex situation where the curvature of
+the function is uniformly lower bounded by c.
+Hence, it is expected that the complexity of SLMC increases with the lack of curvature, i.e. is an
+increasing function of r.
+In section 4 we recall some important consequences of the KL inequality obtained in Lemma 15 of
+[21]. In particular, the growth of any function that satisfies Hr
+KL(c, L) is lower and upper bounded by
+a positive power of the distance to its minimizer.
+If inequality (7) holds for a constant c, then it holds for all positive values less than c. For that
+reason, in section 5 we assume c ≤
+�
+8L
+(1+r)
+�1+r
+.
+Assumption on the prior π0
+We state below the important consequence of a “population” Hr
+KL(c, L)
+assumption, but before, let us state some mild assumptions on π0.
+Hypothesis Hπ0(ℓ0) π0 is a log-concave C2-function such that minθ∈Rd − log π0(θ) > 0 and θ �→
+∇ log π0(θ) is ℓ0-Lipschitz.
+Since the prior distribution is chosen by the user, our Hπ0(ℓ0) hypothesis is not restrictive and
+some typical examples satisfy these conditions, such as Gaussian, Weibull and Gamma, both with
+shape parameter larger than 1, Gumbel, among others.
+Proposition 1.1. We assume Hπ0(ℓ0) and that there exist (c, r) such that for any x: θ �−→ − log pθ(x)
+satisfies Hr
+KL(c, L), then Uνn satisfies Hr
+KL
+�
+cn1+r, nL + ℓ0
+�
+, and in particular, for any Xi, UXi sat-
+isfies Hr
+KL
+�
+cn1+r, nL + ℓ0
+�
+.
+We introduce the notation a ≲uc b (a ≳uc b) which means a ≤ cb (a ≥ cb) where c is a universal
+constant i.e. a positive constant independent of n and d.
+We assume that the minimizers of the functions UXi are contained in a ball of radius which depends
+of n and d. Additionally, we consider minθ∈RdUXi to be at most of order d.
+Hypothesis Hmin There exists β ≥ 0 such that:
+maxi∥ arg min UXi∥2 ≲uc
+√
+d logβ(n)
+and
+maxi minθ∈Rd UXi(θ) ≲uc d.
+Assumption Hmin is not restrictive. In dimension d = 1, it holds for many concentrated i.i.d.
+samples (Xi)1≤i≤n with a suitable sub-Gaussian like behaviour for which the Laplace transform of
+min UXi is upper bounded as:
+E[exp(λmin UXi)] ≤ exp(σ2λk),
+∀λ > 0.
+6
+
+The previous upper bound implies that, in this case, β involved in Hmin is given by β = k−1
+k . We
+recover in particular the situation where β = 1/2 when k = 2. For larger dimensions, the result may
+be extended using that ∥X∥2
+2 ≤ d max1≤j≤d(Xj)2, where Xj is the j-th component of X. We should
+keep in mind from this last discussion that even if Hmin is stated (and makes sense) for any value of
+β > 0, it holds in general for β ≤ 1.
+This Hmin hypothesis together with Hπ0(ℓ0) lead to an almost similar behaviour of the minimizer
+and the minimum of Uνn. Details appear in Proposition 4.4.
+1.5
+Long-time entropy convergence
+We introduce for any time t ≥ 0 the density of Law(θt) w.r.t. µn, which is given by:
+ft(θ) = nt(θ)
+µn(θ),
+and n0 is chosen such that ∥f0∥∞ < +∞. The following hypothesis guarantees this result which will
+be proved in Proposition 3.5.
+Hypothesis Hn0(L, ℓ0) A positive constant σ2 exists such that n0 = N(0, σ2Id). Moreover, there
+exist two universal constants c1 and c2 such that 0 < c1 ≤ c2 < 1 and
+c1
+nL + ℓ0
+≤ σ2 ≤
+c2
+nL + ℓ0
+.
+Futhermore, in Proposition 3.5, as an immediate consequence of the boundedness of ∥f0∥∞, we
+obtain that J0 ≲uc nd1+r log2β(1+r)(n) + d log
+� d
+n
+�
+.
+The next result assesses a mixing property in terms of decrease of the entropy and therefore states
+the convergence of nt towards the correct measure µn.
+Theorem 1.1. Assume Hπ0(ℓ0), Hmin, Hn0(L, ℓ0) and that each θ �→ − log pθ(Xi) satisfies Hr
+KL(c, L),
+then
+• Uνn satisfies a Poincar´e inequality of constant CP (µn), indistinctly denoted as CP .
+• Define cn,d := n4 �
+d log2β(n)
+�1+r
+and On,d :=
+� C1d
+n
+� dr
+2 exp
+�
+C2n
+�
+d log2β(n)
+�1+r�
+, where C1
+and C2 are universal constants, then for any t > 0:
+Jt ≲uc
+�
+J0 + cn,d
+αn
+�
+1 +
+�CP
+αn
++
+�
+CP
+�
+e
+√
+CP
+√a + CP
+3αn
+�
++ On,d
+�
+(1 + t)1/4e−
+√
+Cp
+√a (√1+t−1).
+(8)
+• For any ε > 0, if αn =
+1
+n(d log2β(n))
+1+r , then:
+t ≳uc
+�
+d log2β(n)
+�(1+r)2 �
+log2(ε−1) + n2 �
+d log2β(n)
+�2(1+r)
++ d2 log2 d
+�
+=⇒ Jt ≤ ε.
+If we denote tε the smallest value such that Jtε ≤ ε, then the choice of αn =
+1
+n(d log2β(n))
+1+r
+guarantees that the mean number of jumps αntε of the process (Xt)0≤t≤tε is the minimum possible.
+In order to proof the main result, we first present in Section 2 the classical tools related to the
+Markov semi-group, which could be skipped by the experienced reader in the subject. In Section 3
+we prove the main result. Sections 4 and 5 are reserved to the technical results of the Hr
+KL(c, L)
+hypothesis and Uνn, and the Markov Dynamics respectively.
+7
+
+2
+Markov tools
+It is straightforward to verify that the joint evolution of (θt, Xt)t≥0 exists and is weakly unique (in
+law) with the help of the Martingale Problem (MP below). For this purpose, we preliminary define
+the operator L that acts on any function f ∈ C2(Rd × X) as:
+Lf(θ, x) = −⟨∇θUx(θ), ∇θf(θ, x)⟩ + ∆θf(θ, x)
+�
+��
+�
+:=L1f(θ,x)
++ αn
+n
+n
+�
+i=1
+[f(θ, Xi) − f(θ, x)
+�
+��
+�
+L2f(θ,x)
+],
+(9)
+for all (θ, x) ∈ Rd × X.
+The operator L is divided into two terms, L1 acts on the component θ and is associated to the
+diffusion part, while L2 is the jump operator that acts on the x component. Thanks to the finiteness
+of the number of observations (X1, . . . , Xn), we can apply the results of Section 4 and 5 of chapter 4
+of [17] and deduce the following result:
+Proposition 2.1. Assume that for any x ∈ X, Ux is C2(Rd) and ∇θUx is Lx-Lipschitz, then for any
+initial distribution ν on Rd × X, the martingale problem (L, ν) is well-posed.
+The associated (weakly) unique process (θt, Xt)t≥0 is a Feller Markov process associated to the
+semi-group L. In particular, the θ component verifies the S.D.E. (3).
+If we denote by L⋆ the adjoint operator of L in L2(Rd) × νn, the backward Kolmogorov Equation
+yields:
+∂tmt(θ, x) = L⋆mt(θ, x).
+(10)
+Using the ellipticity of the semi-group L on the θ coordinate, we can use the result of [25] and
+deduce that for any t > 0, nt ∈ C∞(Rd, R) and the irreducibility yields ∀t ≥ 0, nt > 0. We will prove
+in Proposition 3.5 some sufficient conditions that implies ∥f0∥∞ = ∥ n0(θ)
+µn(θ)∥∞ < +∞ and an important
+and standard consequence of the maximum principle, is as follows: if ∥f0∥∞ ≤ M, then
+∀t ≥ 0,
+∥ft∥∞ ≤ M.
+We defer the details of this result to the Proposition 3.5 as they are not central to our analysis and
+are rather technical.
+Thanks to this master equation, it is possible to compute the derivative of the semi-group on some
+time dependent function of θ. For this purpose, we introduce two keystone operators. The first one
+describes the infinitesimal action on the θ coordinate under the average effect of Xt at time t that
+applies ∀f ∈ C2(Rd, R) as:
+Gtf(θ) = −
+n
+�
+i=1
+⟨∇θf(θ), ∇θUXi(θ)⟩mt(Xi|θ) + ∆θf(θ).
+(11)
+The second one is very close to the first one except that the average effect of Xt is replaced by the
+targeted ideal distribution νn. It leads to the definition ∀f ∈ C2(Rd, R):
+Gf(θ) = −
+n
+�
+i=1
+⟨∇θf(θ), ∇θUXi(θ)⟩νn(Xi) + ∆θf(θ) = −⟨∇θf(θ), ∇θUνn(θ)⟩ + ∆θf(θ).
+(12)
+This derivative is given in the next result, whose proof is deferred to the appendix.
+Lemma 2.1. Let be ht a twice differentiable function with uniformly bounded first and second order
+derivatives on Rd, then for t > 0:
+∂t
+��
+Rd ht(θ)dnt(θ)
+�
+=
+�
+Rd ∂t{ht(θ)}dnt(θ) +
+�
+Rd Gtht(θ)dnt(θ),
+(13)
+where Gt is the diffusion operator under the average effect of Xt, defined in Equation (11).
+8
+
+3
+Proof of the main results
+3.1
+Evolution of the entropy Jt
+The entropy satisfies the following differential inequality.
+Proposition 3.1. Assume Hmin, Hπ0(ℓ0) and for each Xi, θ → − log pθ(Xi) satisfies Hr
+KL(c, L), then
+a ”universal” constant C (independent from n and d) exists such that ∀t > 0:
+∂t{Jt} ≤ −
+�
+Rd
+�����∇θ
+��
+nt(θ)
+µn(θ)
+������
+2
+2
+dµn(θ) + CI
+1
+3
+t n
+11
+3
+�
+d log2β(n)
+�1+r
+.
+Proof. We shall use the standard preliminary estimate that may be derived from Equation (3.14) of
+[29] for elliptic diffusions to apply Lemma 2.1 to ft = log(ntµ−1
+n ). From Equation (13), we have:
+∂t{Jt} =
+�
+Rd ∂t
+�
+log
+� nt(θ)
+µn(θ)
+��
+dnt(θ) +
+�
+Rd Gt log
+� nt(θ)
+µn(θ)
+�
+dnt(θ),
+The first term vanishes since:
+�
+Rd ∂t
+�
+log
+� nt(θ)
+µn(θ)
+��
+dnt(θ)
+=
+�
+Rd
+∂t{nt(θ)}
+nt(θ)
+dnt(θ)
+=
+�
+Rd ∂t {nt(θ)} dθ
+=
+∂t
+��
+Rd dnt(θ)
+�
+=
+0.
+Then, the derivative is reduced to the second term, and we are led to:
+∂t{Jt}
+=
+�
+Rd Gt log
+� nt(θ)
+µn(θ)
+�
+dnt(θ),
+=
+�
+Rd G log
+� nt(θ)
+µn(θ)
+�
+dnt(θ)
+�
+��
+�
+J1,t
++
+�
+Rd (Gt − G) log
+� nt(θ)
+µn(θ)
+�
+dnt(θ)
+�
+��
+�
+J2,t
+.
+(14)
+We study the two terms J1,t and J2,t separately.
+• Study of J1,t. Since G is a diffusion operator and µn is the invariant measure associated to G,
+then we can use the classical link between J1,t and the Dirichlet form (see [3]):
+�
+Rd G log
+� nt(θ)
+µn(θ)
+�
+dnt(θ)
+=
+�
+Rd
+nt(θ)
+µn(θ) G log
+� nt(θ)
+µn(θ)
+�
+dµn(θ)
+=
+−4
+�
+Rd
+�����∇θ
+��
+nt(θ)
+µn(θ)
+������
+2
+2
+dµn(θ).
+(15)
+• Study of J2,t. We use the difference between G and Gt, for any twice differentiable function f:
+(Gt − G) f(θ)
+=
+−
+n
+�
+i=1
+⟨∇θf(θ), ∇θUXi(θ)⟩ [mt(Xi|θ) − νn(Xi)]
+=
+−
+n
+�
+i=1
+⟨∇θf(θ), ∇θUXi(θ)⟩
+�mt(Xi|θ)
+νn(Xi)
+− 1
+�
+νn(Xi).
+9
+
+Then, the term J2,t may be computed as:
+|J2,t|
+=
+����
+�
+Rd (Gt − G) log
+� nt(θ)
+µn(θ)
+�
+dnt(θ)
+����
+=
+�����
+�
+Rd
+n
+�
+i=1
+⟨∇θ log
+� nt(θ)
+µn(θ)
+�
+, ∇θUXi(θ)⟩
+�mt(Xi|θ)
+νn(Xi)
+− 1
+�
+νn(Xi) dnt(θ)
+����� .
+Using the Cauchy-Schwartz inequality with respect to the measure νn(Xi) × dnt(θ) in the first
+line, 2ab ≤ a2 + b2 in the second line and ∇ log f = 2∇ log √f = 2 ∇√f
+√f
+in the third line, we
+obtain that:
+|J2,t|
+≤
+��
+Rd
+����∇θ log
+� nt(θ)
+µn(θ)
+�����
+2
+2
+dnt(θ)
+� 1
+2 ��
+Rd
+n
+�
+i=1
+��∇θUXi(θ)
+��2
+2
+�mt(Xi|θ)
+νn(Xi)
+− 1
+�2
+νn(Xi) dnt(θ)
+� 1
+2
+≤
+3
+4
+�
+Rd
+����∇θ log
+� nt(θ)
+µn(θ)
+�����
+2
+2
+dnt(θ) + 1
+3
+�
+Rd
+n
+�
+i=1
+��∇θUXi(θ)
+��2
+2
+� mt(Xi|θ)
+νn(Xi)
+− 1
+�2
+νn(Xi) dnt(θ)
+≤
+3
+�
+Rd
+�����∇θ
+��
+nt(θ)
+µn(θ)
+������
+2
+2
+dµn(θ) + 1
+3
+�
+Rd
+n
+�
+i=1
+��∇θUXi(θ)
+��2
+2
+�mt(Xi|θ)
+νn(Xi)
+− 1
+�2
+νn(Xi) dnt(θ).
+Using Equation (15) and the previous line yields:
+∂t{Jt}
+≤
+−
+�
+Rd
+�����∇θ
+��
+nt(θ)
+µn(θ)
+������
+2
+2
+dµn(θ) + 1
+3
+�
+Rd
+n
+�
+i=1
+��∇θUXi(θ)
+��2
+2
+�mt(Xi|θ)
+νn(Xi)
+− 1
+�2
+νn(Xi) dnt(θ)
+�
+��
+�
+:=∆t
+,
+(16)
+We then focus on the second term of the right hand side. For this purpose, we consider a non-
+negative function g(t), which will be fixed later and we split ∆t into two terms as:
+∆t
+=
+�
+Rd
+n
+�
+i=1
+��∇θUXi(θ)
+��2
+2
+�
+1∥∇θUXi (θ)∥2≤g(t) +
+1∥∇θUXi (θ)∥2>g(t)
+� �mt(Xi|θ)
+νn(Xi)
+− 1
+�2
+νn(Xi) dnt(θ)
+≤
+g2(t)It +
+�
+Rd
+n
+�
+i=1
+��∇θUXi(θ)
+��2
+2
+1∥∇θUXi (θ)∥2>g(t)
+�mt(Xi|θ)
+νn(Xi)
+− 1
+�2
+νn(Xi) dnt(θ),
+where It has been introduced in Equation (6) and measures the closeness of mt(Xi|θ) to νn. Finally,
+for the last term we observe that 0 ≤ mt(Xi|θ) ≤ 1 and
+��� mt(Xi|θ)
+νn(Xi) − 1
+��� = n
+��mt(Xi|θ) − 1
+n
+�� ≤ n, which
+implies that:
+∆t ≤ g2(t)It + n2 1
+n
+�
+Rd
+n
+�
+i=1
+∥∇θUXi(θ)∥2
+2
+1∥∇θUXi (θ)∥2>g(t)dnt(θ)
+�
+��
+�
+:= ˜∆t
+.
+(17)
+The Cauchy inequality leads to:
+˜∆t
+≤
+�
+1
+n
+�
+Rd
+n
+�
+i=1
+∥∇θUXi(θ)∥4
+2 dnt(θ)
+� 1
+2 �
+1
+n
+�
+Rd
+n
+�
+i=1
+1∥∇θUXi (θ)∥2>g(t)dnt(θ)
+� 1
+2
+=
+�
+1
+n
+n
+�
+i=1
+E
+�
+∥∇θUXi(θt)∥4
+2
+�� 1
+2 �
+1
+n
+n
+�
+i=1
+P (∥∇θUXi(θt)∥2 > g(t))
+� 1
+2
+.
+(18)
+We then use Proposition 4.1 and obtain that:
+˜∆t
+≤
+�
+1
+n
+n
+�
+i=1
+E
+��
+2(nL + ℓ0)U 2
+Xi(θt)
+��
+� 1
+2 �
+1
+n
+n
+�
+i=1
+P
+�
+2(nL + ℓ0)UXi(θt) > g2(t)
+�
+� 1
+2
+≤
+2(nL + ℓ0)
+�
+nE[U 2
+νn(θt)]
+� 1
+2
+�
+1
+n
+n
+�
+i=1
+2(nL + ℓ0)
+g2(t)
+E [UXi(θt)]
+� 1
+2
+≤
+[2(nL + ℓ0)]
+3
+2 n
+1
+2 E
+�
+U 2
+νn(θt)
+� 1
+2 E [Uνn(θt)]
+1
+2
+g(t)
+,
+10
+
+where we used the Markov’s inequality and the relation ∥.∥2 ≤ ∥.∥1 in Rn. We apply Proposition 5.1
+with α = 2 and α = 1 and obtain that a constant C > 0 exists (whose value may change from line to
+line) such that:
+˜∆t
+≤
+C
+n
+7
+2
+�
+d log2β(n)
+� 3(1+r)
+2
+g(t)
+.
+We use this last bound in (17) and we deduce that:
+∆t ≤ g2(t)It + C
+n
+11
+2
+�
+d log2β(n)
+� 3(1+r)
+2
+g(t)
+.
+Optimizing this last bound with respect to g(t) leads to the upper bound:
+∆t ≤ CI
+1
+3
+t n
+11
+3
+�
+d log2β(n)
+�1+r
+,
+∀t ≥ 0.
+3.2
+Evolution of the weighted L2 distance It
+The quantity It involved in Proposition 3.1 measures how close to νn the conditional distribution of
+Xt|θt is. To study It, we first remark that it may be rewritten in a simpler way.
+It
+=
+�
+Rd
+n
+�
+i=1
+�mt(Xi|θ)
+νn(Xi)
+− 1
+�2
+νn(Xi) dnt(θ)
+=
+�
+Rd
+n
+�
+i=1
+�m2
+t(Xi|θ)
+ν2n(Xi)
+− 2mt(Xi|θ)
+νn(Xi)
++ 1
+�
+νn(Xi) dnt(θ)
+=
+�
+Rd
+n
+�
+i=1
+�m2
+t(Xi|θ)
+νn(Xi)
+− 2mt(Xi|θ) + νn(Xi)
+�
+dnt(θ)
+=
+�
+Rd
+� n
+�
+i=1
+m2
+t(Xi|θ)
+νn(Xi)
+− 1
+�
+dnt(θ)
+=
+�
+Rd
+n
+�
+i=1
+m2
+t(Xi|θ)
+νn(Xi) dnt(θ) − 1.
+Using that mt(Xi|θ)nt(θ) = mt(θ, Xi) and νn(Xi) = 1
+n for i = 1, 2, . . ., n, we obtain that:
+It = n
+�
+Rd
+n
+�
+i=1
+m2
+t(θ, Xi)
+nt(θ)
+dθ − 1.
+(19)
+The next proposition then assesses how fast It decreases to 0 as t −→ +∞.
+Proposition 3.2. For any t ≥ 0:
+It ≤ I0e−2αnt ≤ (n − 1)e−2αnt.
+(20)
+Proof. Our starting point is Equation (19). We compute its derivative with respect to t:
+∂t{It}
+=
+2n
+�
+Rd
+n
+�
+i=1
+mt(θ, Xi)
+nt(θ)
+∂tmt(θ, Xi)dθ − n
+�
+Rd
+n
+�
+i=1
+m2
+t(θ, Xi)
+n2
+t(θ)
+∂tnt(θ)dθ
+=
+2n
+�
+Rd
+n
+�
+i=1
+mt(Xi|θ)∂tmt(θ, Xi)dθ − n
+�
+Rd
+n
+�
+i=1
+m2
+t(Xi|θ)∂tnt(θ)dθ.
+11
+
+Using the Kolmogorov backward equation in the first line and L = L1 + L2 in the second one where
+L1 and L2 are defined in Equation (9), we have:
+∂t{It}
+=
+2n
+�
+Rd
+n
+�
+i=1
+Lmt(Xi|θ) mt(θ, Xi)dθ − n
+�
+Rd
+n
+�
+i=1
+m2
+t(Xi|θ)∂tnt(θ)dθ
+=
+2n
+�
+Rd
+n
+�
+i=1
+L1mt(Xi|θ) mt(θ, Xi)dθ
+�
+��
+�
+:=I3,t
++ 2n
+�
+Rd
+n
+�
+i=1
+L2mt(Xi|θ) mt(θ, Xi)dθ
+�
+��
+�
+:=I1,t
+−n
+�
+Rd
+n
+�
+i=1
+m2
+t(Xi|θ)∂tnt(θ)dθ
+�
+��
+�
+:=I2,t
+.
+(21)
+Then, ∂t{It} may be splitted into three terms that are studied separately.
+• Study of I1,t. We observe that:
+L2mt(Xi|θ) = αn
+n
+n
+�
+j=1
+[mt(Xj|θ) − mt(Xi|θ)] = αn
+n − αn mt(Xi|θ).
+(22)
+We then use this last equation in the definition of I1(t) and obtain that:
+I1,t
+=
+2n
+�
+Rd
+n
+�
+i=1
+L2mt(Xi|θ) mt(θ, Xi)dθ
+=
+2αn
+�
+Rd
+n
+�
+i=1
+mt(θ, Xi)dθ − 2αnn
+�
+Rd
+n
+�
+i=1
+mt(Xi|θ)mt(θ, Xi)dθ
+=
+2αn − 2αnn
+�
+Rd
+n
+�
+i=1
+m2
+t(θ, Xi)
+nt(θ)
+dθ
+=
+−2αnIt.
+(23)
+• Study of I2,t. Using the definition of nt, we obtain that:
+I2,t
+=
+−n
+�
+Rd
+n
+�
+i=1
+m2
+t(Xi|θ)∂tnt(θ)dθ
+=
+−n
+�
+Rd
+n
+�
+i=1
+m2
+t(Xi|θ)∂t
+
+
+n
+�
+j=1
+mt(θ, Xj)
+
+ dθ
+=
+−n
+�
+Rd
+n
+�
+j=1
+n
+�
+i=1
+m2
+t (Xi|θ)∂tmt(θ, Xj)dθ
+=
+−n
+�
+Rd
+n
+�
+j=1
+� n
+�
+i=1
+Lm2
+t (Xi|θ)
+�
+mt(θ, Xj)dθ
+=
+−n
+�
+Rd
+n
+�
+i=1
+Lm2
+t(Xi|θ) dnt(θ).
+where we used the Kolmogorov backward equation in the fourth line and again the definition of
+nt in the last line. Again, the decomposition L = L1 + L2 yields:
+I2,t
+=
+−n
+�
+Rd
+n
+�
+i=1
+L1m2
+t(Xi|θ) dnt(θ) − n
+�
+Rd
+n
+�
+i=1
+L2m2
+t(Xi|θ) dnt(θ).
+12
+
+We repeat some similar computations as those developed in Equation (22) to study the action
+of the jump component induced by L2 on m2
+t . We obtain that:
+L2m2
+t(Xi|θ) = αn
+n
+n
+�
+k=1
+[m2
+t(Xk|θ) − m2
+t(Xi|θ)] = αn
+n
+n
+�
+k=1
+m2
+t (Xk|θ) − αn m2
+t(Xi|θ).
+We use this last equation and obtain that:
+I2,t
+=
+−n
+�
+Rd
+n
+�
+i=1
+L1m2
+t (Xi|θ) dnt(θ) − αn
+�
+Rd
+n
+�
+i=1
+n
+�
+k=1
+m2
+t(Xk|θ) dnt(θ)
++αnn
+�
+Rd
+n
+�
+i=1
+m2
+t(Xi|θ) dnt(θ)
+=
+−n
+�
+Rd
+n
+�
+i=1
+L1m2
+t (Xi|θ) dnt(θ) − αnn
+�
+Rd
+n
+�
+k=1
+m2
+t(Xk|θ) dnt(θ)
++αnn
+�
+Rd
+n
+�
+i=1
+m2
+t(Xi|θ) dnt(θ)
+=
+−n
+�
+Rd
+n
+�
+i=1
+L1m2
+t (Xi|θ) dnt(θ).
+(24)
+• Study of I2,t + I3,t. We observe that this sum involves only L1 (see Equation (9). We first
+compute:
+L1mt(Xi|θ) = −⟨∇θUXi(θ), ∇θmt(Xi|θ)⟩ + ∆θmt(Xi|θ),
+and similarly:
+L1m2
+t(Xi|θ) = −⟨∇θUXi(θ), ∇θm2
+t(Xi|θ), ⟩ + ∆θm2
+t(Xi|θ)
+= −2mt(Xi|θ)⟨∇θUXi(θ), ∇θmt(Xi|θ)⟩ + 2∥∇θmt(Xi|θ)∥2
+2 + 2mt(Xi|θ)∆θmt(Xi|θ).
+Using these two equations into I2,t + I3,t and mt(Xi|θ)nt(θ) = mt(θ, Xi), we get:
+I2,t + I3,t
+n
+= 2
+�
+Rd
+n
+�
+i=1
+⟨∇θmt(Xi|θ), ∇θUXi(θ)⟩mt(θ, Xi)dθ
+− 2
+�
+Rd
+n
+�
+i=1
+∥∇θmt(Xi|θ)∥2
+2 nt(θ)dθ − 2
+�
+Rd
+n
+�
+i=1
+∆θmt(Xi|θ) mt(θ, Xi)dθ
+− 2
+�
+Rd
+n
+�
+i=1
+⟨∇θmt(Xi|θ), ∇θUXi(θ)⟩mt(θ, Xi)dθ + 2
+�
+Rd
+n
+�
+i=1
+∆θmt(Xi|θ) mt(θ, Xi)dθ
+= −
+�
+Rd
+n
+�
+i=1
+∥∇θmt(Xi|θ)∥2
+2 dnt(θ) ≤ 0.
+Gathering this last inequality with (23) into Equation (21) yields:
+∂t{It} ≤ −2αnIt.
+We conclude with a direct application of the Gronwall lemma while observing that I0 ≤ n − 1.
+3.3
+Functional (weak) log-Sobolev inequalities
+3.3.1
+Related works on functional inequalities
+A straightforward consequence of Proposition 3.1 and Proposition 3.2 is the following differential
+inequality on the relative entropy Jt:
+∂t{Jt} ≤ −
+�
+Rd
+�����∇θ
+��
+nt(θ)
+µn(θ)
+������
+2
+2
+dµn(θ) + cn,de− 2αn
+3
+t,
+(25)
+13
+
+where cn,d is defined as:
+cn,d ≲uc n4 �
+d log2β(n)
+�1+r
+.
+(26)
+At this stage, we should observe that a standard approach consists in finding a functional inequality
+that relates the key Dirichlet form E(f) defined by:
+E(f) =
+�
+Rd ∥∇θf(θ)∥2
+2dµn(θ),
+(27)
+to Entµn(f 2), the entropy itself with respect to µn. These approaches rely on the initial works of [23]
+where Logarithmic Sobolev Inequality (LSI for short) were introduced. The consequences of LSI to
+exponential ergodicity has then been an extensive field of research and we refer to [3] for an overview
+on this topic. A popular sufficient condition that ensures LSI is the log strong-convexity of the targeted
+measure (see among other [2]) and an impressive amount of literature has been focused on the existing
+links between these functional inequalities, ergodicity of the semi-group, transport inequalities and
+Lyapunov conditions. We refer to [8, 1] (these two works are far from being exhaustive). The great
+interest of LSI has then been observed in machine learning and statistics more recently as testified by
+the recent works in Monte Carlo samplings of [31, 34]. A popular way to extend LSI from the strongly
+convex situation to a more general case relies on the “strong convexity outside a ball” hypothesis using
+the perturbation argument of the seminal contributions of [26]. If this method proves to be suitable
+for the study of the simulated annealing process in [33], [26], it appears to be doubtful for the study
+of sampling problems with convex potentials that satisfies Hr
+KL(c, L) as this settings do not imply an
+asymptotic strong convexity of θ �−→ U(θ) for large values of ∥θ∥2. That being said, and maybe an
+even worst consequence of such approach, is the unavoidable dependency on the dimension for the LSI
+constant when using a perturbation approach, which leads to a serious exponential degradation of the
+convergence rates with the dimension of the ambient space.
+To overcome these difficulties, we have chosen to use a slightly different functional inequality that
+may be considered as an innocent modification of LSI, but that indeed appears to be well suited
+to weakly log-concave setting described through an Hr
+KL(c, L) assumption.
+For this purpose, we
+shall use weak log-Sobolev inequalities (WLSI for short below) that have been introduced in [37]
+and whose interest has been extensively studied in many works to obtain exponentially sub-linear
+rates of mixing, see among others for example [7].
+To derive such inequalities, our starting point
+will be the contribution of [10] that makes the link between Lyapunov conditions and WLSI. Our
+approach based on Hr
+KL(c, L) certainly shares some similarities with the recent work of [6] where
+some functional inequalities (Poincar´e and Transport inequalities) are obtained within a framework of
+variable curvature bound.
+3.3.2
+Weak log Sobolev inequalities
+We briefly introduce the key theoretical ingredients, that are exhaustively described in [3]. We intro-
+duce the following assumption, that will be suitable for the setting of bounded functions.
+Definition 3.1 (Weak Log-Sobolev Inequality ). For any measurable space (Ω, F, µ) and for any nice
+function f, let us define:
+Entµ(f 2) :=
+�
+Ω
+f 2 log(f 2)dµ −
+�
+Ω
+f 2dµ log
+��
+Ω
+f 2dµ
+�
+.
+The measure µ satisfies a WLSI if a non-increasing function ϕWLS : (0, +∞) �→ R+ exists such that
+for any f ∈ C
+1
+b (Ω):
+Entµ(f 2) ≤ ϕWLS(s)E(f) + s Osc2(f),
+(28)
+where Osc(f) := sup f − inf f.
+Before establishing how to use this functional inequality, we first state the important relationship
+between Poincar´e Inequality and WLSI.
+14
+
+Proposition 3.3. Assume that µ satisfies a Poincar´e Inequality of constant CP , i.e. for any smooth
+integrable function f:
+Cp(µ)V arµ(f) = Cp(µ)
+�
+Ω
+(f − µ[f])2dµ ≤
+�
+Ω
+|∇f|2dµ,
+then if log c =
+3
+14e2
+� 1
+e + 1
+2
+�
++ 1 + log
+� 14
+3
+�
+, then µ satisfies a WLSI with:
+ϕWLS(s) =
+�
+0,
+s > 1
+e + 1
+2
+32
+CP log
+� c
+s
+�
+,
+s ≤ 1
+e + 1
+2
+.
+For the sake of readability, we introduce a universal a > 0 such that:
+ϕWLS(s) =
+�
+0,
+s > 1
+e + 1
+2
+a
+1+log( 1
+s)
+CP
+,
+s ≤ 1
+e + 1
+2
+.
+(29)
+Proof of Proposition 3.3. The proof of how the Poincar´e Inequality implies the WLSI in the bounded
+setting described in Definition 28 is given for the sake of completeness. Technical details are skipped
+and we refer to the references below. We use the measure-capacity inequality (see [3], Section 8.3).
+We know that the Poincar´e Inequality implies a capacity inequality (Proposition 8.3.1 of [3]) with a
+constant equal to 2CP . Then, we can apply Theorem 2.2 of [7] that induces a WLSI which is based
+on the function ϕWLS given in the statement of the proposition.
+3.3.3
+Weak log Sobolev inequalities under Hr
+KL(c, L)
+Of course, in the previous result, the only important dependency will be the one induced by CP , which
+will deserve an ad-hoc study under Assumption Hr
+KL(c, L). The numbers 32 and log(c) will be dealt
+with as “universal constants” in what follows.
+The next proposition states two lower bounds on the Poincar´e constant within the Hr
+KL(c, L)
+framework. The first one always holds, regardless the value of (X1, . . . , Xn) that may be been randomly
+sampled. The second one has to be considered with high probability, with respect to the sampling
+process (X1, . . . , Xn).
+Proposition 3.4. Assume Hmin,Hn0(L, ℓ0), Hπ0(ℓ0) and for any x, θ �→ − log pθ(x) satisfies Hr
+KL(c, L),
+then:
+i) For any sample (X1, . . . , Xn), it holds:
+CP (µn) ≳uc
+1
+�
+d log2β(n)
+�(1+r)2
+ii) Assume that θ �→ Pθ is injective and θ0 exists such that (X1, . . . , Xn) ∼ Pθ0. If locally around
+θ0, θ �→ |θ − θ0|−αW1(Pθ, Pθ0) does not vanish, then:
+E(X1,...,Xn)∼Pθ0[CP (µn)] ≳uc
+�
+n
+Ld log n
+�α
+.
+We are finally led to upper bound the oscillations of the function involved in the WLSI introduced
+in (28), i.e. we are looking for an upper bound of Osc2 ��
+nt
+µn
+�
+for any time t > 0. For this purpose,
+we observe that the Markov semi-group induces that ft =
+nt
+µn = Ptf0 where f0 =
+n0
+µn .
+The next
+proposition implies the boundedness of ft over Rd when n0 is chosen as a Gaussian distribution with
+a carefully tuned covariance matrix.
+Proposition 3.5. Assume Hmin,Hn0(L, ℓ0), Hπ0(ℓ0) and that, for any x, θ �→ − log pθ(x) satisfies
+Hr
+KL(c, L), then:
+15
+
+i) Two positive constants C1 and C2 exist, which are independent from n and d and such that:
+∥f0∥∞ ≲uc
+�C1d
+n
+� dr
+2
+exp
+�
+C2nd1+r log2β(1+r)(n)
+�
+.
+ii) As a consequence:
+Osc2(
+�
+ft) ≤ Osc2(
+�
+f0) ≲uc
+�C1d
+n
+� dr
+2
+exp
+�
+C2nd1+r log2β(1+r)(n)
+�
+.
+iii) Moreover, a straightforward consequence of i) is:
+J0 =
+�
+Rd log (f0(θ)) dn0(θ) ≲uc nd1+r log2β(1+r)(n) + d log
+� d
+n
+�
+.
+3.4
+Entropic convergence of the SLMC
+The purpose of this paragraph is to prove the main result of the paper, i.e. Theorem 1.1 that guarantees
+the convergence of the SLMC algorithm.
+Proof of Theorem 1.1. Our starting point is the semi-group inequality (25) associated with the func-
+tional WLSI inequality (28). Using cn,d defined in (26), we obtain for any s > 0:
+∂t{Jt} ≤ −E
+�� nt
+µn
+�
++ cn,de− 2αn
+3
+t
+≤ −
+Jt
+ϕWLS(s) +
+s
+ϕWLS(s)Osc2
+�� nt
+µn
+�
++ cn,de− 2αn
+3
+t
+≤ −
+Jt
+ϕWLS(s) +
+s On,d
+ϕWLS(s) + cn,de− 2αn
+3
+t,
+where we applied Proposition 3.5 in the last line with On,d ≲uc
+� C1d
+n
+� dr
+2 exp
+�
+C2nd1+r log2β(1+r)(n)
+�
+and C1 and C2 two universal constants. We then choose s (that depends on t) such that:
+st = e−A√t+1
+with
+A > 1
+that will be chosen later on.
+We observe that st < e−1 + 1/2, so that Equation (29) of Proposition 3.3 yields:
+ϕWLS(st) = a
+1 + log
+�
+1
+st
+�
+CP
+= a1 + A√1 + t
+CP
+.
+We introduce ψ(t) = exp
+�
+CP
+a
+� t
+0
+du
+1+A√1+u
+�
+and deduce that
+ψ(t) = exp
+
+CP
+a
+2A(√1 + t − 1) − 2 log
+�
+1+A√1+t
+1+A
+�
+A2
+
+ ≤ exp
+�2CP
+aA (
+√
+1 + t − 1)
+�
+.
+We now apply the Gronwall Lemma:
+∂t {ψ(t)Jt} =
+�
+CP
+a(1 + A√1 + t)Jt + J′
+t
+�
+ψ(t)
+≤
+�
+CP On,d
+a
+e−A√t+1
+1 + A√1 + t + cn,de− 2αn
+3
+t
+�
+ψ(t)
+≤ CP On,d
+a
+e−(A− 2CP
+aA )√1+t + cn,de
+2CP
+aA (√1+t−1)− 2αn
+3
+t.
+16
+
+We denote by t0 the positive real value that solves the equation 2CP
+aA
+√1 + t0 = αnt0
+3 . We then observe
+that:
+� t
+0
+e
+2CP
+aA (√1+u−1)− 2αn
+3
+udu ≤
+� t0
+0
+e
+2CP
+aA
+√1+udu +
+� +∞
+t0
+e− αn
+3 udu
+≤ t0e
+2CP
+aA
+√1+t0 + 3
+αn
+= t0e
+αnt0
+3
++ 3
+αn
+.
+If A is chosen such that A > 2CP
+aA , we then deduce that:
+Jt ≤
+�
+J0 + cn,dt0e
+αnt0
+3
++ 3cn,d
+αn
+�
+ψ(t)−1 + CP On,d
+a
+ψ(t)−1
+� t
+0
+e−
+�
+A− 2CP
+aA
+�√1+udu
+≤
+�
+J0 + cn,dt0e
+αnt0
+3
++ 3cn,d
+αn
+�
+ψ(t)−1 +
+2CP On,d
+a
+�
+A − 2CP
+aA
+�2 ψ(t)−1,
+where we used in the previous line the bound:
+� t
+0
+e−b√1+udu ≤
+� +∞
+0
+e−b√1+udu ≤ 2
+b2 .
+To obtain the lowest upper bound, we are led to choose A such that 2CP
+aA as large as possible and
+below A, which naturally drives to the choice:
+2CP
+aA = A
+2 =⇒ A =
+2
+√a
+�
+CP .
+Using this value of A in the previous bound, we observe that t0 ≤ 3√CP
+αn
+√a + CP
+α2n , so that a constant C
+exists such that:
+Jt ≤ C
+�
+J0 + cn,d
+αn
+�
+1 +
+�CP
+αn
++
+�
+CP
+�
+e
+√
+CP
+√a + CP
+3αn
+�
++ On,d
+�
+(1 + t)1/4e−
+√
+Cp
+√a (√1+t−1).
+(30)
+In Proposition 3.4 we obtained CP ≥
+κ
+(d log2β(n))
+(1+r)2 . If instead of using the constant CP , we use
+directly
+κ
+(d log2β(n))
+(1+r)2 with κ < 1, then all the previous computations remain the same only replacing
+CP by its lower bound and:
+Jt ≤ C
+
+
+J0 + cn,d
+αn
+e
+√κ
+�
+1
+√a +
+1
+3αn
+�
+(d log2β(n))(1+r)2/2 + On,d
+
+
+ (1 + t)1/4e
+−
+√κ(√1+t−1)
+√a(d log2β(n))(1+r)2/2 .
+(31)
+Using the values of On,d, cn,d and the upper bound of J0, we finally observe that if αn =
+1
+n(d log2β(n))
+1+r , then:
+t ≥ ℵ
+�
+d log2β(n)
+�(1+r)2 �
+log2(ε−1) + n2 �
+d log2β(n)
+�2(1+r)
++ d2 log2 d
+�
+=⇒ Jt ≤ ε.
+4
+Technical results on KL and Uνn
+4.1
+Growth properties under the Kurdyka-�Lojasiewicz inequality
+We remind here some important consequences of the KL inequality that implies several relationships
+between the function and the norm of its gradient. The proof of these inequalities may be found in
+Lemma 15 of [21] (a small mistake appears and we correct the statement with a factor 2 in our work).
+17
+
+Proposition 4.1. Assume that a function V satisfies Hr
+KL(c, L), then:
+2c
+1 − r
+�
+V 1−r(θ) − min(V )1−r�
+≤ ∥∇V (θ)∥2
+2 ≤ 2L [V (θ) − min(V )] ,
+∀θ ∈ Rd.
+It is furthermore possible to assess a minimal and maximal growth property of any function that
+satisfies Hr
+KL(c, L), which is necessarily lower and upper bounded by a positive power of the distance
+to its minimizer.
+Proposition 4.2. Assume that a function V satisfies Hr
+KL(c, L), then, ∀θ ∈ Rd:
+V 1+r(θ) − min(V )1+r ≥ (1 + r)c
+2
+∥θ − arg min V ∥2
+2,
+and
+V (θ) − min(V ) ≤ L
+2 ∥θ − arg min V ∥2
+2.
+A straightforward consequence of the first inequality is then
+Proposition 4.3. Assume that a function V satisfies Hr
+KL(c, L), then, ∀θ ∈ Rd:
+V (θ) ≥ 2−
+r
+1+r
+�
+min(V ) +
+�(1 + r)c
+2
+�
+1
+1+r
+∥θ − arg min V ∥
+2
+1+r
+2
+�
+.
+4.2
+Properties of Uνn
+Proof of Proposition 1.1. First, we observe that if each θ �→ ∇ log pθ(Xi) is L-Lipschitz and θ �→
+∇ log π0 is ℓ0-Lipschitz, then the triangle inequality implies that
+∥∇Uνn(θ1) − ∇Uνn(θ2)∥2 ≤ (nL + ℓ0)∥θ1 − θ2∥2.
+Second, we consider the lower-bound property on the curvature and observe that:
+λ∇2Uνn(θ) =
+inf
+e∈Rd:|e|=1 eT (∇2Uνn)(θ)e ≥ 1
+n
+n
+�
+i=1
+inf
+e∈Rd:|e|=1 eT (∇2UXi)(θ)e.
+The log concavity of the prior yields
+λ∇2Uνn(θ) ≥ 1
+n
+n
+�
+i=1
+λ∇2(−n log pθ(Xi)) =
+n
+�
+i=1
+λ∇2(− log pθ(Xi)).
+Then, the Hr
+KL(c, L) property applied to each term of the sum above and minθ∈Rd − log π0(θ) > 0
+yields
+λ∇2Uνn(θ) ≥ c
+n
+�
+i=1
+[− log pθ(Xi)]−r ≥ cnr
+n
+�
+i=1
+U −r
+Xi (θ) = cn1+r
+�
+1
+n
+n
+�
+i=1
+U −r
+Xi (θ)
+�
+.
+From the Jensen inequality, we finally deduce that:
+λ∇2Uνn(θ) ≥ cn1+r
+�
+1
+n
+n
+�
+i=1
+U −r
+Xi (θ)
+�
+≥ cn1+rU −r
+νn (θ).
+We conclude that Uνn satisfies Hr
+KL
+�
+cn1+r, nL + ℓ0
+�
+. For UXi, the proof is similar.
+Proposition 4.4. We assume Hπ0(ℓ0), Hmin and that for any x: θ �−→ − log pθ(x) satisfies Hr
+KL(c, L),
+then:
+∥ arg min Uνn∥2 ≲uc d
+1+r
+2 logβ(1+r)(n)
+and
+minθ∈Rd Uνn(θ) ≲uc nd log2β(n).
+18
+
+Proof. Proposition 1.1 shows that Uνn satisfies Hr
+KL
+�
+cn1+r, nL + ℓ0
+�
+. Therefore, we can apply Propo-
+sition 4.2 with θ = 0 and deduce that:
+∥ arg min Uνn∥2
+2 ≤
+2
+(1 + r)cn1+r
+�
+U 1+r
+νn (0) − min U 1+r
+νn
+�
+.
+To obtain an upper bound of Uνn(0) we first bound UXi(0) using Proposition 4.2, for all i, as follows:
+UXi(0) ≤ min UXi + nL + ℓ0
+2
+∥ arg min UXi∥2
+2 ≲uc d + nd log2β(n) ≲uc nd log2β(n),
+then Uνn(0) ≲uc nd log2β(n). We deduce that:
+∥ arg min Uνn∥2
+2 ≤
+2
+(1 + r)cn1+r U 1+r
+νn (0) ≲uc d1+r log2β(1+r)(n).
+The second part comes from min Uνn ≤ Uνn(0).
+5
+Smoothness and boundedness of the semi-group
+Proof of Proposition 3.4. i). The proof relies on an argument set up with a ”fixed” sample (X1, . . . , Xn).
+Our starting point is Proposition 4.2 and the consequences of the Kurdyka-�Lojasiewicz inequality.
+Since Hπ0(ℓ0) and θ �→ − log pθ(Xi) satisfies Hr
+KL(c, L), then Proposition 1.1 shows that Uνn satisfies
+Hr
+KL
+�
+cn1+r, nL + ℓ0
+�
+. Therefore, we can apply Proposition 4.2 and deduce that:
+∥θ − arg min Uνn∥2
+2 ≤
+2
+(1 + r)cn1+r
+�
+U 1+r
+νn (θ) − min U 1+r
+νn
+�
+≤
+2
+(1 + r)cn1+r U 1+r
+νn (θ).
+If Id refers to the identity map, we use the fact that for any distribution µ, we have V ar[µ] ≤ µ[∥Id−a∥2
+2]
+for any a ∈ Rd so that a straightforward consequence with a = arg min Uνn is then:
+V ar(µn) ≤
+�
+Rd ∥θ − arg min Uνn∥2
+2dµn(θ) ≤
+2
+(1 + r)cn1+r µn[U 1+r
+νn ].
+We then use the ergodic behaviour of (θt)t≥0 and observe that there exists a constant C independent
+from n and d such that:
+V ar(µn) ≤
+2
+(1 + r)cn1+r lim sup
+t≥0
+E[U 1+r
+νn (θt)]
+≤ C
+�
+d log2β(n)
+�(1+r)2
+,
+where the last inequality comes from Proposition 5.1. We now use the Bobkov bound on the Poincar´e
+constant for log-concave distribution (see Theorem 1.2 of [4]) and deduce that a universal constant K
+exists such that:
+CP (µn) ≥
+1
+4K2V ar(µn).
+Using the upper bound of the variance, we deduce that a universal κ > 0 exists such that:
+CP (µn) ≥
+κ
+�
+d log2β(n)
+�(1+r)2 .
+ii). For the second point, we consider a situation on average over the samples and the result uses the
+concentration of the posterior distribution around its mean. We know from Theorem 3 of [21] that a
+constant c > 0 exists such that:
+E(X1,...,Xn)∼Pθ0[Var(µn)] ≤ cǫ2
+n,d,
+with ǫn,d =
+�
+Ld log n
+n
+�α−1
+. The result follows using the Jensen inequality and the Bobkov bound.
+19
+
+Proof of Proposition 3.5. i). We first establish the boundedness of f0.
+From our assumptions, we
+apply Proposition 1.1 and obtain that Uνn satisfies Hr
+KL
+�
+cn1+r, nL + ℓ0
+�
+. If θ⋆
+n = arg min Uνn, we
+then deduce from Proposition 4.2 that:
+f0(θ) = n0(θ)
+µn(θ) = Zne−
+∥θ∥2
+2
+2σ2 +Uνn(θ)
+(2π)d/2σd
+≤ Zne−
+∥θ∥2
+2
+2σ2 +Uνn(θ⋆
+n)+ (nL+ℓ0)
+2
+∥θ−θ⋆
+n∥2
+2
+(2π)d/2σd
+.
+(32)
+We compute an upper bound of Zn and use the lower bound of Uνn induced by Proposition 4.3:
+Zn =
+�
+Rd e−Uνn(θ)dθ
+≤
+�
+Rd e
+−2
+−
+r
+1+r
+�
+Uνn(θ⋆
+n)+n(
+(1+r)c
+2
+)
+1
+1+r ∥θ−θ⋆
+n∥
+2
+1+r
+2
+�
+dθ
+≤ e−2
+−
+r
+1+r Uνn(θ⋆
+n)
+�
+Rd e−nar∥θ∥
+2
+1+r
+2
+dθ,
+with ar = ((1+r)c)
+1
+1+r
+2
+. Using the well known equality:
+�
+Rd e−a|θ|ℓdθ =
+dπd/2Γ(d/ℓ)
+ℓad/ℓΓ(d/2 + 1),
+∀a > 0,
+∀ℓ > 0.
+we then deduce with a = nar and ℓ =
+2
+1+r that:
+Zn ≤ e−2
+−
+r
+1+r Uνn(θ⋆
+n)
+�
+Rd e−nar∥θ∥
+2
+1+r
+2
+dθ ≤ d(1 + r)
+2
+πd/2
+(nar)
+d(1+r)
+2
+Γ
+�
+d(1+r)
+2
+�
+Γ
+� d
+2 + 1
+� .
+From standard relationships on the Gamma function:
+Zn ≤ 2
+�21+rπ
+cn1+r
+� d
+2
+d
+dr
+2 .
+(33)
+We gather Equations (32) and (33) and obtain that:
+f0(θ) ≤ 2eUνn(θ⋆
+n)
+�
+2
+cσ2n1+r
+� d
+2
+d
+dr
+2 e−
+∥θ∥2
+2
+2σ2 + (nL+ℓ0)
+2
+∥θ−θ⋆
+n∥2
+2.
+For all σ2 <
+1
+nL+ℓ0 , a straightforward optimization on θ yields :
+∥f0∥∞ ≤ 2eUνn(θ⋆
+n)
+�
+2
+cσ2n1+r
+� d
+2
+d
+dr
+2 exp
+�
+(nL + ℓ0)
+2(1 − σ2(nL + ℓ0))∥θ⋆
+n∥2
+2
+�
+.
+Then, the choice
+c1
+nL+ℓ0 ≤ σ2 ≤
+c2
+nL+ℓ0 , where 0 < c1 ≤ c2 < 1 in Hn0(L, ℓ0) and the bounds of ∥θ⋆
+n∥2
+2
+and Uνn(θ⋆
+n) in Proposition 4.4 lead to :
+∥f0∥∞ ≤ 2
+�C1d
+n
+� dr
+2
+exp
+�
+C2nd1+r log2β(1+r)(n)
+�
+,
+where C1 and C2 are universal constants.
+ii). This result is an almost standard consequence of the maximum principle for a Markov semi-group
+property with a Brownian diffusion. For any bounded measurable h > 0, we observe that Pth > 0
+using the Markov property, and we are led to define gt as the following function gt := √Pth. We then
+introduce θ(t) and θ(t) as:
+θ(t) = arg max gt(θ)
+and
+θ(t) = arg min gt(θ).
+20
+
+The chain rule yields:
+d
+dtOsc(gt)
+=
+d
+dt
+�
+gt(θ(t)) − gt(θ(t))
+�
+=
+dgt
+dt (θ(t)) +
+�
+∇gt(θ(t)), dθ(t)
+dt
+�
+− dgt
+dt (θ(t)) −
+�
+∇gt(θ(t)), dθ(t)
+dt
+�
+.
+(34)
+We compute:
+dgt
+dt (θ)
+=
+1
+2√Pth
+dPth
+dt (θ)
+=
+1
+2√PthGtPth(θ)
+=
+1
+2
+�
+Pth(θ)
+�
+−
+n
+�
+i=1
+⟨∇θPth(θ), ∇θUXi(θ)⟩mt(Xi|θ) + ∆θPth(θ)
+�
+.
+(35)
+Now, we use that θ(t) = arg max gt = arg max Pth, (a similar argument holds for θ(t)):
+∇θgt(θ(t)) = 0,
+∇θPth(θ(t)) = 0
+and
+∆θPth(θ(t)) ≤ 0.
+then:
+d
+dtOsc(gt)
+=
+dgt
+dt (θ(t)) − dgt
+dt (θ(t))
+=
+∆θPth
+2√Pth(θ(t)) − ∆θPth
+2√Pth(θ(t))
+(36)
+≤
+0.
+We have therefore shown that Osc(√Pth) is decreasing in t ≥ 0, which ends the proof.
+Proof of Lemma 2.1. We proceed as in Proposition 3 of [33] to justify the use of the Lebesgue domi-
+nated convergence theorem for the derivation of the integral involved in our statement. We can then
+deduce that:
+∂t
+��
+Rd ft(θ)dnt(θ)
+�
+=
+�
+Rd ∂t{ft(θ)}dnt(θ) +
+�
+Rd ft(θ)∂t{nt(θ)}dθ.
+We leave the first term unchanged and now focus on the second term:
+�
+Rd ft(θ)∂t{nt(θ)}dθ
+=
+�
+Rd ft(θ)∂t
+� n
+�
+i=1
+mt(θ, Xi)
+�
+dθ
+=
+�
+Rd
+n
+�
+i=1
+ft(θ)∂t{mt(θ, Xi)}dθ
+=
+�
+Rd
+n
+�
+i=1
+Lft(θ) mt(θ, Xi)dθ,
+where we used the definition of nt in the first step and Kolmogorov backward equation (10) in the last
+one. Since the function ft(θ) does not depend on x, we observe that L2ft(θ) = 0 and we only need to
+compute the remaining term L1ft(θ):
+�
+Rd ft(θ)∂t{nt(θ)}dθ
+=
+�
+Rd
+n
+�
+i=1
+L1ft(θ) mt(θ, Xi)dθ
+(37)
+=
+�
+Rd
+n
+�
+i=1
+[−⟨∇θft(θ), ∇θUXi(θ)⟩ + ∆θft(θ)] mt(θ, Xi)dθ
+=
+−
+�
+Rd
+n
+�
+i=1
+⟨∇θft(θ), ∇θUXi(θ)⟩mt(Xi|θ)dnt(θ) +
+�
+Rd ∆θft(θ)dnt(θ)
+=
+�
+Rd Gtft(θ)dnt(θ),
+(38)
+21
+
+where we used the fact that mt(θ, Xi) = mt(Xi|θ)nt(θ).
+5.1
+Moments upper bounds
+Proposition 5.1. Assume Hn0(L, ℓ0), Hπ0(ℓ0), Hmin and that for each Xi, θ �→ − log pθ(Xi) satisfies
+Hr
+KL(c, L). Then:
+i) Three positive constants C1, C2 and C3, independent from n and d, exist such that for any t > 0:
+E
+�
+e
+(1+r)nc
+1
+1+r
+16
+(∥θt∥2
+2+1)
+1
+1+r
+�
+≤ C1
+�
+d log2β(n)
+�
+r
+1+r eC2nd log2β(n) + Cd
+3e
+(1+r)nc
+1
+1+r
+16
+.
+ii) For any t > 0 and for any α ≥ 1:
+E[U α
+νn(θt)] ≲uc nα �
+d log2β(n)
+�α(1+r)
+.
+Proof of i). We consider the function f(θ) = exp
+� a
+2(∥θ∥2
+2 + 1)ρ�
+where 0 < ρ < 1, which is twice
+differentiable. The gradient of f is computed as:
+∇f(θ) = aρ(∥θ∥2
+2 + 1)ρ−1f(θ)θ.
+The Laplace operator is given as:
+∆f(θ) = aρ(∥θ∥2
+2 + 1)ρ−2f(θ)
+�
+aρ(∥θ∥2
+2 + 1)ρ∥θ∥2
+2 + (d + 2ρ − 2)∥θ∥2
+2 + d
+�
+.
+We then deduce that for any θ ∈ Rd:
+Gtf(θ)
+=
+−
+n
+�
+i=1
+⟨∇UXi, ∇f(θ)⟩mt(Xi|θ) + ∆f(θ)
+=
+aρ(∥θ∥2
+2 + 1)ρ−2f(θ)
+�
+− (∥θ∥2
+2 + 1)
+n
+�
+i=1
+⟨θ, ∇θUXi(θ)⟩mt(Xi|θ)
++aρ(∥θ∥2
+2 + 1)ρ∥θ∥2
+2 + (d + 2ρ − 2) ∥θ∥2
+2 + d
+�
+≤
+aρ(∥θ∥2
+2 + 1)ρ−2f(θ)
+�
+− (∥θ∥2
+2 + 1)
+n
+�
+i=1
+(UXi(θ) − UXi(0)) mt(Xi|θ)
++aρ(∥θ∥2
+2 + 1)ρ+1 + d
+�
+∥θ∥2
+2 + 1
+� �
+≤
+aρ(∥θ∥2
+2 + 1)ρ−1f(θ)
+�
+−
+n
+�
+i=1
+(UXi(θ) − UXi(0)) mt(Xi|θ) + a��(∥θ∥2
+2 + 1)ρ + d
+�
+,
+where we used the convexity of Ux for any position x.
+Let us establish the bounds of UXi(θ) and UXi(0).
+We denote by θi = arg min UXi and from
+Hypothesis Hmin, there exist two positive constants K1 and K2 independent on n and d such that:
+maxi ∥θi∥2
+2 ≤ K1d log2β(n)
+and
+maxi UXi(θi) ≤ K2d.
+We apply Proposition 4.2 to each non-negative function UXi that satisfies Hr
+KL
+�
+cn1+r, nL + ℓ0
+�
+, then
+we obtain that:
+UXi(θ) ≥ n
+�(1 + r)c
+2
+�
+1
+1+r
+∥θ − θi∥
+2
+1+r
+2
+.
+Since
+2
+1+r > 1, the Jensen inequality yields (u + v)
+2
+1+r ≤ 2
+1−r
+1+r
+�
+u
+2
+1+r + v
+2
+1+r
+�
+, for all (u, v) ∈ R2
++ and
+we deduce that:
+∥θ − θi∥
+2
+1+r
+2
+≥ 2
+r−1
+1+r ∥θ∥
+2
+1+r
+2
+− ∥θi∥
+2
+1+r
+2
+≥ 2
+r−1
+1+r ∥θ∥
+2
+1+r
+2
+−
+�
+K1d log2β(n)
+�
+1
+1+r .
+22
+
+Then we use this inequality to obtain a lower bound of UXi:
+UXi(θ) ≥ 2n
+�(1 + r)c
+8
+�
+1
+1+r
+∥θ∥
+2
+1+r
+2
+− n
+�(1 + r)c
+2
+�
+1
+1+r
+(K1d log2β(n))
+1
+1+r .
+Moreover an upper bound of max UXi(0) comes from Proposition 1.1 and 4.2 as follows:
+UXi(0) ≤ UXi(θi) + nL + ℓ0
+2
+∥θi∥2
+2 ≤ K2d + K1(nL + ℓ0)d log2β(n)
+2
+.
+Using the previous bounds and the fact that �n
+i=1 mt(Xi|θ) = 1, it yields:
+n
+�
+i=1
+(UXi(θ) − UXi(0)) mt(Xi|θ)
+≥
+2n
+�(1 + r)c
+8
+�
+1
+1+r
+∥θ∥
+2
+1+r
+2
+− n
+�(1 + r)c
+2
+�
+1
+1+r
+(K1d log2β(n))
+1
+1+r − K2d − K1(nL + ℓ0)d log2β(n)
+2
+≥
+nc
+1
+1+r
+4
+∥θ∥
+2
+1+r
+2
+− nc
+1
+1+r (K1d log2β(n))
+1
+1+r − K2d − K1(nL + ℓ0)d log2β(n)
+2
+,
+where we used some uniform upper bounds when r ∈ [0, 1). We then choose ρ =
+1
+1+r and we deduce
+that:
+Gtf(θ)
+≤
+a
+1 + r (∥θ∥2
+2 + 1)−
+r
+1+r f(θ)
+�
+−nc
+1
+1+r
+4
+∥θ∥
+2
+1+r
+2
++ nc
+1
+1+r (K1d log2β(n))
+1
+1+r + K2d
++K1(nL + ℓ0)d log2β(n)
+2
++
+a
+(1 + r)(∥θ∥2
+2 + 1)
+1
+1+r + d
+�
+≤
+a
+1 + r (∥θ∥2
+2 + 1)−
+r
+1+r f(θ)
+�
+−
+�
+nc
+1
+1+r
+4
+−
+a
+(1 + r)
+�
+∥θ∥
+2
+1+r
+2
++ nc
+1
+1+r (K1d log2β(n))
+1
+1+r
++(K2 + 1)d + K1(nL + ℓ0)d log2β(n)
+2
++
+a
+(1 + r)
+�
+,
+where we used (∥θ∥2
+2 + 1)
+1
+1+r ≤ ∥θ∥
+2
+1+r
+2
++ 1 in the second line.
+We now fix a = n(1+r)c
+1
+1+r
+8
+and deduce that:
+Gtf(θ)
+f(θ)
+≤
+n2c
+2
+1+r
+64
+(∥θ∥2
+2 + 1)−
+r
+1+r
+�
+−∥θ∥
+2
+1+r
+2
++ 8(K1d log2β(n))
+1
+1+r +
++8(K2 + 1)d + 4K1(nL + ℓ0)d log2β(n)
+nc
+1
+1+r
++ 1
+�
+.
+(39)
+We then study two complementary situations and below, we denote by Kn,d the radius of the key
+compact set involved by the previous Lyapunov contraction:
+K
+2
+1+r
+n,d = Cd log2β(n).
+• When ∥θ∥2 is large enough (∥θ∥2 ≥ Kn,d), we observe that a large enough C > 0 independent from
+n and d exists such that:
+∥θ∥
+2
+1+r
+2
+≥ Cd log2β(n) =⇒ Gtf(θ)
+f(θ)
+≤ −
+n2 �
+d log2β(n)
+�
+1
+1+r c
+2
+1+r
+128
+= −an,d.
+(40)
+23
+
+• When ∥θ∥2 is upper bounded (∥θ∥2 ≤ Kn,d), we use the upper bound stated in Equation (39) and
+obtain that a universal C1 (whose value may change from line to line) exists such that :
+∥θ∥
+2
+1+r
+2
+≤ Cd log2β(n) =⇒
+Gtf(θ) ≤ C1n2f(θ)
+�
+8(K1d log2β(n))
+1
+1+r + 8(K2 + 1)d + 4K1(nL + ℓ0)d log2β(n)
+nc
+1
+1+r
++ 1
+�
+≤ C1n2d log2β(n) exp
+�
+(C + 1)c
+1
+1+r nd log2β(n)
+8
+�
+≤ bn,deδn,d.
+(41)
+We then use Equations (40) and (41) as follows. We define the function ψn,d as ψn,d(t) = E[f(θt)] and
+use Lemma 2.1:
+ψ′
+n,d(t)
+=
+E[Gtf(θt)]
+=
+E
+�
+Gtf(θt)
+�
+1∥θt∥2≥Kn,d +
+1∥θt∥2≤Kn,d
+��
+≤
+E
+�
+−an,df(θt)1∥θt∥2≥Kn,d + bn,deδn,d
+1∥θt∥2≤Kn,d
+�
+≤
+−an,dψn,d(t) + an,d
+sup
+∥θ∥2≤Kn,d
+f(θ) + bn,deδn,d
+≤
+−an,dψn,d(t) + (an,d + bn,d)eδn,d.
+We apply the Gronwall Lemma and obtain that:
+∀t > 0
+ψn,d(t) ≤
+�
+1 + bn,d
+an,d
+�
+eδn,d + ψn,d(0)e−an,dt.
+(42)
+Using that n0 is a Gaussian distribution, which was fixed in Hn0(L, ℓ0) hypothesis, we find an
+upper bound for ψn,d(0) = E[f(θ0)] =
+�
+Rd f(θ)dn0(θ) as follows :
+ψn,d(0)
+=
+�
+2πσ2�− d
+2
+�
+Rd e
+a
+2(∥θ∥2
+2+1)
+1
+1+r −
+∥θ∥2
+2
+2σ2 dθ
+≤
+�
+2πσ2�− d
+2 e
+a
+2
+�
+Rd e−
+∥θ∥2
+2
+2 ( 1
+σ2 −a)dθ,
+if σ2 ≤
+1
+a =
+8
+n(1+r)c
+1
+1+r then the integral above is finite. Since c2 < 1 ≤
+8L
+(1+r)c
+1
+1+r , it guarantees
+σ2 < 1
+a, then:
+ψn,d(0)
+≤
+�
+1 − aσ2�− d
+2 e
+a
+2
+≤
+Cd
+3e
+(1+r)nc
+1
+1+r
+16
+,
+where C3 is a constant independent from n and d.
+Finally, using the value of an,d and bn,d in (42), we deduce that:
+E
+�
+e
+(1+r)nc
+1
+1+r
+16
+(∥θt∥2
+2+1)
+1
+1+r
+�
+≤ C1
+�
+d log2β(n)
+�
+r
+1+r eC2nd log2β(n) + Cd
+3e
+(1+r)nc
+1
+1+r
+16
+,
+∀t > 0.
+where C2 is another universal constant, which concludes the proof.
+Proof of ii). We consider α > 1 and below, C > 0 refers to a “constant” independent from n and d,
+whose value may change from line to line. Our starting point is the upper bound of the exponential
+moments obtained in i). Proposition 1.1 shows that Uνn satisfies Hr
+KL
+�
+cn1+r, nL + ℓ0
+�
+, then thanks
+to Proposition 4.2:
+E[U α
+νn(θt)] ≤ E
+��
+min Uνn + Cn∥θt − θ∗
+n∥2
+2
+�α�
+≤ E
+��
+min Uνn + Cn∥θ∗
+n∥2
+2 + Cn∥θt∥2
+2
+�α�
+,
+24
+
+where θ∗
+n = arg min Uνn.
+By using Proposition 4.4 and the inequality derived from the Jensen inequality (a+b)β ≤ cβ(aβ+bβ)
+for (a, b) ∈ R2
++ and β ≥ 1, we obtain that:
+(min Uνn+ Cn∥θ∗
+n∥2
+2 + Cn∥θt∥2
+2
+�α
+≤ C
+�
+nd log2β(n) + nd1+r log2β(1+r)(n) + n∥θt∥2
+2
+�α
+≤ Cnα
+��
+d log2β(n)
+�α(1+r)
++ ∥θt∥2α
+2
+�
+≤ Cnα
+��
+d log2β(n)
+�α(1+r)
++ k−α(1+r) logα(1+r)
+�
+ek∥θt∥
+2
+1+r
+2
+��
+≤ Cnα
+��
+d log2β(n)
+�α(1+r)
++ k−α(1+r) logα(1+r)
+�
+eα(1+r)−1+k∥θt∥
+2
+1+r
+2
+��
+.
+The Jensen inequality and the concavity of x �→ logp(x) on [ep−1, +∞[ when p ≥ 1 yield
+E[U α
+νn(θt)]
+≤ Cnα
+��
+d log2β(n)
+�α(1+r)
++ k−α(1+r)E
+�
+logα(1+r)
+�
+eα(1+r)−1+k∥θt∥
+2
+1+r
+2
+���
+≤ Cnα
+��
+d log2β(n)
+�α(1+r)
++ k−α(1+r) logα(1+r)
+�
+E
+�
+eα(1+r)−1+k∥θt∥
+2
+1+r
+2
+���
+≤ Cnα
+��
+d log2β(n)
+�α(1+r)
++ k−α(1+r)
+�
+α(1 + r) − 1 + log E
+�
+ek∥θt∥
+2
+1+r
+2
+��α(1+r)�
+≤ Cnα
+��
+d log2β(n)
+�α(1+r)
++ k−α(1+r)
+�
+α(1 + r) − 1 + log E
+�
+ek(∥θt∥2
+2+1)
+1
+1+r
+��α(1+r)�
+,
+where we used in the last inequality that ∥θ∥2
+2 ≤ ∥θ∥2
+2 + 1.
+We then apply i) in Proposition 5.1, we choose k = (1+r)nc
+1
+1+r
+16
+and obtain that:
+E[U α
+νn(θt)]
+≤ Cnα
+
+
+�
+d log2β(n)
+�α(1+r)
++
+1
+nα(1+r)
+�
+1 + log E
+�
+e
+(1+r)nc
+1
+1+r
+16
+(∥θt∥2
+2+1)
+1
+1+r
+��α(1+r)
+
+≤ C
+�
+nα �
+d log2β(n)
+�α(1+r)
++ 1
+nαr
+�
+1 + log
+�
+C1
+�
+d log2β(n)
+�
+r
+1+r eC2nd log2β(n) + Cd
+3e
+(1+r)nc
+1
+1+r
+16
+��α(1+r)
+
+≤ Cnα �
+d log2β(n)
+�α(1+r)
+,
+where we used in the previous lines simple algebra and log(a+b) ≤ log(2)+log(a)+log(b) when a ≥ 1
+and b ≥ 1. This concludes the proof.
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+27
+

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+HYPERBOLIC AND SATELLITE LORENZ LINKS
+OBTAINED BY TWISTING
+THIAGO DE PAIVA AND JESSICA S. PURCELL
+Abstract. A Lorenz link is equivalent to a T-link, which is a positive
+braid built by concatenating torus braids of increasing size. When each
+torus braid except the largest is obtained by full twists, then the T-link
+can be described as the Dehn filling of a parent link. In this paper, we
+completely classify when such parent links are hyperbolic. This gives
+a classification of the geometry of T-links obtained by full twists when
+the amount of twisting is large, although the bound on the number
+of required twists is not effective. We also present effective results on
+hyperbolicity for two families of T-links obtained by twisting. Finally,
+we identify families of satellite T-links obtained by half-twists.
+1. Introduction
+Lorenz links are the closed periodic orbits of a system of equations in-
+vestigated by Lorenz in the 1960s [18]. They exhibit interesting dynamics
+that has led to significant further investigation over the years, in the fields of
+dynamics, geometry, and topology; see for example [9]. These links can be
+described as links on an embedded branched surface in R3, called the Lorenz
+template, due to work of Guckenheimer and Williams [12], and Tucker [22].
+Birman and Williams were the first to investigate Lorenz links through the
+lens of knot theory, in the 1980s [2], and the first to show such links are
+closed positive braids. Birman and Kofman [1] showed that Lorenz links are
+equivalent to T-links, which are positive braids with a particular form; see
+Section 2 below. Thus techniques from braid theory can be brought to bear
+upon Lorenz links via T-links.
+We are interested in the complement of these links, and in particular their
+geometrisation. Thurston showed in the 1980s that all knots in the 3-sphere
+are either torus knots, satellite, or hyperbolic [20], and we refer to this as
+the knot’s geometric type. The geometric type of Lorenz links has been
+considered since work of Birman and Williams in the 1980s [2]. They showed
+that all torus knots are Lorenz knots, and satellites obtained as certain cables
+of Lorenz knots are Lorenz knots. Hyperbolic geometry has been considered
+by Gomes, Franco, and Silva [10, 11], who proved hyperbolicity of Lorenz
+links satisfying certain conditions based on the Lorenz template. Satellite
+links have received additional attention, by El Rifai [7], de Paiva [4], and
+de Paiva and Purcell [6].
+1
+arXiv:2301.01934v1  [math.GT]  5 Jan 2023
+
+2
+THIAGO DE PAIVA AND JESSICA S. PURCELL
+In spite of this work, there remains no systematic way of determining
+whether a Lorenz link is hyperbolic, toroidal, or satellite using its description
+either on the Lorenz template, or as a closed braid in the form of a T-link.
+These descriptions uniquely determine a link, and hence uniquely determine
+its geometric type, so it is natural to ask for a simple description of geometric
+type based on the description. We focus on T-links in this paper.
+The paper [6] begins a classification of the geometry of T-links, by finding
+examples that are satellite and also by identifying certain “parent links”,
+which give classes of T-links under Dehn filling. While the work in that
+paper finds examples of satellite and hyperbolic links, it is incomplete for
+two reasons:
+(1) First, the hyperbolic geometry of the parent links is used to determine
+geometry of T-links for many examples. But the classification of the
+hyperbolic geometry of the parent links is incomplete.
+(2) Second, because the results are obtained by Dehn filling, they apply
+only to links that admit full twists as T-link parameters, which are
+not required for general T-links.
+In this paper, we extend the classification of geometry of T-links as
+follows. First, we complete the classification of item (1) above: Theorem 3.8
+completely classifies when parent links of fully twisted T-links are hyperbolic.
+This can be seen as an extension of work of Lee [16, Proposition 5.7], who
+proved a similar result for twisted torus knots. Positive twisted torus knots
+are T-links with only one additional torus braid besides the largest. Lee’s
+result essentially proves Theorem 3.8 in the case of only one additional link
+component in the parent. Our result applies to any number of additional
+link components in the parent.
+Theorem 3.8 leads to new infinite families of hyperbolic T-links, determined
+only by parameters in a braid describing the link.
+Theorem 1.1. Fix relatively prime integers q < p, and let a1, . . . , an be
+integers less than p and increasing in value.
+There exists B ≫ 0 with
+the following property. Consider the T-link obtained from the (p, q)-torus
+knot by full twisting at least B times in regions with a1, a2, . . . , an strands,
+respectively. This Lorenz link is hyperbolic if and only if either all ai < q, or
+there is ai > q that is not a multiple of q.
+The T-links of Theorem 1.1 must be obtained by full twisting, and we
+currently do not have a concrete, universal bound on the number of full
+twists that are required in general; this is the constant B in the above result.
+In Section 4 we improve this: We present two theorems that guarantee
+hyperbolicity of T-links with full twists, given only their parameters, where
+the bounds on numbers of full twists required are explicit and relatively
+simple. The results are Theorem 4.3 and Theorem 4.5.
+It seems much more difficult to address item (2), especially in the hyperbolic
+case. There are some partial results known, for example by de Paiva for
+torus knots [5]. In this paper, we give more results in the satellite case. We
+
+LORENZ LINKS OBTAINED BY TWISTING
+3
+extend the results on satellite knots, requiring full twists in [6], to families of
+T-links with both full twists and half twists, which gives many more families
+in a very natural way.
+Theorem 1.2. For q < p integers, let K be a T-link obtained from the
+(p, q)-torus link by half-twisting in circles encircling less than q strands, or
+encircling multiples of q strands. Then S3 − K is satellite.
+The precise statement is Theorem 5.4.
+1.1. Acknowledgements. This work was partially supported by the Aus-
+tralian Research Council, grant DP210103136.
+2. Results on braids
+This section reviews results on braids that will be used throughout. As
+usual, let σi be the standard generator of the braid group, giving a positive
+crossing between the i-th and (i + 1)-th strands.
+For 1 < p, q, define the (p, q)-torus braid as:
+(σ1 . . . σp−1)q
+Note that within the braid group on p strands, its closure is the torus link
+T(p, q). When p, q are coprime, this is a torus knot, but we will not always
+restrict to coprime p and q unless specifically stated.
+We will also consider such braids within larger braid groups. When r < p,
+the (r, s) braid within the braid group on p strands is still defined to be
+(σ1 . . . σr−1)s, but now note this has p − r strands with no crossings lying to
+the right of the braid, viewing the braid arranged from top to bottom.
+Let r1, . . . , rk and si, . . . , sk be integers such that 2 ≤ r1 < · · · < rk, and
+si > 0 for all i. The T-link T((r1, s1), . . . , (rk, sk)) is defined to be the closure
+of the braid
+(σ1σ2 . . . σr1−1)s1(σ1σ2 . . . σr2−1)s2 . . . (σ1σ2 . . . σrk−1)sk.
+Thus T((r1, s1), . . . , (rk, sk)) is obtained by concatenating the braids (ri, si)
+within the braid group on rk strands, and then taking the closure.
+Taking closures of torus braids and related braids allows additional sym-
+metries and restrictions on the braid. For example, we will use the following
+standard result on torus knots and links.
+Lemma 2.1. Let 1 < p, q be integers. Then the torus link T(p, q) is equiva-
+lent to the torus link T(q, p) via a homeomorphism of S3 fixing the Heegaard
+torus containing T(p, q) and switching the two solid tori bounded by F.
+The proof of Lemma 2.1 is well known, and appears in many knot theory
+texts. We visualise the proof in Figure 1.
+The next result generalises [6, Lemma 2.7]. There the result only holds
+when each si is a multiple of ri. Here we extend more generally.
+
+4
+THIAGO DE PAIVA AND JESSICA S. PURCELL
+p
+q
+p
+q
+Figure 1. The equivalence of T(p, q) and T(q, p) is given by
+rotating 180◦ in the diagonal axis shown for the Heegaard
+torus for S3. This exchanges the solid tori in the standard
+genus-1 Heegaard splitting for S3.
+Proposition 2.2. Let 0 < r1 < · · · < ri−1 < q < ri+1 < · · · < rn < p be
+integers. Then, for k > 0, the T-link
+K = T((r1, s1), . . . , (ri−1, si−1), (q, qk), (ri+1, si+1), . . . , (rn, sn), (p, q))
+is equivalent to the T-link
+K′ = T((r1, s1), . . . , (ri−1, si−1), (ri+1, si+1), . . . , (rn, sn), (p + qk, q)).
+Note that Proposition 2.2 allows us to assume there are no full twists on
+q strands in a T-link of the form T(· · · , (p, q)).
+Proof. The braid (q, qk) is obtained by performing k full twists on q strands.
+We know that these full twists commute in the braid group. Thus in the
+braid representing K, we may isotope (q, qk) to the top of the braid, leaving
+the rest of the braid unchanged.
+Now perform the isotopy of K of Lemma 2.1, switching p and q in the
+(p, q)-torus link. The rotation in the diagonal shown in Figure 1 takes the
+(vertical) braids (r1, s1) ∗ · · · ∗ (rn, sn) to inverted braids, forming a tangle in
+the horizontal direction on a quadrilateral representing the projection torus.
+(The form of this tangle is not important for the argument here, but more
+details can be found in [6, Lemma 2.3].) The result is a link of the form
+T(q, p) with a tangle along the horizontal p-strands. The first such tangle
+is the braid (q, qk), which is unchanged by this isotopy because it is a full
+twist (see, for example, Birman and Kofman [1, Corollary 3]). Then the link
+diagram is formed by the braid (q, p) followed by (q, qk). These two braids
+can be combined to form the braid (q, p + qk). Now apply the inverse of the
+isotopy of Figure 1. This changes the link from T(q, p + qk) with tangles
+along the p horizontal strands to a link of the form T(p + qk, p) with these
+tangles returned to their form as braids (r1, s1) ∗ · · · ∗ (rn, sn). The result is
+the link K′.
+□
+Proposition 2.3. Let p, q, and r be positive integers with 0 < q ≤ r < p.
+Consider the (p, q) torus link, which is the closure of the braid on p strands
+given by (σ1 . . . σp−1)q. There is an ambient isotopy of S3 taking this to the
+
+LORENZ LINKS OBTAINED BY TWISTING
+5
+Figure 2. Illustration of Proposition 2.3 in the case that
+q = 2, r = 4, p = 7, for an arbitrary tangle shown as a
+gray box. The left-most picture shows the original link. The
+(r + 1)-st strand, shown in blue, can be pulled tight beneath
+the diagram, resulting in the middle picture. The right-most
+picture shows the result after isotoping strands (r + 1) to p.
+closure of the braid on r strands given by
+(σr−1 . . . σr−q+1)p−r(σ1 . . . σr−1)q.
+Moreover, an ambient isotopy realising the equivalence fixes the portion of
+the braid (σ1 . . . σp−1)q corresponding to the r left-most strands at the top the
+braid. Thus, we may replace a neighbourhood of these strands above the braid
+(σ1 . . . σp−1)q with any tangle τ on r strands, and we find that the resulting
+link is ambient isotopic to the closure of the link obtained by concatenating the
+braid on r strands (σr−1 . . . σr−q+1)p−r, with τ, and then with (σ1 . . . σr−1)q.
+See Figure 2.
+Proof. Because r ≥ q, the (r + 1)-st strand at the top of the braid only runs
+under the q overcrossing strands in the braid corresponding to the (p, q)
+torus link. It then runs around the braid closure back to the top, returning
+to the r − q + 1 position. Together with a horizontal line from the r − q + 1
+position to the r + 1 position, this strand bounds a disc in S3, lying under
+the plane of projection. Use this disc to push the strand in S3 to become a
+horizontal strand lying below the plane of projection, running from the r + 1
+position, then behind q strands, to the r + 1 − q position. Adjust slightly,
+pulling the right side up, so that the result is a closed braid; see Figure 2,
+middle. Note that the resulting braid consists of only p − 1 strands. This
+isotopy generalises the isotopy given by Lee in [17, Figure 6], and by de Paiva
+in [5, Figure 1].
+
+6
+THIAGO DE PAIVA AND JESSICA S. PURCELL
+This move can be repeated for all the p − r strands to the right of the
+(r+1)-st strand. When finished, we obtain a link on r strands as claimed.
+□
+2.1. Braid index. Recall that the braid index of a knot K, which we will
+denote β(K), is the minimal number of strands required to form a braid
+with closure isotopic to K. We will repeatedly use the following result of
+Franks and Williams [8] on braid index of the closure of a positive braid.
+Theorem 2.4 (Corollary 2.4 of [8]). Let B be a positive braid on p strands
+that contains a full twist
+∆2 = (σ1 . . . σp−1)p.
+Then B has braid index p.
+□
+Lemma 2.5. Let p, q, d and r be positive integers such that q ≤ r < p and
+d + q ≥ r. Let Br be a positive braid on r strands, and let Bp denote the
+braid on p strands obtained by adding p − r trivial strands to the right of the
+braid Br. Then the closure of the braid on p strands
+Bp(σ1 . . . σr−1)d(σ1 . . . σp−1)q
+has braid index equal to r.
+Proof. By Proposition 2.3, the closure of the given braid on p strands is
+equivalent to the closure of the braid on r strands
+B′ = (σr−1 . . . σr−q+1)p−rBr(σ1 . . . σr−1)d(σ1 . . . σr−1)q.
+Because this is a positive braid, and because d + q ≥ r, the braid B′ has at
+least one positive full twist on r strands. Thus Theorem 2.4 implies that the
+closure of B (and B′) has braid index equal to r.
+□
+Corollary 2.6. Suppose 0 < r1 < · · · < rn < p are integers, s1, . . . , sn and
+q are positive integers, and suppose q ≤ rn ≤ sn + q. Then the T-link
+K = T((r1, s1), . . . , (rn, sn), (p, q))
+has braid index equal to rn.
+Proof. Let Brn be the braid on rn strands obtained as the concatenation of
+torus braids (r1, s1) . . . (rn−1, sn−1), where we view each (ri, si) as a braid
+on rn strands by adding rn − ri trivial strands to the right of the braid
+(ri, si) = (σ1 . . . σri−1)si. Then the given T-link is the closure of the braid
+Brn(σ1 . . . σrn−1)sn(σ1 . . . σp−1)q.
+Since q ≤ rn ≤ sn + q, the result follows from Lemma 2.5.
+□
+The next definition is from Williams [23].
+Definition 2.7. A generalized q-cabling of a link L is a link L′ contained in
+the interior of a tubular neighbourhood L × D2 of L such that
+(1) each fiber D2 intersects L′ transversely in q points; and
+(2) all strands of L′ are oriented in the same direction as L itself.
+
+LORENZ LINKS OBTAINED BY TWISTING
+7
+Williams showed the following result on generalised q-cablings for knotted
+L in [23].
+Theorem 2.8 (Theorem 1 of Williams [23]). The braid index is multiplicative
+under generalized cabling. That is, if L is a link with each component a
+non-trivial knot and L′ is a generalized q-cabling of L then β(L′) = qβ(L),
+where β(∗) is the braid index of ∗.
+□
+This result was extended to unknotted L in the case of positive braids by
+de Paiva in [3]. The following result is from that paper.
+Lemma 2.9 (Lemma 2.3 of [3]). Let L′ be a generalized q-cabling of the
+unknot L, with L given by a positive braid on n strands, where n > 1. Also,
+assume the knot inside L is given by a positive braid. Then L′ has braid
+index equal to q.
+□
+3. Parents of T-links
+In this section, we build the “parent links” mentioned in the introduction.
+Dehn filling on such links produces T-links with full twists. By classifying
+when such links are hyperbolic, and applying Thurston’s hyperbolic Dehn
+filling theorem, we show that, in an appropriate sense, most T-links with
+only full twists are hyperbolic. This is an extension of work by de Paiva and
+Purcell [6]. There, the same links were constructed, and some conditions
+were given to guarantee hyperbolicity. Here, we strengthen the result by
+completely characterising when such links are hyperbolic.
+Definition 3.1. Let p, q be relatively prime integers such that 1 < q < p.
+Consider the (p, q)-torus braid on p strands, and its closure, the torus link
+T(p, q). Let F denote the Heegaard torus on which T(p, q) lies. Let a be
+an integer with 0 < a < p. Denote by Ja an unknot lying horizontally with
+respect to the (p, q)-torus braid, positioned just above the crossings of the
+braid, bounding a disc such that the interior of that disc meets F transversely
+in a single arc intersecting the a leftmost strands of the braid.
+More generally, given a1, . . . , an satisfying 1 < a1 < · · · < an < p, take
+disjoint unknots Ja1, . . . , Jan as above, positioned so that the i-th is pushed
+vertically above the (i + 1)-th with respect to the braid, so that all are
+disjoint. Figure 3 shows an example.
+Proposition 3.2. Let p, q be relatively prime integers with 1 < q < p. Let
+an, . . . , a1 be integers such that 1 < a1 < · · · < an < p, with n > 1. Also,
+assume that there is ai > q which is not a multiple of q. Then the link
+K = T(p, q) ∪ Jan ∪ · · · ∪ Ja1 is atoroidal.
+In [6], it is shown that K is hyperbolic if all the ai > q are not multiples
+of q. Here, we show only one needs not be a multiple of q for hyperbolicity.
+Proof. Suppose S3 − N(K) admits an essential torus T. Then T bounds a
+solid torus V that must contain at least one component of K.
+
+8
+THIAGO DE PAIVA AND JESSICA S. PURCELL
+Figure 3. Shows T(7, 2) augmented at the top right by J2,
+J3, and J4.
+First we show that we may choose V to contain T(p, q). For suppose V is
+disjoint from T(p, q). Then it must contain at least one Jaj. The component
+Jaj must have positive wrapping number in V , for otherwise T(p, q) and Jaj
+would have zero linking number, which is a contradiction. Because there
+is no essential torus in the exterior of the unknot in S3, it follows in this
+case that T is unknotted in S3. Therefore, T bounds a second solid torus V ′
+containing T(p, q). Thus in all cases we may assume T bounds a solid torus
+containing T(p, q).
+As an ≥ q, by Proposition 2.3, the torus knot T(p, q) is isotopic to a
+closed braid with an strands so that under the isotopy, the largest unknot
+Jan becomes the braid axis. Because the isotopy moves only the right-most
+p − an strands, all unknots Ja1, . . . , Jan are untouched by the isotopy.
+The torus T is then contained in the solid torus S3 − N(Jan), and bounds
+a solid torus V containing T(p, q). It follows that Jan is disjoint from V .
+The torus T must intersect the disc Dan bounded by Jan in a series of
+circles, with each circle bounding a meridian of V . Each meridian of V
+can be isotoped to meet the same number of strands of T(p, q), as follows.
+The boundary of a meridian defines an unknot in S3, and all such unknots
+are isotopic in S3 − N(K), where the isotopy is obtained by pushing the
+boundary of the meridian disc along the torus T. Because T(p, q) forms a
+braid, it meets these discs monotonically. Let b denote the number of times
+that a meridian of V intersects the strands of T(p, q) on the disc Dan. Note
+b > 1, or else T would be boundary parallel.
+Note also that V winds some number of times around the solid torus
+S3 − N(Jan), and note that each meridian of this solid torus meets exactly
+an strands of T(p, q), since this is the number of strands in the closed
+braid isotopic to T(p, q) obtained from Proposition 2.3. Since V meets each
+
+LORENZ LINKS OBTAINED BY TWISTING
+9
+meridian of S3 − N(Jan) a total of an times, and each meridian of V meets
+T(p, q) a total of b times, b must divide an.
+It follows that T(p, q) is a generalised b-cabling of L, where L is the core
+of the solid torus V .
+Observe that T is embedded in exterior of the torus knot S3 − N(T(p, q)).
+By work of Tsau [21], there are no essential tori in a torus knot exterior.
+Because b > 1, it follows that T must be compressible to its outside. That
+is, V is unknotted in S3. Thus, Lemma 2.9 implies that T(p, q) has braid
+index equal to b.
+On the other hand, the torus knot T(p, q) with 1 < q < p has braid index
+equal to q; for example this follows from Franks and Williams’ Theorem 2.4.
+Then, b = q, and b divides an. Hence, q divides an.
+By hypothesis, there is ai ∈ {a1, . . . , an} which is greater than q and not
+a multiple of q. Since ai > q, it must be the case that Jai is disjoint from the
+solid torus V . Since T(p, q) intersects the disc Dai bounded by Jai a total
+of ai times, and T(p, q) is a generalised q-cabling of L, it must be the case
+that L intersects the disc ai/q times. However, q does not divide ai. This is
+a contradiction.
+□
+Lemma 3.3. Let p, q be relatively prime integers with 1 < q < p. Let
+an, . . . , a1 be integers such that 1 < a1 < · · · < an < p with n > 1. Then
+the link K = T(p, q) ∪ Jan ∪ · · · ∪ Ja1 has no annuli with boundaries in two
+different components.
+Proof. Suppose that S3 − N(K) has an annulus A with boundaries ∂1A
+and ∂2A that lie in two different components, C1 and C2, respectively, of
+∂(S3 − N(K)).
+Case 1: Consider first that C1 and C2 are Jaj and Jak, respectively, for
+some j ̸= k ∈ {1, . . . , n}.
+Note ∂1A and ∂2A are isotopic in S3−N(K). The linking number between
+Cj and ∂jA is zero if and only if ∂jA is the longitude of Cj, in which case
+Cj and ∂jA are isotopic, for j = 1, 2.
+Suppose ∂1A is the longitude of C1, but ∂2A is not the longitude of C2.
+Since ∂1A and ∂2A are isotopic, C1 and C2 would have nonzero linking
+number in this case, but this is not possible. Similarly ∂2A cannot be the
+longitude of C2 if ∂1A is not the longitude of C1.
+Thus either ∂1A is the longitude of C1 and ∂2A is the longitude of C2, or
+neither is a longitude. If both are longitudes, then C1 and C2 are isotopic,
+which is not possible. Thus neither are longitudes.
+Then the linking number between C2 and ∂2A is positive. However, C1
+and C2 have zero linking number, so ∂1A and C2 must have zero linking
+number. But ∂2A is isotopic to ∂1A, and so ∂1A and C2 have nonzero linking
+number equal to the linking number of C2 and ∂2A. This is a contradiction.
+Case 2: Now suppose that C1 and C2 are Jaj and T(p, q), respectively,
+for some j ∈ {1, . . . , n}. Again ∂1A and ∂2A are isotopic.
+
+10
+THIAGO DE PAIVA AND JESSICA S. PURCELL
+Suppose first that ∂2A wraps at least one time along the longitude of
+C2 = T(p, q).
+Then ∂2A has positive linking number with each of the
+components Jak, because T(p, q) has positive linking number with each. But
+the linking number between ∂2A and Jak for Jak ̸= C1 is zero, because C1
+has linking number zero with each such component, and ∂2A has the same
+linking number with C1 as ∂1A. This is a contradiction.
+Thus ∂2A is a meridian of C2 = T(p, q). So ∂2A and T(p, q) have linking
+number equal to one. The curve ∂1A is some torus knot T(a, b) on ∂N(C1).
+If a is equal to zero, then ∂1A is a meridian of C1. Because a meridian
+of C1 has linking number zero with C2 = T(p, q), it follows that ∂1A and
+T(p, q) have linking number equal to zero. However, this is not possible as
+∂1A and ∂2A are isotopic. So, a ̸= 0. The linking number between ∂1A and
+C2 = T(p, q) is equal to a · aj, where C1 = Jaj. Because ∂2A and T(p, q)
+have linking number 1, and ∂1A and T(p, q) have linking number identical
+to ∂2A and T(p, q), it follows that a · aj = 1. This is impossible since aj > 1.
+Therefore, no such annulus exists.
+□
+Lemma 3.4. Let K be as in Proposition 3.2. Then K has no essential
+annuli with both boundary components in ∂N(T(p, q)).
+Proof. Suppose that S3 −N(K) has an essential annulus A with both bound-
+ary components in ∂N(T(p, q)).
+The exterior of a torus knot has just one essential annulus by work of
+Tsau [21]. By work of Lee, [16, Lemma 5.1] that essential annulus would be
+punctured by Jai, where ai > q is not a multiple of q. Thus A is not essential
+in S3 − N(T(p, q)). Thus A is compressible, boundary compressible, or
+boundary parallel in S3 − N(T(p, q)). Observe that a boundary compressible
+annulus is in fact boundary parallel, using the fact that S3 − N(T(p, q)) is
+irreducible and boundary irreducible.
+Consider first that A is boundary parallel to an annulus B in ∂N(T(p, q)).
+Then A ∪ B bounds a solid torus V in S3 − N(T(p, q)). Since A is not
+boundary parallel in S3 − N(K), at least one Jaj must be inside V . In
+addition, Jaj has wrapping number greater than zero in V , or else T(p, q)
+and Jaj would have linking number equal to zero, which is a contradiction.
+But Jaj is an unknot, whose complement admits no essential tori (e.g. [13,
+page 15]). Thus V is also unknotted in S3. This implies that B is a meridional
+annulus of ∂N(T(p, q)). If ∂V is boundary parallel to Jaj, then Jaj is the
+core of ∂V . Hence, the linking number between T(p, q) and Jaj would be
+one, which is not possible. Thus, as ∂V is not boundary parallel to Jaj, ∂V
+is an essential torus for S3 − N(K). This contradicts Proposition 3.2.
+Assume now that A is compressible in S3 − N(T(p, q)). Then there is a
+compression disk D for A in S3 − N(T(p, q)). Surgering A along D yields
+two discs, D1 and D2, such that ∂A = ∂D1 ∪ ∂D2. Since S3 − N(T(p, q))
+is boundary irreducible, ∂Di bounds a disk Ei on ∂N(T(p, q)). Thus, by
+pushing Ei slightly off of ∂N(T(p, q)) in S3 −N(K), we obtain a compressing
+
+LORENZ LINKS OBTAINED BY TWISTING
+11
+disc for A in S3−N(K), which contradicts our assumption that A is essential.
+Therefore, A is not compressible.
+Thus A cannot have both boundary components on ∂N(T(p, q)).
+□
+Lemma 3.5. Let K be as in Proposition 3.2. Then K has no essential
+annulus with both boundary components on one ∂N(Jaj).
+Proof. Suppose that S3 −N(K) has an essential annulus A with both bound-
+ary components on ∂N(Jaj). Since S3 −N(Jaj) is a solid torus, and the solid
+torus admits no essential annuli, A is not essential in S3 − N(Jaj). Thus A
+is either compressible or boundary parallel in S3 − N(Jaj).
+Case A: Suppose A is boundary parallel, parallel to an annulus B in
+∂N(Jaj). Then A ∪ B bounds a solid torus V in S3 − N(Jaj). Since A is not
+boundary parallel in S3 − N(K), at least one component C of K must be
+inside V .
+Case A1:
+Consider first that C = T(p, q). Then T(p, q) has wrapping
+number greater than zero in V , for otherwise Jaj and T(p, q) would have zero
+linking number, a contradiction. Note this implies that ∂V is incompressible
+to its inside.
+Suppose that some circle Jak with j ̸= k lies in S3 − V . Then we may
+isotope Jak to lie outside of W = N(Jaj) ∪ V , which is a regular solid torus
+neighbourhood of the unknot Jaj. Denote by ω the winding number of Jak in
+S3 − W. If ω = 0, then the linking number between Jak and T(p, q) is zero.
+Thus, ω ̸= 0. But then this implies that the linking number between Jaj and
+Jak is nonzero, a contradiction. Thus all circles Ja1, . . . , Jai−1, Jai+1, . . . , Jan
+are inside V in this case. Because at least two components of K lie inside V ,
+∂V is not boundary parallel to the inside.
+The core of V forms a torus knot T(a, b) on N(Jaj). Note b > 0 or else
+T(p, q) runs around a longitude of N(Jaj) and hence has linking number zero
+with Jaj, a contradiction.
+Suppose b = ±1, so the core of V has the form of the trivial knot T(a, ±1).
+Then there exists a disc in S3 − N(K) that is a longitude for ∂N(Jaj) whose
+boundary can be divided into two arcs, one of which meets A ⊂ ∂V in
+a nontrivial arc, and the other meets ∂N(Jaj). See Figure 4. This is an
+essential boundary compression disc for A, contradicting the fact that A is
+essential.
+Since |b| > 1, ∂V = ∂N(T(a, b)) is incompressible and not boundary
+parallel to the outside, i.e. in the solid torus S3 − Jaj.
+This implies that in all cases ∂V is essential in S3 − N(K) contradicting
+Proposition 3.2.
+Case A2:
+The torus knot T(p, q) cannot lie inside V by the previous
+case. So some C = Jak with j ̸= k lies inside V . The wrapping number of
+Jak inside V must be different from zero as Jak and T(p, q) have positive
+linking number. Since Jak and Jaj have zero linking number, V must be a
+longitude of ∂N(Jaj). If Jak is the core of V , then Jaj and Jak are isotopic
+
+12
+THIAGO DE PAIVA AND JESSICA S. PURCELL
+Figure 4. A disc with boundary an arc on each of A ⊂ ∂V
+and ∂N(Jaj).
+in S3 − N(T(p, q)), a contradiction. So Jak is not the core of V . But then
+∂V is incompressible and not boundary parallel to the inside in S3 − K,
+and incompressible and not boundary parallel to the outside in S3 − K,
+contradicting Proposition 3.2.
+Case B:
+Suppose A is compressible in S3 − N(Jaj). Then there is a
+compression disk D for A in S3 − N(Jaj). Surgering A along D yields two
+discs, D1 and D2, such that ∂A = ∂D1 ∪ ∂D2. If one of ∂D1 or ∂D2 bounds
+a disk E on ∂N(Jaj), then by considering a disc with boundary in A close
+to E, we see that A is also compressible in S3 − N(K), a contradiction. So
+suppose that neither ∂D1 nor ∂D2 bounds a disk on ∂N(Jaj). Then D1 and
+D2 are discs in the solid torus S3 − N(Jaj) with nontrivial boundary on
+∂N(Jaj) and hence both are meridians of S3 − N(Jaj), i.e. with ∂D1 and
+∂D2 forming longitudes of ∂N(Jaj). Undoing the surgery along D, it follows
+that A is boundary parallel in S3 − N(Jaj). Thus we have a contradiction
+to Case A.
+Therefore, S3 − N(K) has no essential annulus with both boundary com-
+ponents in one ∂N(Jaj).
+□
+Proposition 3.6. The link K as in Proposition 3.2 has no essential annuli.
+Proof. By Lemma 3.3, any essential annulus has both boundary components
+on the same component of K. By Lemma 3.4, the two boundary components
+cannot lie on ∂N(T(p, q)). By Lemma 3.5 the two boundary components
+cannot lie on one of the ∂N(Jaj). Thus no such annulus exists.
+□
+Theorem 3.7. Let p, q be relatively prime integers with 1 < q < p. Let
+an, . . . , a1 be integers such that 1 < a1 < · · · < an < p with n > 1. Also,
+assume that there is ai > q which is not a multiple of q. Then, the link
+K = T(p, q) ∪ Jan ∪ · · · ∪ Ja1 is hyperbolic.
+Proof. By de Paiva and Purcell [6, Lemma 5.1], the link exterior is irre-
+ducible and boundary irreducible. By Proposition 3.2, it is atoroidal. By
+Proposition 3.6, it is anannular. Therefore it is hyperbolic by Thurston’s
+hyperbolisation theorem for Haken manifolds [20].
+□
+Combining Theorem 3.7 and de Paiva and Purcell [6, Theorem 5.6], we
+completely classify the geometric types of the links T(p, q) ∪ Ja1 ∪ . . . Jan.
+
+LORENZ LINKS OBTAINED BY TWISTING
+13
+Theorem 3.8. Let p, q be relatively prime integers with 1 < q < p. Let
+a1, . . . , an be integers such that 1 < a1 < · · · < an < p. Then the link
+K = T(p, q) ∪ Ja1 ∪ . . . Jan is hyperbolic if and only if either all ai < q, or
+there is ai > q which is not a multiple of q.
+Proof. When n = 1, the link K = T(p, q) ∪ Ja1 is the Dehn-filling parent of
+a twisted torus knot; this has been treated by Lee [15, 16]. If n = 1 and
+a1 = q, then [15, Theorem 1] implies that infinitely many Dehn surgeries
+along Ja1 yield non-hyperbolic knots. Therefore, Thurston’s hyperbolic Dehn
+filling theorem [19] implies K is not hyperbolic. In fact, the proof of [15,
+Theorem 1] implies K is annular. If n = 1 and a1 is not a multiple of q, then
+K is hyperbolic by [16, Proposition 5.7].
+In the case n > 1, if there is ai > q that is not a multiple of q, then K is
+hyperbolic by Theorem 3.7.
+If n > 1 and all ai are less than q, then no ai is a multiple of q, and K is
+hyperbolic by [6, Theorem 5.6].
+Finally, if n > 1, there is some ai > q and all ai > q are multiples of q,
+then K is satellite by [6, Theorem 5.6].
+□
+Corollary 3.9. Let p, q be relatively prime integers with 1 < q < p, and let
+a1, . . . , an and s1, . . . , sn be integers such that 1 < a1 < · · · < an < p and
+si > 0 for all i. Then, there exists B ≫ 0 such that if each si > B, the
+T-link
+T((a1, a1s1), . . . , (an, ansn), (p, q))
+is hyperbolic if and only if either all ai < q, or there is ai > q which is not a
+multiple of q.
+Proof. By Theorem 3.8, the link K = T(p, q) ∪ Ja1 ∪ · · · ∪ Jan is hyperbolic
+if and only if the ai satisfy the hypotheses of the corollary. Obtain the
+given T-link by Dehn filling the link components Ja1, . . . , Jan along slopes
+1/s1, . . . , 1/sn, respectively. When the link K is hyperbolic, the Dehn filling
+remains hyperbolic by Thurston’s hyperbolic Dehn filling theorem [19] pro-
+vided the si are sufficiently large. On the other hand, Dehn filling a satellite
+K yields a satellite T-link, by de Paiva and Purcell [6, Theorem 5.6], and in
+the case n = 1 and a1 = q, Dehn filling yields an annular link by Lee [15].
+□
+Note that Theorem 1.1 in the introduction follows immediately from
+Corollary 3.9.
+4. Hyperbolicity with effective full twist bounds
+While Corollary Corollary 3.9 is quite broad, unfortunately the constant B
+in that theorem is not explicit, and so it may be difficult to apply in practice.
+In this section we find explicit parameters which produce hyperbolic T-knot
+obtained by full twists. Because we are considering full twists exclusively in
+this section, Proposition 2.2 implies that we may assume that none of the ai
+are equal to q.
+
+14
+THIAGO DE PAIVA AND JESSICA S. PURCELL
+Proposition 4.1. Let a1, . . . , an, s1, . . . , sn, and p, q, k be integers satisfying
+the following hypotheses:
+• p and q are relatively prime,
+• 1 < a1 < · · · < an, and 0 < q < an < p,
+• each si > 0, and sn ≥ 2,
+• p and an are relatively prime,
+• k ≥ 2.
+Then the T-knot K = T((a1, a1s1), . . . , (an, ansn), (p, q + kp)) is atoroidal.
+Proof. Suppose that the exterior of K in S3 admits an essential torus T. By
+work of Ito [14, Theorem 1.2(3)], because K is the closure of a braid with at
+least two positive full twists on p strands, the torus T does not intersect the
+braid axis C. Moreover, the knot inside T is given by a braid. Thus there
+exists some integer d > 1 such that K is a generalized d-cabling of a knot L,
+where L is the core of the solid torus bounded by T. As a consequence, d
+must divide p.
+After (−1/k)-Dehn surgery along the braid axis C, the knot K becomes
+the T-knot
+K′ = T((a1, a1s1), . . . , (an, ansn), (p, q))
+and the torus T becomes a new torus T ′. This will bound a solid torus V ′ in
+S3, with core L′. Because q < an < ansn + q, the knot K′ has braid index
+equal to an by Corollary 2.6.
+If L′ is trivial, then an is equal to d by Lemma 2.9. However, this is not
+possible since gcd(p, an) = 1.
+So L′ is knotted. Then by Theorem 2.8, an is equal to dβ(L′), where
+β(L′) is the braid index of L′. But then d divides p and d divides an, again
+contradicting gcd(p, an) = 1.
+Therefore, the exterior of K admits no essential torus.
+□
+We will combine the previous result with the following from [5], which
+gives information on torus knots.
+Theorem 4.2 (Theorem 1.2 of de Paiva [5]). Let p, q, a1, . . . , an, s1, . . . , sn
+be positive integers such that 1 < q < p and 1 < a1 < · · · < an < p with
+ai ̸= q. If gcd(p, q) = 1 and in addition one of the following hold:
+• q < an, or
+• q > an and p is not of the form bq + 1 for some b > 0, or
+• q > an and p = bq + 1 for some b > 0, but s1 > 1, or
+• q > an, p = bq + 1 for some b > 0, and s1 = 1, but a2 ̸= a1 + 1,
+then T((a1, s1a1), (a2, s2a2), . . . , (an, snan), (p, q)) is not a torus knot.
+□
+Theorem 4.3. Let a1, . . . , an, s1, . . . , sn, and p, q, k be integers satisfying
+the following hypotheses:
+• p and q are relatively prime,
+• 1 < a1 < · · · < an and 1 < q < an < p,
+• each si > 0 and sn ≥ 2,
+
+LORENZ LINKS OBTAINED BY TWISTING
+15
+• p and an are relatively prime,
+• k ≥ 2, n ≥ 2.
+Then if in addition, one of the following hold:
+• q ̸= 1,
+• or s1 > 1,
+• or a2 ̸= a1 + 1,
+then the T-link K = T((a1, a1s1), . . . , (an, ansn), (p, q + kp)) is hyperbolic.
+Proof. Because gcd(p, q) = 1, K is a knot. By Proposition 4.1, K is atoroidal,
+so not a satellite knot.
+The T-knot K is equivalent to the T-knot
+T((a1, s1), . . . , (an, ansn), (q + kp, p))
+by [1, Corollary 3]. The integer q + kp does not have the form bp + 1 if and
+only if q is different from 1. So under these conditions, K is not a torus knot
+by Theorem 4.2.
+Therefore, by Thurston’s hyperbolisation Theorem for knots [20], K is
+hyperbolic.
+□
+Proposition 4.4. Let a1, . . . , an, s1, . . . , sn, and p, q, k be integers satisfying
+the following hypotheses:
+• p and q are relatively prime,
+• 1 < a1 < · · · < an, and 1 < q < an < p,
+• each si > 0 and both sn and sn−1 are at least 2.
+Suppose also that one of the following holds:
+• q < an−1 and an and an−1 are relatively prime, or
+• q > an−1 and an and q are relatively prime.
+Then the knot K = T((a1, a1s1), . . . , (an, ansn), (p, q)) is atoroidal.
+Proof. Suppose the exterior of K in S3 admits an essential torus T.
+By Proposition 2.3, K is equivalent to the knot given by the closure of
+the braid
+B = (σan−1 . . . σan−q+1)p−an · τ · (σ1 . . . σan−1)snan+q,
+where τ is the concatination of braids (a1, a1s1) . . . (an−1, an−1sn−1).
+Since B has at least two positive full twists on an strands, it follows from
+[14, Theorem 1.2(3)] that T does not intersect the braid axis C of B. Thus
+there is an integer d > 0 such that K is a generalized d-cabling of the core L
+of the solid torus bounded by T. Hence d divides an.
+Perform (−1/sn)-Dehn surgery along the braid axis C to obtain the braid
+B′ = (σan−1 . . . σan−q+1)p−an · τ · (σ1 . . . σan−1)q.
+Its closure gives K′ = T((a1, a1s1), . . . , (an−1, an−1sn−1), (an, q)). The torus
+T becomes a new essential torus T ′ in the exterior of K′.
+The torus T ′ bounds a solid torus with core L′, which is either trivial or
+knotted.
+
+16
+THIAGO DE PAIVA AND JESSICA S. PURCELL
+Suppose first the case that q < an−1. Then q ≤ an−1 ≤ an−1sn−1 + q, so
+Corollary 2.6 implies that K′ has braid index equal to an−1. If L′ is the
+trivial knot, then an−1 is equal to d by Lemma 2.9. This implies that d
+divides both an and an−1, contradicting the assumption in this case that
+these are relatively prime. Similarly, if L′ is knotted, then Theorem 2.8
+implies that an−1 is a multiple of d, with the same contradiction.
+Now suppose q > an−1. Then K′ has braid index q by Franks and Williams,
+Theorem 2.4. If L′ is trivial, then as above, Lemma 2.9 implies q equals d, and
+therefore d divides both an and q, contradicting the hypothesis. Similarly, if
+L′ is knotted, Theorem 2.8 implies q is a multiple of d, and again d divides
+both an and q, which is a contradiction.
+□
+Theorem 4.5. Let a1, . . . , an, s1, . . . , sn, and p, q, k be integers satisfying
+the following hypotheses:
+• p and q are relatively prime,
+• 1 < a1 < · · · < an and 1 < q < an < p,
+• each si > 0 and both sn and sn−1 are at least 2.
+Suppose also that one of the following holds:
+• q < an−1 and an and an−1 are relatively prime, or
+• q > an−1 and an and q are relatively prime.
+Then K = T((a1, a1s1), . . . , (an, ansn), (p, q)) is hyperbolic.
+Proof. By Proposition 4.4, the knot K is atoroidal. By Theorem 4.2, using
+the fact that q < an, K is anannular.
+Therefore, K is hyperbolic.
+□
+5. Satellite T-links obtained by Half-twists
+In this section we switch from discussions of hyperbolic links to satellite
+links. We find families of Lorenz links that are satellites using half-twists,
+rather than full-twists. Previous work by de Paiva and Purcell found con-
+ditions that ensure a T-link is satellite, namely [6, Theorem 4.3]. Lee has
+similar results for the case of twisted torus knots [15, Theorem 1]. We extend
+these results.
+Definition 5.1. Suppose B is a diagram given as a closed braid; we consider
+the braid to have strands running vertically on the plane of projection. A
+positive half-twist on the strands from a to b is the braid
+∆a,b = (σa . . . σb)(σa . . . σb−1) . . . (σa).
+This can be thought of as cutting the braid between the a-th and b-th strands,
+rotating in the anticlockwise direction by 180◦, and gluing back. In braid
+theory literature, the positive half-twist on all strands is well known as the
+Garside fundamental braid. A negative half-twist is defined similarly, only
+the rotation is in the clockwise direction. See Figure 5.
+
+LORENZ LINKS OBTAINED BY TWISTING
+17
+Figure 5. An example of half twists when r = 2, q = 3, t =
+1. Left: A positive half-twist ∆1,rq, a negative half-twist
+∆1,(r−t)q and a positive half-twist ∆(r−t)q+1,tq. The green
+circle indicates the braid axis. Middle: The negative half-
+twist cancels crossings above. Right: The additional positive
+half-twist gives the braid (rq, tq).
+Lemma 5.2. Let r, q, s be positive integers, and suppose s is not a multiple
+of r. The (rq, sq)-torus braid is obtained by the following procedure. Start
+with the trivial braid on rq strands; let J1,rq be an unknot encircling all rq
+strands. Let t be an integer such that 0 < t < r and s = t + kr for some
+integer k. Insert a positive half-twist ∆1,rq, followed by a negative half-twist
+∆1,(r−t)q and a positive half-twist ∆(r−t)q+1,rq. Finally, perform 1/k-Dehn
+filling on J1,rq. The result is the (rq, sq)-torus braid.
+Proof. The process is illustrated in Figure 5. The positive half-twist ∆1,rq
+yields a braid
+(σ1σ2 . . . σrq−1)(σ1 . . . σrq−2) . . . (σ1),
+encircled by J1,rq. Perform the negative half-twist ∆1,(r−t)q. This concate-
+nates the previous braid with
+(σ−1
+(r−t)q−1 . . . σ−1
+2 σ−1
+1 )(σ−1
+(r−t)q−1 . . . σ−1
+2 ) . . . (σ−1
+(r−t)q−1).
+This braid cancels with the positive half-twist along the first (r − t)q strands,
+as shown in Figure 5, middle. Finally, the positive half-twist ∆(r−t)q+1,rq
+concatenates a positive half-twist along the last tq strands, giving the braid
+(σ1 . . . σrq−1)tq = (rq, tq),
+still augmented by the unlink J1,rq.
+To obtain the braid (rq, sq), perform 1/k Dehn filling on J1,rq, removing
+that link component and inserting an additional krq overstrands into the
+braid, for a total of tq+krq = sq overstrands, giving the desired (rq, sq)-torus
+braid.
+□
+Lemma 5.3. Let r, q, s be positive integers, with s not a multiple of r.
+Consider the torus braid (rq, sq). At the top of the braid, consider r disjoint
+discs arranged horizontally, each encircling q strands of the braid, and similar
+discs at the bottom of the braid. The boundary of each disc at the top connects
+
+18
+THIAGO DE PAIVA AND JESSICA S. PURCELL
+via a cylinder, embedded in the complement of the braid and enclosing q
+strands, to the boundary of a disc at the bottom of the braid.
+Moreover, the solid cylinders enclosed by these cylinders, containing q
+strands each, forms the (r, s)-torus braid.
+Proof. Let t be an integer such that 0 < t < r and s = t+kr for some integer
+k. By Lemma 5.2, the (rq, sq) torus braid is formed from k full twists on
+rq strands, followed by a positive half-twist ∆1,rq, then a negative half-twist
+−∆1,(r−t)q and a positive half-twist ∆(r−t)q+1,rq. Each half-twist is on a
+multiple of q strands.
+Observe that the cylinders described above can be arranged to completely
+contain any half-twist on q strands. For a half-twist on a multiple of q strands,
+say xq strands, x disjoint cylinders enter the top of the half-twist, and then
+are half-twisted themselves, remaining disjoint, to exit the bottom of the
+half-twist. Thus the cylinders remain embedded as claimed when passing
+through half-twists. Finally, each full twist also preserves the cylinders,
+sending each through a full twist.
+To see that the braid formed by the solid cylinders is as claimed, observe
+that the cylinders form k full twists, followed by one positive half-twist on all
+strands. The (r−t) left-most cylinders then pass through a negative half-twist,
+and the remaining t right-most cylinders pass through a positive half-twist.
+As in Lemma 5.2, this creates braid on r strands, with rk overstrands at
+the top coming from the full twists, followed by t overstrands coming from
+the concatenation of half-twists. Thus this is an (r, rk + t) = (r, s)-torus
+braid.
+□
+Theorem 5.4. Let p, q be integers such that 1 < q < p, and let (a1, b1), . . . ,
+(an, bn) be pairs of integers such that 1 < a1 < · · · < an ≤ q and bi > 0
+for i = 1, . . . , n. Finally let r1, . . . , rm and s1, . . . , sm be integers such that
+q < r1q < · · · < rmq < p, and si > 0 for i = 1, . . . , m. Then the T-link
+K = T((a1, b1), . . . , (an, bn), (r1q, s1q), . . . , (rmq, smq), (p, q))
+is satellite with companion the T-link T((r1, s1), . . . , (rm, sm+1)) and pattern
+given by the closure of the braid
+(a1, b1) . . . (an, bn)(σq−1 . . . σ1)p−rm
+�
+q, q
+� m
+�
+i=1
+risi
+�
++ qrm
+�
+Proof. As before, we think of the T-link as the closure of a braid on p
+strands arranged vertically, the concatenation of braids (a1, b1), . . . , (an, bn),
+(r1q, s1q), . . . , (rmq, smq), (p, q) in that order.
+First apply Proposition 2.3 to change the closed braid of the T-link to a
+closed braid B′ on rmq strands. This isotopy fixes all of the rmq strands at
+the top left of the original braid; thus it does not affect any of the braids
+(aj, bj) or (riq, siq), for any i, j. In other words, B′ is the braid given by
+concatenating (σrmq−1 . . . σrmq−q+1)p−rm with braids (a1, b1), . . . , (an, bn),
+(r1q, s1q), . . . , (rmq, smq), and finally the braid (σ1 . . . σrmq−1)q.
+
+LORENZ LINKS OBTAINED BY TWISTING
+19
+By Lemma 5.3, there are rm disjoint embedded cylinders in the complement
+of the portion of the braid starting just above the braid (a1, b1), and ending
+just below the braid (σ1 . . . σrmq−1)q at the bottom. These cylinders each
+enclose q strands. They extend around the braid closure to give rm disjoint
+embedded cylinders running to the top of the braid, each enclosing q strands,
+arranged right to left across the top of the braid.
+The only portion of the braid that is not already enclosed in one of these
+cylinders is the braid (σrmq−1 . . . σrmq−q+1)p−rm lying at the top. This is
+a braid whose left-most strand is the (rmq − q + 1)-th strand, and whose
+right-most strand is the rmq-th strand. In other words, this is a braid on the
+right-most q strands of the rmq-strand braid. Thus the right-most cylinder,
+enclosing q strands, can be extended to enclose this braid. Then all cylinders
+connect to form a closed embedded torus Σ, encircling q strands of the braid.
+The torus Σ bounds a solid torus containing q strands, which we check
+has the claimed form of the companion in the theorem statement. This solid
+torus forms a braid on rm strands. By Lemma 5.3, each (riq, siq)-torus braid
+from the original T-link causes the solid cylinder to form a braid (ri, si).
+The braids (aj, bj) and (σrmq−1 . . . σrmq−q+1)p−rm lie completely inside the
+solid cylinder, so they do not affect the braid it forms. Finally, consider
+the braid (σ1 . . . σrmq−1)q at the bottom of B′. This is formed by q strands
+running over all the rmq strands. When the collection of solid cylinders
+encounter this braid, the left-most solid cylinder encircles exactly these q
+strands, and runs over all others to lie on the right-most side. Thus it
+forms a (rm, 1)-torus braid. So the solid torus enclosing q strands has the
+form of the closure of a braid (r1, s1) . . . (rm, sm), (rm, 1). This is the T-link
+T((r1, s1), . . . (rm, sm + 1)) as claimed. Since it forms a nontrivial knot in
+S3, Σ is an incompressible torus.
+Finally we check the form of the pattern. Starting at the top-left of
+the braid B′, the torus Σ encloses the braid (a1, b1) . . . (an, bn), which will
+form part of the braid describing the pattern. As Σ follows the companion
+into each of the braids (ri, si), all the q strands will make one full twist
+each time Σ runs completely through an overstrand. There are si of these,
+i = 1, . . . m − 1, plus sm + 1 for the (rm, sm + 1) braid that the compan-
+ion runs over. These will occur in some order, with Σ also enclosing the
+braid (σq−1 . . . σ1)p−rm, coming from the top right of B′, at some point.
+Because full twists commute in the braid group, we may write the braid
+as (a1, b1) . . . (an, bn)(σq−1 . . . σ1)p−rm · τ where τ is an appropriate number
+of full twists. To obtain the appropriate number of full twists, we need
+to consider the homological longitude of the companion. The pattern is
+the braid obtained when we apply a homeomorphism taking the solid torus
+bounded by the companion to an unknotted solid torus, with homological
+longitude mapped to a standard longitude of the unknot. The effect is to
+add �m−1
+i=1 (ri − 1)si + (rm − 1)(sm + 1) additional full twists, for a total of
+
+20
+THIAGO DE PAIVA AND JESSICA S. PURCELL
+�m
+i=1 risi + rm full twists. Thus the pattern can be written as the braid
+(a1, b1) . . . (an, bn)(σq−1 . . . σ1)p−rm(q, q(
+�
+risi) + qrm)
+□
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+

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+1
+EMAHA-DB1: A New Upper Limb sEMG Dataset
+for Classification of Activities of Daily Living
+Naveen Kumar Karnam, Anish Chand Turlapaty, Member, IEEE, Shiv Ram Dubey, Senior Member, IEEE and
+Balakrishna Gokaraju, Member, IEEE
+Abstract—In this paper, we present electromyography analysis
+of human activity - database 1 (EMAHA-DB1), a novel dataset
+of multi-channel surface electromyography (sEMG) signals to
+evaluate the activities of daily living (ADL). The dataset is
+acquired from 25 able-bodied subjects while performing 22 activ-
+ities categorised according to functional arm activity behavioral
+system (FAABOS) (3 - full hand gestures, 6 - open/close office
+draw, 8 - grasping and holding of small office objects, 2 - flexion
+and extension of finger movements, 2 - writing and 1 - rest). The
+sEMG data is measured by a set of five Noraxon Ultium wireless
+sEMG sensors with Ag/Agcl electrodes placed on a human hand.
+The dataset is analyzed for hand activity recognition classification
+performance. The classification is performed using four state-of-
+the-art machine learning classifiers, including Random Forest
+(RF), Fine K-Nearest Neighbour (KNN), Ensemble KNN (sKNN)
+and Support Vector Machine (SVM) with seven combinations of
+time domain and frequency domain feature sets. The state-of-the-
+art classification accuracy on five FAABOS categories is 83.21%
+by using the SVM classifier with the third order polynomial
+kernel using energy feature and auto regressive feature set
+ensemble. The classification accuracy on 22 class hand activities
+is 75.39% by the same SVM classifier with the log moments in
+frequency domain (LMF) feature, modified LMF, time domain
+statistical (TDS) feature, spectral band powers (SBP), channel
+cross correlation and local binary patterns (LBP) set ensemble.
+The analysis depicts the technical challenges addressed by the
+dataset. The developed dataset can be used as a benchmark for
+various classification methods as well as for sEMG signal analysis
+corresponding to ADL and for the development of prosthetics and
+other wearable robotics.
+Index Terms—Machine learning, Classification Algorithms,
+Surface Electromyography (sEMG), Activities of Daily Living
+(ADL), Features, Dataset and Benchmark.
+I. INTRODUCTION
+P
+ERFORMING hand movements during activities of daily
+living (ADL) [1] without any difficulty provides func-
+tional independence and a decent quality of life [2]. However,
+it is quite difficult to perform simple hand movements for
+individuals affected by the following disorders: upper limb
+disabilities [3], [4], disorders related to aging [5], neuromus-
+cular disorders [6], and stroke related disabilities [7], [8], [9],
+[10]. Human computer interfaces and human robot interfaces
+N.K. Karnam and A.C. Turlapaty are with the Biosignal Analysis Lab at
+the Indian Institute of Information Technology, Sri City, A.P., India (email:
+anish.turlapaty@iiits.in).
+S.R. Dubey is with the Computer Vision and Biometrics Laboratory at
+Indian Institute of Information Technology, Allahabad, Prayagraj-211015,
+U.P., India (email: srdubey@iiita.ac.in).
+B. Gokaraju is with the Visualizations and Computing Advanced Research
+Center (ViCAR) and Department of Computational Data Science an Engineer-
+ing, North Carolina A and T State University, Greensboro, North Carolina
+(email: bgokaraju@ncat.edu).
+can support the rehabilitation process to recover from the
+above mentioned disorders. For instance, hand gesture-based
+interfaces based on computer vision techniques for identifying
+and classifying gestures are currently under development [11].
+Moreover, many researchers have explored robotic control
+using visual gestures [12], [13], [14]. However, vision based
+control methods are inadequate to determine the appropriate
+control for actuation and the amount of force exerted by a
+muscle during action. One approach to quantify the upper
+limb activity is to use wearable sensors such as inertial
+motion sensors (IMUs) including accelerometers, gyroscopes
+and magnetometers. These sensors are utilised to measure
+and monitor limb activities, quantify muscle motor deficits
+[15], and classify the types of physical activity [16], [17].
+Although wearable sensors can recognize human activity, they
+are deficient in precise identification of hand gestures, finer
+finger movements and the amount of muscle strength used to
+execute the movement [18].
+Alternatively, hand movement classification and the limb
+control [19], [20] through surface electromyography (sEMG)
+signals facilitates the design of prosthetic devices, exoskeleton
+arms, advanced realistic bio-mechanical models, and rehabili-
+tation therapies [21]. In these applications, utilization of multi-
+modal signals is also very common. In the literature, fusion of
+the IMU’s and sEMG signals [22], [23], [24] for hand activity
+classification and estimation of the continuous orientation of
+the forearm is analyzed. The electroencephalography signals
+(EEG)) and sEMG signals are also fused to decode the
+intention of the person. This fusion process can generate better
+control signals compared to a standalone sEMG signal based
+control [25], [26], [27]. In order to obtain better classification
+accuracies the features from sEMG signals can be fused with
+those from the vision based image classification network [28].
+In practice, the multi-modal methods increase the complexity
+of the hardware as well as software systems, hence they pose
+difficulty for different real-life applications. Hand movement
+analysis and classification through standalone sEMG signals
+is gaining attention [29], [30], [31], [32], [33] and is the focus
+of our current work.
+In this paper, we present electromyography analysis of
+human activity - database 1 (EMAHA-DB1), a novel sEMG
+dataset on ADL for the Indian population. Following are the
+salient features of EMAHA-DB1:
+• There are several sEMG datasets available that include
+activities such as hand gestures, hand movements, wrist
+movements, and grasping objects. These datasets are
+mainly collected for western populations and there is no
+arXiv:2301.03325v1  [eess.SP]  9 Jan 2023
+
+2
+dataset for ADL from the Indian population. EMAHA-
+DB1 fills this gap.
+• There is a tradition of anthropometric data collection in
+India [34], [35]. For any population, there is an influence
+of anthropometrics on their kinematics and kinetics [36],
+[37]. EMAHA-DB1 will compliment existing anthropo-
+metrics, kinematics and kinetics datasets [30], [37] which
+will be helpful in conducting upper limb rehabilitation
+therapies, physiological studies and clinical studies for
+Indian population.
+• The ADL performance is analyzed by grouping the
+actions according to the functional arm activity behav-
+ioral observation system (FAABOS [38]). The functional
+taxonomy provided by Uswatte et al. quantifies group of
+hand actions based on the behavioral significance.
+• There are publicly available ADL datasets such as the
+NinaPro [39], the BioPatRec [40], the Ramikushaba [41]
+and the UCI Gesture [42] that have not covered a few
+important ADL categories. The hand activities are usually
+performed in an experimental set up with a fixed duration
+for each of the activities, however we have considered
+different durations for distinct activities to approximate
+corresponding durations of real time hand movements.
+• The dataset can be used to benchmark classification
+algorithms or perform statistical studies. The developed
+dataset consists of a larger number of subjects and a
+higher number of activity repetitions compared to any
+other publicly available ADL datasets.
+The main contributions of the paper are:
+1) In this work, we have carried out muscle activity mea-
+surements corresponding to activities of daily living and
+collected a novel multichannel sEMG data from Indian
+population.
+2) The EMAHA-DB1 dataset is organized according to
+custom FAABOS functional categories to perform anal-
+ysis using state-of-the-art machine learning classifiers.
+Specifically the sEMG signals are analyzed to classify
+into the functional groups as well as individual activities.
+3) We also perform extensive feature analysis with respect
+to the FAABOS functional categories.
+The rest of the paper is organised as follows: Section II
+details about the proposed EMAHA-DB1 dataset; Section
+III presents experiments in machine learning frameworks;
+Section IV demonstrates the experimental results; and Section
+V provides a conclusion along with the future scope.
+II. EMAHA-DB1: PROPOSED SEMG DATASET
+A. Data Collection
+1) Study participants: The institutional ethics committee
+of Indian Institute of Information Technology Sri City (No.
+IIITS/EC/2022/01) approved the proposed data collection pro-
+tocol developed in general accordance with the declaration of
+Helsinki and specific accordance with the “National Ethical
+Guidelines for Biomedical and Health Research involving hu-
+man participants” of India. Twenty-five healthy subjects with
+no history of upper limb pathology, including 22 males and 3
+females, participated in the sEMG data collection process. The
+TABLE I: List of hand activities
+Activity No.
+Activity description
+A0
+Hand at rest (sitting)
+A1
+Tossing a coin (sitting)
+A2
+Finger snapping (sitting)
+A3
+Pulling an empty draw - Posterior view (sitting)
+A4
+Pulling a draw with weight (2kg) - Posterior view (sitting)
+A5
+Pulling an empty draw - Anterior view (sitting)
+A6
+Pulling a draw with weight (2kg) - Anterior view (sitting)
+A7
+Pushing an empty draw - Posterior view (sitting)
+A8
+Pushing a draw with weight (2kg) - Posterior view (sitting)
+A9
+Clasping both hands (sitting)
+A10
+Hand clapping (sitting)
+A11
+Grasping and holding 1L water bottle (sitting)
+A12
+Grasping and holding small hammer (sitting)
+A13
+Grasping and holding small saw (sitting)
+A14
+Writing the phrase ”Bio signal lab” on paper with pen -
+lateral grasp (sitting)
+A15
+Writing the phrase ”Bio signal lab” on board with marker
+- lateral grasp (standing)
+A16
+Lifting a small bucket with 4L water (standing)
+A17
+Typing the phrase ”Bio signal lab” on keyboard using
+single finger (sitting)
+A18
+Drinking tea/water from a cup - lateral grasp (sitting)
+A19
+Picking up the phone, placing it to his/her ear and hanging
+up the phone on table (sitting)
+A20
+Grasping and holding a book (sitting)
+A21
+Grasping and holding a tennis ball (sitting)
+TABLE II: Sensor placement on hand muscle
+Channel No.
+Sensor No.
+Hand muscle name
+1
+21621
+Brachio Radialis (BR) muscle
+2
+21623
+Flexor Carpi Radialis(FCR) muscle
+3
+21624
+Flexor Carpi Ulnaris (FCU) muscle
+4
+21625
+Biceps Brachii (BB) muscle
+5
+21626
+Abductor Pollicis Brevis (APB) muscle
+average age is 28±6 years. Before the first session of activities,
+each of the participants gave written informed consent and the
+data collection process is completely non-invasive.
+2) Experimental setup and acquisition protocol: The 22
+activities performed by each subject are listed in Table I.
+Each of the hand muscle activity is recorded with a 5-channel
+Noraxon Ultium wireless sEMG sensor setup [43] as shown
+in Fig. 1. Five self-adhesive Ag/AgCL dual electrodes were
+placed at the centre of the five most representative muscle sites
+of the right arm as shown in Fig. 1. Each subject is instructed
+to sit comfortably with one elbow resting on a table and an arm
+flexed 90◦ compared to the forearm. The muscle locations are
+selected according to the atlas in chapter 17 [44] and is given
+in Table II. At the beginning of each session, the participant’s
+hands are cleaned with an alcohol based wet wipe.
+Prior to each session, the subject is acquainted with the
+experiment protocol including a video demonstration of the
+proposed activities. The total duration of each session is up-to
+one hour per subject depending on adaptability. Each activity
+is performed for a maximum duration of 10s and repeated
+10 times. There is a rest period of 5s between each of the
+repetitions and a 30s gap between the sessions of different
+activities. Each of the activities consists of two phases: (1) an
+action and (2) rest. However, some of the activities included
+an extra release phase. During the action phase, the subject
+performs the corresponding activity; during the release phase,
+the subject transitions from the action state to rest state; and
+during the rest phase, the subject completely relaxes each of
+
+3
+Fig. 1: Learning steps from sEMG dataset collection to classification of hand activities
+TABLE III: Phase-wise durations of each activity.
+No.
+TX TA TR TT
+No.
+TX TA TR TT
+No.
+TX TA TR TT
+A1
+3
+5
+0
+8
+A8
+3
+5
+0
+8
+A15 3
+15
+2
+20
+A2
+3
+5
+0
+8
+A9
+3
+5
+0
+8
+A16 5
+5
+3
+13
+A3
+3
+5
+0
+8
+A10 3
+5
+0
+8
+A17 3
+10
+2
+15
+A4
+3
+5
+0
+8
+A11 3
+5
+3
+11
+A18 5
+5
+3
+13
+A5
+3
+5
+0
+8
+A12 3
+5
+3
+11
+A19 5
+5
+3
+13
+A6
+3
+5
+0
+8
+A13 3
+5
+3
+11
+A20 5
+5
+3
+13
+A7
+3
+5
+0
+8
+A14 3
+10
+2
+15
+A21 5
+5
+3
+13
+his/her muscles. The time duration for each activity is given
+in Table III, where TX, TA, TR, and TT are the rest, action,
+release, and total duration, respectively.
+3) Comparisons with existing datasets : The characteristics
+of EMAHA-DB1 data are compared against those of existing
+sEMG hand activity datasets in TABLE IV. Apart from those
+mentioned in salient features in Introduction, a few additional
+and distinct characteristics of the EMAHA-DB1 are: 1) the
+experiments are designed such that hand activities performed
+consists of three phases of action (contraction/relaxation of
+muscles), release (retreating of action), and rest (relaxing of
+muscles), 2) the measurements are acquired with a minimal
+number of sensors hence requires lower computational re-
+sources compared to the existing datasets.
+B. Data Preparation
+1) Activity
+segmentation:
+For
+the
+sEMG
+signals
+in
+EMAHA-DB1, the preliminary annotations for onset and
+offset of the actions are performed based on the respective
+durations of action phases shown in Table III. To improve the
+quality of activity labels, based on the procedure developed in
+[46], an improved signal segmentation process (listed below)
+is implemented:
+1) Initially, for each trial of each activity performed by each
+subject, the multi-channel signal is rectified.
+2) For each of these trials, the maximum and minimum
+values are identified to determine the range R.
+3) The signal strengths at R/
+√
+2 (3dB amplitude) are
+considered the thresholds on either side.
+4) The first signal strength, past the preliminary onset,
+crossing the 3dB threshold is identified for each channel.
+The earliest location among the threshold crossings from
+the five channels is considered the onset of action.
+5) The trial data is parsed backwards from the end of the
+action. The first point from the end i.e., the final 3dB
+crossing is identified for each channel. The right most
+location among the crossings from these channels is
+labelled the offset of action.
+6) Finally, the signal samples between the onset and the
+offsets are annotated as the action and assigned the
+corresponding activity number, and the remaining signal
+is considered to be rest state.
+The above procedure is illustrated in Fig. 2 for a single trial
+of ADL. It is observed that signal segmentation improves the
+annotation process of activity vs. rest which further improves
+veracity of the classification process.
+2) FAABOS categories: The EMAHA-DB1 is mapped ac-
+cording to function arm activity behavioral observation system
+(FAABOS) [38], [29]. Specifically, actions in the EMAHA-
+DB1 are reorganized into the following five major groups:
+1) No object action, 2) object holding, 3) object grasping, 4)
+Flexion and Extension of Fingers, and 5) writing. The action
+categories that are mapped into these groups are listed in Table
+V.
+
+Multi-channel sEMG signals
+Output hand activity classification
+A sample of hand
+movements performed in
+Learning
+Activities of Daily Living
+kinematic
+(ADL)
+characteristics of -
+uu m d
+the hand activities
+ML classifier
+Testing
+training (KNN, RF,
+SKNN, SVM3)
+Grasping and holding
+Grasping and holding
+small hammer
+a book
+Preprocessing
+O
+Training
+Notch filtering at 50Hz
+Feature set visualisation by
+sEMG Data acquired for
+Low pass filtering with
+fc = 500Hz
+t-SNE plot
+ADL by Noraxon Ultium
+Wavelet denoising
+Feature extraction with six
+sEMG sensor setup
+4
+feature sets of F0, F1, F2, F3,
+Trial wise
+segmentation
+3
+F4, F5, and F6
+2
+LO
+1
+0
+-1
+Data train and test split-up
+3
+Train data with
+Test data with
+4
+trial no.
+trial no. 2, 5
+1,3,4,6,8,9 and 10
+and 7
+6
+Datset is curated and
+-4
+-2
+0
+Dimension1
+labelled using audio cue
+ Relabelled by an
+algorithm4
+TABLE IV: Comparisons of basic data characteristics with benchmark datasets
+Dataset
+Name
+Action categories
+Sensor
+No.
+of
+Subjects
+(S)
+No.
+of
+activities
+(NA)
+(including
+rest)
+No.
+of
+channels
+(Nc)
+Sampling
+frequency
+(Ns)
+(samples
+per sec)
+Rest
+dura-
+tion
+(TX)(s)
+Action
+dura-
+tion
+(TA)(s)
+Release
+dura-
+tion
+(TR)(s)
+No.
+of
+repe-
+titions
+(NR)
+Total
+no.
+of
+pat-
+terns
+(N)
+NinaPro
+DB1 [39]
+Gestures, Wrist move-
+ments,
+and
+Grasping
+Objects
+Otto Bock
+27
+53
+10
+100
+3
+5
+-
+10
+14310
+NinaPro
+DB2 [39]
+Gestures, Wrist move-
+ments, Grasping Ob-
+jects, and Finger press-
+ing movements
+Delsys Trigno
+wireless
+40
+50
+12
+2000
+3
+5
+-
+6
+12000
+NinaPro
+DB4 [45]
+Gestures, Wrist move-
+ments,
+and
+Grasping
+Objects
+Cometa Mini-
+Wave
+10
+53
+12
+2000
+3
+5
+-
+6
+3180
+BioPatRec
+DB2 [40]
+Gestures,
+and
+Wrist
+and hand movements
+Thalmic
+myoarm band
+17
+27
+8
+2000
+3
+3
+-
+3
+1377
+UCI Ges-
+ture [42]
+Wrist and hand move-
+ments
+Myo Thalmic
+bracelet
+36
+7
+8
+1000
+3
+3
+-
+4
+1008
+Rami-
+kushaba
+DB6 [41]
+Hand movements
+Delsys DE
+2.x series
+EMG sensors
+11
+40
+7
+4000
+3-5
+5
+-
+6
+2640
+EMAHA-
+DB1
+(Our
+dataset)
+Daily activities -
+Grasping and
+holding, writing, and
+draw open/close
+Noraxon Ul-
+tium
+sEMG
+sensor
+25
+22
+5
+2000
+3-5
+5-15
+3-5
+10
+5500
+Fig. 2: Illustration of manual segmentation of sEMG signals for a trial of ADL
+TABLE V: FAABOS groups of activities.
+Group label
+Group Name
+Activity No.
+0
+Rest
+A0
+1
+No object action
+A2, A9 and A10
+2
+Hold object
+A3, A4, A5, A6, A7 and A8
+3
+Object grasping
+A11, A12, A13, A16, A18,
+A19, A20 and A21
+4
+Flexion
+and
+Exten-
+sion of Fingers
+A1 and A17
+5
+Writing
+A14 and A15
+III. METHODOLOGY
+A. Problem Statement
+The total number of sEMG patterns in the EMAHA-DB1
+is N = S × NA × NR, where S is the total number of
+subjects, NA is the number of different activities, and NR
+corresponds to the number of repetitions per action per subject.
+The proposed sEMG dataset can be represented as:
+x = {xn}N
+n=1
+(1)
+where each observation array xn consists of multiple channels
+as:
+xn = {xn,m}NC
+m=1,
+n = 1, · · · , N
+(2)
+TABLE VI: Summary of extracted features
+Feature
+Set
+Features
+Feature Length
+F0 [47] Mean Absolute Value (MAV), Temporal Spec-
+tral Energies (TSE) and Spectral Band Ener-
+gies (SBE)
+1×NC, 4×NC,
+and 4 × NC
+F1 [48] MAV, Zero Crossings (ZC), Slope Changes
+(SC), and Wavelength (WL)
+1×NC, 1×NC,
+1×NC, and 1×
+NC
+F2 [49] F1 and Auto Regression Coefficients (ARC)
+9 × NC and 2 ×
+NC
+F3 [50] F1, Myopulse Rate (MPR), Willison Ampli-
+tude (WAMP), and Cardinality
+9×NC, 1×NC,
+1×NC, and 1×
+NC
+F4 [51] Log moments in frequency domain (LMF)
+5 × NC
+F5 [52] F4, modified LMF, Time domain statistics
+(TDS), Spectral Band Powers (SBP), Max
+channel cross correlations, and Local Binary
+Patterns (LBP)
+5 × NC, 10 ×
+NC, 4 × NC,
+4×NC, 2×NC,
+and 2 × NC
+F6 [53] Root Mean Square (RMS), Time Dependent
+Power spectrum Descriptors (TD-PSD) [51],
+Difference Absolute Standard Deviation Value
+(DASDV), and Difference Absolute Mean
+Value (DAMV)
+1×NC, 6×NC,
+1×NC, and 1×
+NC
+where NC is the number of channels (from different elec-
+trodes) and each of these channels consists of an array as:
+xn,m = {xn,m(i)}NT
+i=1
+(3)
+where NT = Ns × TT is the number of values in one trial of
+duration TT and Ns is the sampling rate (samples/sec). For a
+given trial, for feature extraction purposes, the signal is divided
+into Nseg segments. Each segment sg consists of an array as:
+sj
+g = {xn,m(i)}Ng
+i=1
+j = 1, · · · Nseg
+(4)
+where Ng is the number of samples in one segment such that
+NT = Nseg × Ng.
+The objective of this study is to map the segmented sEMG
+signals to the corresponding activity labels (i.e., tg - targets),
+
+Channel 3
+cue-ON
+6
+Cue-OFF
+OFF
+40
+SEMG
+20
+0
+2000
+4000
+6000
+8000
+10000
+12000
+14000
+16000
+Channel 4
+10
+cue-OFFl
+cue-ON
+NO
+OFF
+sEMG Amp
+５
+0
+8000
+2000
+4000
+6000
+10000
+12000
+14000
+16000
+0
+Rest vs. Action
+Action-OFF
+SEMG
+Action-
+norm
+0
+2000
+4000
+6000
+8000
+10000
+12000
+14000
+16000
+0
+Sample Index5
+(a)
+(b)
+(c)
+(d)
+(e)
+Fig. 3: Performance comparison (a) with different Feature Ensembles with Cubic SVM (Polynomial SVM of order 3), (b) with different classifiers for the benchmark feature
+ensemble F5, (c) against benchmark frameworks, (d) against benchmark frameworks in terms of various metrics, and (e) against FAABOS categories frameworks.
+TABLE VII: Numerical setup for classifiers.
+Classifier
+Model Setup
+Fine KNN
+No.of neighbours = 5, Distance Metric = Cityblock, Dis-
+tance weight = Squared Inverse
+Ensemble
+KNN
+No.of learning cycles = 30, learners = KNN, Subspace
+dimension = 25
+Cubic SVM
+Polynomial kernel, Order = 3, Box constraint = 1, Multi-
+class Method = one-vs-one
+Random Forest No.of bags for bootstrapping = 300
+which can be formulated as:
+f{sg} → tg
+(5)
+where tg denotes targets (group labels) as specified in TABLE
+V or individual activity labels as specified in TABLE I. The
+mapping function in (5) is implemented by a supervised
+classifier. For the mapping function, appropriate features are
+required that represent the underlying inverse kinematic rela-
+tionships between the sEMG signals and the corresponding
+activity performed.
+B. Feature Extraction
+In this work, the following feature sets are adapted from
+[47]: F0, F1, F2, F3, F4, and F5 with an additional feature
+set F6 consisting of root mean square (RMS), time dependent
+power spectrum descriptors (TD-PSD), difference absolute
+standard deviation value (DASDV), and difference absolute
+mean value (DAMV). Note the features are computed for each
+segment and concatenated to build the full feature vector. The
+extracted feature sets are summarized in Table VI.
+C. Supervised Classifiers
+In this paper, four algorithms including random forest (RF),
+fine K-nearest neighbour (FKNN), ensemble KNN (sKNN)
+and cubic support vector machine (SVM3) are considered for
+sEMG signal classification task. The classifiers are trained and
+TABLE VIII: Feature ensemble vs benchmark classifier setup.
+FE FL
+BF
+Classifier
+FE FL
+BF
+Classifier
+F0
+9 × NC
+B0 [47]
+Fine KNN
+F4
+5 × NC
+B4 [54]
+SVM3
+F1
+4 × NC
+B1 [48]
+SVM3
+F5
+27×NC
+B5 [52]
+SVM3
+F2
+11×NC
+B2 [49]
+Fine KNN
+F6
+9 × NC
+B6 [39]
+RF
+F3
+12×NC
+B3 [50]
+SVM3
+tested with subject-wise data and the average performance is
+reported. The hyperparameter settings for different machine
+learning algorithms used in this work are summarized in Table
+VII. The performance of classifiers is evaluated using the
+standard metrics such as cross validation accuracy (α), testing
+accuracy (β), Kappa coefficient (κ), precision (γ), recall (ρ)
+and F-1 score (F1).
+IV. CLASSIFICATION EXPERIMENTS, RESULTS &
+ANALYSIS
+The developed EMAHA-DB1 sEMG dataset is analyzed
+using the state-of-the-art classification and feature extraction
+methods as detailed below.
+A. Pre-processing and Data Split-up
+Based on the procedure described in [55], the recorded
+sEMG data is pre-processed as follows. First, the sEMG data is
+filtered to remove power line noise at 50Hz. Then a first order
+Butterworth low pass filter is applied at a cut-off frequency of
+500Hz. Finally, wavelet denoising of order 8 with the symlet
+as the mother wavelet is applied. The data from each subject
+is split trials-wise into 70% for training and 30% for testing as
+per the splitting method in [55]. A non overlapping moving
+window segment of Ng = 200 samples is considered with
+duration Tseg = 100ms. The number of features obtained per
+segment sg are summarized in Table VIII.
+
+80
+Cross Validation (α)
+Testing (B)
+75
+(%)
+Accuracy
+70
+65
+60
+F2
+F3
+F1
+F4
+F5
+F6
+F0
+Feature sets80
+Cross Validation (α)
+Testing (β)
+75
+%
+70
+Accuracy
+65
+60
+55
+RF
+SKNN
+SVM3
+FKNN
+Classifiers80
+Cross Validation (α)
+Testing (β)
+75
+(%)
+Accuracy
+70
+65
+60
+B1
+B2
+B3
+B4
+B5
+B0
+B6
+Feature sets0.8
+-Precision ()kappa ()
+*F, score (F,)
+Recall (p)
+0.75
+Metric
+0.7
+rmance
+0.65
+0.6
+0.55
+B2
+B3
+B4
+B5
+B6
+B0
+B1
+Benchmarks85
+Feature set F0
+Feature set F5
+Feature set F2
+80
+75
+70
+65
+RF
+SKNN
+SVM3
+FKNN
+Classifier6
+TABLE IX: Muscle vs action mapping.
+Muscle
+Major functionality of the mus-
+cle
+Biceps Brachii (BB) muscle
+Flexes elbow joint, Supinates fore-
+arm and hand at radioulnar joint
+Brachio Radialis (BR) muscle
+Flexes elbow joint
+Flexor Carpi Radialis (FCR) muscle
+Flexes and abducts hand at wrist
+Flexor Carpi Ulnaris (FCU) muscle
+Flexes and adducts wrist
+Abductor Pollocis Brevis (APB) muscle Abducts joints of thumb
+B. Experiments
+In this paper, as mentioned earlier two case studies are
+carried out as follows, 1) classification of individual action
+categories listed in Table I, in this case study, the performance
+is analyzed with respect to feature ensembles, classifiers,
+benchmark classification frameworks and finally feature vi-
+sualization; 2) classification of FAABOS categories listed in
+Table V, in the second case study, the performance is analyzed
+with respect to feature ensembles followed by an analysis of
+the most relevant features with respect to the muscle sites.
+C. Case Study 1: Results and Analysis
+1) Comparison with feature ensembles: The feature sets
+F0-F6 are analyzed in this comparison study. Each of the
+feature set is utilised as input for SVM3 and their performance
+metrics α and β are evaluated. As shown in Fig. 3a, the best
+performance is produced by the feature set F5 (α = 77.42 and
+β = 75.39). The next best feature ensemble F2 lags behind by
+0.3% at α = 78.06 and β = 75.09. The feature ensemble F6
+has produced the least classification performance (α = 66.68
+and β = 66.79).
+2) Comparison with classifiers: In this experiment, the
+classification performance of the standard machine learning
+algorithms such as the RF, FKNN, sKNN and SVM3 using
+the F5 feature set is analyzed. As shown in Fig. 3b, the best
+performance is produced by the SVM3 classifier (α = 77.42
+and β
+= 75.39) and then by FKNN (α = 74.83 and
+β = 72.42). The least performance is obtained with SKNN
+classifier (α = 58.4 and β = 58.3). Thus, it is observed from
+this experiment that for the feature set F5 the SVM3 classifier
+outperforms other benchmark classifiers.
+3) Comparison with benchmark algorithms: The most suit-
+able classification framework for the EMAHA-DB1 is deter-
+mined by comparisons with the existing sEMG benchmark
+classification methods consisting of respective combinations
+of a feature ensemble and a classification framework as listed
+in Table VIII. The benchmark Bi indicates the classification
+framework with feature set Fi for i = 0, 1, · · · , 6. The param-
+eter setups of the different classifiers used in the numerical
+analysis are also shown in Table VII. The performance of these
+classifiers is analyzed based on the cross validation accuracy
+(α) and the test accuracy (β) with the corresponding results
+shown in Fig. 3c. The benchmark B5 has achieved state-of-
+the-art performance with α = 77.42 and β = 75.39. The lowest
+performance among the compared benchmarks is for B6 with
+α = 74.2 and β = 69.04. The other performance metrics (i.e., κ,
+γ, ρ, and F1) of the benchmark frameworks are shown in Fig.
+3d. The benchmark framework F5 has achieved highest values
+for each of the performance metrics, i.e., κ = 0.73, γ=0.66,
+ρ=0.71, and F1 = 0.68. The runner-up is B6 framework with
+metric values κ = 0.66, γ= 0.60, ρ= 0.64, and F1 = 0.66.
+4) Feature Visualization by t-SNE: The following analysis
+is meant for the 22 individual action categories however
+carried out FAABOS group wise. Among the feature sets F0
+to F6, it is observed that F5 is the best performing feature
+set, hence used for sequential feature selection analysis (SFS).
+From SFS, the most relevant features for each group of hand
+activities are identified and further used for analysis with
+t-distributed Stochastic Neighbourhood Embedding (t-SNE)
+[56]. The top 6 feature columns of 84, 85, 96, 97, 101, and
+105 are used in this study. The columns with higher ranking
+are 84 and 85 that correspond to the features of mean and
+variance respectively (from TDS feature set), and 96, 97, 101,
+and 105 that correspond to the spectral bands [0 (Ns/8)] and
+[(Ns/8) (Ns/4)] of SBP feature set [47]. The flexion and
+extension of elbow and wrist flexion and extension are mainly
+supported by the muscle groups BB, BR, FCR and FCU [57]
+as given in TABLE IX. The action categories in group 2 and
+group 3 involve the common muscle movements including
+elbow flexion and extension, wrist flexion and extension and
+pronation and supination as shown in TABLE X. Hence From
+Fig. 4b and 4c, the clusters for some of the actions overlap
+due to involvement of similar muscle groups across actions
+with same basic muscle movements. The actions within group
+1, group 4 and group 5 are clearly separable which can be
+observed from Fig. 4a, Fig. 4d and Fig. 4e, respectively.
+D. Case Study 2: Results and Analysis
+1) Comparison of FAABOS categories with feature ensem-
+bles: The sEMG signals from the EMAHA-DB1 are classified
+based on FAABOS categories specified in Table V. The six
+FAABOS categories of sEMG signals are trained and tested
+with the top three feature sets such as F0, F2 and F5 and
+the corresponding results are plotted in Fig. 3e. The best
+performance is produced by the SVM3 classifier (α = 86.54
+and β = 83.21) with the feature set F2. The next best
+performance is produced by the same SVM3 classifier (α =
+85.85 and β = 83.14), but with the feature set F5 having a
+slight variation of 0.07%. The least performance is observed
+for feature set F0 with SKNN classifier (α = 85.66 and β =
+82.39).
+2) Feature Visualization by t-SNE for FAABOS groups:
+This analysis is carried out for 6 FAABOS categories. Among
+the feature sets F0, F2, and F5, it is observed that F2 is the
+best performing feature set and used for SFS analysis. From
+SFS, the top 6 feature columns 1, 3, 4, 5, 19, and 23 are
+used in this study. The columns with higher ranking are 1,
+3, 4, and 5 that correspond to mean absolute value (MAV),
+19 corresponds to the waveform length, and 23 corresponds
+to auto regressive coefficients. The t-SNE plot is generated
+with high ranking column features as shown in Fig. 5. It is
+observed that the action and rest clusters are clearly separable,
+but clusters within action groups are overlapping due to similar
+muscle group involvement. Based on a recent review of sEMG
+studies of muscle groups and their functions [58], the muscles
+
+7
+(a)
+(b)
+(c)
+(d)
+(e)
+Fig. 4: t-SNE plots of feature set for (a) group 1, (b) group 2, (c) group 3, (d) group 4,
+and (e) group 5, respectively.
+Fig. 5: t-SNE plot of feature set for six FAABOS groups
+FCR, FCU, BR and BB are mapped to the major functions
+involved in each of the FAABOS categories in our study and
+detailed in Table X.
+E. Discussion
+The SVM3 method has the best classification performance
+in case of the FAABOS categories (no. classes = 5). This can
+be explained by relatively less number of classes and ability
+of feature ensemble F5 to better capture the representation at
+functional category level. The ML framework’s performance
+TABLE X: FAABOS group vs actions vs muscle mapping.
+Group
+Major actions involved
+Muscles
+No object ac-
+tion (1)
+Wrist flexion & extension and hand digit ma-
+nipulation
+FCR, FCU,
+BR
+Hold
+object
+(2)
+Elbow flexion & extension, Wrist flexion & ex-
+tension, and Forearm Pronation & Supination
+BB,
+BR,
+FCR, FCU
+Object grasp-
+ing (3)
+Elbow flexion & extension, Wrist flexion &
+extension, Forearm Pronation & Supination,
+and hand digit manipulation
+FCR, FCU,
+BR, BB
+Flexion
+and
+Extension
+of
+Fingers (4)
+Wrist flexion & extension and hand digit ma-
+nipulation
+BB,
+BR,
+FCR, FCU
+Writing (5)
+Elbow flexion & extension, Wrist flexion &
+extension, and hand digit manipulation
+FCR,
+BB,
+FCU, APB
+may need further improvement. This performance can be
+explained by relatively higher number of activities and higher
+intra-class correlations. The feature visualizations with t-SNE
+has shown better separability of activities within FAABOS
+groups. A clear separation between rest and action is also
+observed in t-SNE plot across FAABOS groups.
+V. CONCLUSION & FUTURE SCOPE
+In this paper, we have collected a novel sEMG dataset
+(EMAHA-DB1) of 22 activities of daily living from Indian
+population. The EMAHA-DB1 includes a few activities that
+are not considered in existing datasets. The sEMG EMAHA-
+DB1 dataset is compared against the publicly available ex-
+isting sEMG datasets. The dataset is analyzed from different
+perspectives including feature set analysis in time domain and
+frequency domain, individual action classification, FAABOS
+category classification and feature visualization using t-SNE.
+In the above mentioned analysis, the modified LMF, time
+domain statistical (TDS) feature, spectral band powers (SBP),
+channel cross correlation and local binary patterns (LBP)
+ensemble feature set (F5) with Cubic SVM classifier has
+obtained highest test accuracy of β = 75.39%. Additionally,
+in the FAABOS groups classification, the best performance is
+again produced by the cubic SVM classifier (β = 83.21) with
+the feature set consisting of energy features and auto regressive
+coefficients (F2). Finally, the visual analysis using t-SNE
+plots showed that the extracted feature set is able to clearly
+distinguish the ADL activities within a group. The obtained
+results indicate that the EMAHA-DB1 can be successfully
+used as a benchmark for the development of hand gesture
+recognition system, physiological analysis and clinical studies
+of sEMG for ADL.
+In terms of future work, the framework may need further in-
+novation in terms of features to improve the classification per-
+formance; the EMAHA-DB1 is analysed using only machine
+learning classifiers, there is a scope for improvement with deep
+learning; the dataset can also be analysed by decomposing the
+time series with wavelets or empirical mode decomposition
+(EMD) techniques; finally, the EMAHA-DB1 dataset can also
+be analysed for learning the statistical distributions.
+ACKNOWLEDGMENT
+This research is funded by SERB, Govt. of India under
+Project Grant No. CRG/2019/003801.
+
+50
+40
+30
+20
+Dimension
+10
+12
+10
+13
+16
+-20
+18
+-30
+19
+20
+-40
+21
+-50
+-60
+-40
+-20
+20
+40
+60
+Dimension.
+717
+10
+5
+2
+Dimension 
+5
+-10
+-15
+-20
+-15
+-10
+5
+10
+-5
+15
+Dimension.40
+14
+15
+30
+20
+Dimension
+10
+-10
+-20
+-20
+-10
+20
+30
+10
+0
+Dimension 180
+0
+60
+1
+2
+40
+3
+4
+20
+5
+Dimension
+0
+-20
+-40
+-60
+-80
+-100
+-100
+-50
+50
+100
+0
+Dimension 12
+3
+9
+10
+2
+1
+Dimension
+.2
+-3
+-5
+6
+2
+2
+4
+8
+0
+6
+Dimension20
+3
+4
+15
+5
+6
+10
+7
+8
+5
+Dimension
+5
+-10
+-15
+-20
+-10
+10
+15
+-5
+5
+20
+08
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+

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+Relevance Classification of Flood-related Twitter Posts
+via Multiple Transformers
+Wisal Mukhtiar1,†, Waliiya Rizwan1,†, Aneela Habib1,†, Yasir Saleem Afridi1,
+Laiq Hasan1 and Kashif Ahmad2
+1Department of Computer Systems Engineering, University of Engineering and Technology, Peshawar, Pakistan.
+2Department of Computer Science, Munsters Technological University, Cork, Ireland.
+Abstract
+In recent years, social media has been widely explored as a potential source of communication and informa-
+tion in disasters and emergency situations. Several interesting works and case studies of disaster analytics
+exploring different aspects of natural disasters have been already conducted. Along with the great potential,
+disaster analytics comes with several challenges mainly due to the nature of social media content. In this
+paper, we explore one such challenge and propose a text classification framework to deal with Twitter noisy
+data. More specifically, we employed several transformers both individually and in combination, so as to
+differentiate between relevant and non-relevant Twitter posts, achieving the highest F1-score of 0.87.
+1. Introduction
+Natural disasters, which are hazardous events and occur frequently in different parts of the world,
+can have devastating effects on society. Depending on the severity of the disaster, it may result in
+significant damage to the infrastructure and human lives. Rapid response to natural disasters may
+help in mitigating their adverse impact on society. In disasters and emergency situations, access
+to relevant and timely information is key to a rapid and effective response. However, the literature
+reports several situations where access to relevant and timely information may not be possible
+due to several factors [1].
+In recent years, social media outlets, such as Twitter, Facebook, and Instagram, have been
+explored as a source of communication and information dissemination in emergency situations
+[2]. The literature already reports the feasibility and effectiveness of social media for a diversified
+list of tasks in disaster analytics. For instance, Ahmad et al. [3] explored social media outlets as a
+source of information collection and dissemination during natural disasters by proposing a system
+that is able to collect and analyze disaster-related multimedia content from social media. Similarly,
+social media content has also been utilized for disaster severity and damage assessment [4, 5].
+Despite being very effective in disaster analytics, social media data also come with several
+limitations. For instance, social media content contains a lot of noise and irrelevant information.
+This paper targets one of such challenges by proposing several solutions for the Relevance Classi-
+fication of Twitter Posts (RCTP), sub-task introduced in DisasterMM challenge of MediaEval 2022
+MediaEval’22: Multimedia Evaluation Workshop, January 13–15, 2023, Bergen, Norwa,y and Online
+*Corresponding author.
+†These authors contributed equally.
+� kashif.ahmad@mtu.ie (K. Ahmad)
+© 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
+CEUR
+Workshop
+Proceedings
+http://ceur-ws.org
+ISSN 1613-0073
+CEUR Workshop Proceedings (CEUR-WS.org)
+arXiv:2301.00320v1  [cs.CL]  1 Jan 2023
+
+[6]. The task aims at automatically analyzing and classifying flood-related tweets into relevant
+and non-relevant tweets.
+2. Related Work
+Disaster analysis in social media content has been one of the active topics of research in the
+domain over the last few years [2]. During this time, different aspects and applications of disaster
+analytics in social media content have been explored [7]. Some key applications include com-
+munication/information dissemination, damage assessment, response management, sentiment
+analysis, and identification of the needs of affected individuals. The literature already reports
+several interesting works on these applications. For instance, Nguyen et al. [8] utilized social
+media content for damage assessment by analyzing disaster-related visual media posts. Ahmad
+et al. [9] analyzed social media imagery for monitoring road conditions after floods. Moreover,
+a vast majority of the literature demonstrates how social media outlets can be used as means of
+communication in disasters and emergency situations [10, 1].
+In the literature, different types of disasters including natural disasters, such as earthquakes,
+landslides, droughts, wildfires, and floods, as well as man-made disasters, such as accidents, have
+been explored [1, 11]. However, the majority of the works have targeted floods, being one of
+the most common natural disasters. The literature reports several interesting works on flood
+analysis in social media content for different tasks. For instance, Ahmad et al. [9] proposed a
+late fusion-based framework for the automatic detection of passable roads after a flood. For this
+purpose, several deep learning models are trained on flood-related images from social media. Alam
+et al. [4], on the other hand, employed social media imagery for post floods damage severity
+assessment.
+Flood detection and analysis in social content have also been a part of the MediaEval benchmark
+initiative as a shared task for several years. Each time a separate aspect of flood analysis has
+been explored. For instance, in MediaEval 2017 the task aimed at the retrieval of flood-related
+images from social media. The task mainly involved analyzing the water level in different areas to
+differentiate between floods and regular water reservoirs, such as lakes [12]. In MediaEval 2018,
+the task was slightly modified by asking the participants to propose multi-modal classification
+frameworks for flood-related multimedia content [13]. In MediaEval 2019 and 2020, the tasks
+aimed at analyzing flood severity and flood events recognition in social media posts.
+3. Approach
+Figure 1 provides the block diagram of the proposed framework for the RCTP task. The framework
+is composed of three main components namely (i) Pre-processing, (ii) Training and Classification,
+and (iii) Fusion. In the first step, different pre-processing techniques are employed to clean the
+dataset. Three different transformers are then trained on the data to obtain classification scores.
+In the final step, the classification scores of the individual models are combined in a late fusion
+scheme. The details of these steps are provided below.
+
+Figure 1: Block diagram of the proposed approach.
+3.1. Pre-processing
+In the pre-processing step, we employed different techniques for cleaning the dataset. More
+specifically, we removed unnecessary information, such as user names, URLs, emojis, punctuation
+marks, stop words, etc. Besides this, we also performed the necessary pre-possessing tasks that
+are required to transform the raw text into a form that is suitable for the transformers. To achieve
+this, we used the TF.text library1.
+3.2. Classification via Transformers
+After cleaning and pre-processing the data, we trained three different models, namely BERT [14],
+RoBERTa [15], and XLNet [16]. The selection of these models for the task is motivated by their
+proven performance on similar tasks [17]. A brief overview of these models is provided below.
+• BERT: Bidirectional Encoder Representations from Transformers (BERT) is one of the state-
+of-the-art NLP algorithms for text processing. The model is pre-trained on a large collection
+of unlabeled text and can be fine-tuned for different text-analysis applications. The key
+attributes of the model include its bi-directional nature, pre-training with Masked Language
+Modeling (MLM), and Next Structure Prediction (NSP) objectives. In the experiments with
+BERT, we used the Adam optimizer with a learning rate of 0.001 and a batch size of 8 for 3
+epochs.
+• RoBERTa: Robustly Optimized BERT is a modified version of the BERT model with an
+improved training mechanism. More specifically, in RoBERTa the NSP capabilities are
+removed. Moreover, dynamic masking is introduced. In addition, a larger batch size and a
+larger amount of training data were used in the training process. In this work, we used a
+learning rate of 0.001, batch size of 20, and 10 epochs during the fine-tuning of the model
+for the desired task.
+• XLNet: XLNet is another state-of-the-art NLP algorithm. Similar to BERT, XLNet is also
+a bidirectional transformer and uses an improved training approach. In contrast to BERT
+and traditional NLP algorithms, XLNet relies on Permutation Language Modeling (PLM) by
+predicting all the tokens in random order. This allows XLNet to handle dependencies and
+bidirectional relationships in a better way. In this work, we used a learning rate of 0.002, a
+batch size of 32, and 4 epochs during the fine-tuning of the model for the desired task.
+1https://www.tensorflow.org/text/guide/bert_preprocessing_guide#text_preprocessing_with_tftext#
+
+Input Data
+Data Pre-processing
+Classification
+Late Fusion
+Model 1
+F = S1+S2....Sn
+Score obtained with M2
+TextStreams
+Pre-processing
+Model 2
+Mn
+Final Score
+Score
+Model NWe obtained the results in the form of posterior probabilities from these models, which are then
+used in the fusion scheme to obtain the final predicted labels. The fusion method used in this work
+is described in the next section.
+3.3. Fusion
+Our fusion method is based on late fusion, where we combined the classification scores obtained
+with the individual models for the final classification decision as shown in Equ. 1. In the equation,
+𝑆𝑓𝑖𝑛𝑎𝑙 represents the final classification score while 𝑠𝑛 is the score obtained with the nth model.
+We note that in the current implementation, we used a simple fusion method by treating all the
+models equally (i.e., simple aggregation of the individual scores).
+𝑆𝑓𝑖𝑛𝑎𝑙 = 𝑆1 + 𝑆2 + 𝑠3 + .... + 𝑆𝑛
+(1)
+4. Results and Analysis
+Table 1 provides the experimental results of the proposed solutions on the development set. As
+can be been in the table, overall better results are obtained with the BERT model, and surprisingly,
+a lower F1-score is observed for RoBERTa. In the future, we will further investigate the potential
+causes of the lower performance of RoBERTa by exploring different implementations and hyper-
+parameter settings for it. As far as the performance of the fusion methods is concerned, overall
+better results are obtained with the pair of XLNet and BERT. One of the potential reasons for the
+lower performance of the fusion of all the models is the less accurate prediction of RoBERTa, as
+also evident from the performance of the individual models.
+Table 1
+Experimental results of the proposed solutions on the development set.
+Method
+F1-Score
+BERT
+0.94
+RoBERTa
+0.78
+XLNet
+0.93
+Fusion 1 (RoBERTa, BERT, XLNet)
+0.75
+Fusion 2 (BERT, XLNet)
+0.93
+Fusion 3 (RoBERTa, XLNet)
+0.92
+Table 2 provides the official results of the proposed solutions on the test set. In total, three
+different runs were submitted. The first run is based on the fusion of all three models used in this
+work. The remaining two runs are based on the fusion of the models in pairs of two. In run 2,
+BERT and XLNet are combined while in run 3 RoBERTa and XLNet are jointly used. As can be
+seen in the table, better results are obtained for the fusion of the models in pairs of two where the
+best performing pair of two models obtained an improvement of 20% over the fusion of all three
+models.
+
+Table 2
+Experimental results of the proposed solutions on the test set.
+Run
+Precision
+Recall
+F1-Score
+1 (Fusion of BERT, RoBERTa, XLNet)
+0.6738
+0.5431
+0.6014
+2 (Fusion of BERT and XLNet)
+0.8044
+0.6948
+0.7456
+3 (Fusion of RoBERTa and XLNet)
+0.8977
+0.8598
+0.8784
+5. Conclusions
+In this paper, we presented our solutions for the RCTP subtask of DisasterMM challenge posted
+in MediaEval 2022. We proposed a late fusion framework incorporating several state-of-the-art
+transformers for the task. In the current implementation, all the models are treated equally by
+assigning them equal weights (i.e., 1). In the future, we aim to employ merit-based fusion methods
+to further improve the final classification score.
+References
+[1] K. Ahmad, K. Pogorelov, M. Riegler, N. Conci, P. Halvorsen, Social media and satellites, Multimedia
+Tools and Applications 78 (2019) 2837–2875.
+[2] N. Said, K. Ahmad, M. Riegler, K. Pogorelov, L. Hassan, N. Ahmad, N. Conci, Natural disasters
+detection in social media and satellite imagery: a survey, Multimedia Tools and Applications 78 (2019)
+31267–31302.
+[3] K. Ahmad, M. Riegler, A. Riaz, N. Conci, D.-T. Dang-Nguyen, P. Halvorsen, The jord system: Linking
+sky and social multimedia data to natural disasters, in: Proceedings of the 2017 ACM on International
+Conference on Multimedia Retrieval, 2017, pp. 461–465.
+[4] F. Alam, M. Imran, F. Ofli, Image4act: Online social media image processing for disaster response, in:
+Proceedings of the 2017 IEEE/ACM international conference on advances in social networks analysis
+and mining 2017, 2017, pp. 601–604.
+[5] F. Alam, F. Ofli, M. Imran, Crisismmd: Multimodal twitter datasets from natural disasters, in: Twelfth
+international AAAI conference on web and social media, 2018.
+[6] S. Andreadis, A. Bozas, I. Gialampoukidis, A. Moumtzidou, R. Fiorin, F. Lombardo, T. Mavropoulos,
+D. Norbiato, S. Vrochidis, M. Ferri, I. Kompatsiaris, DisasterMM: Multimedia Analysis of Disaster-
+Related Social Media Data Task at MediaEval 2022, in: Proceedings of the MediaEval 2022 Workshop,
+Bergen, Norway and Online, 2023.
+[7] F. Ofli, M. Imran, F. Alam, Using artificial intelligence and social media for disaster response and
+management: an overview, AI and Robotics in Disaster Studies (2020) 63–81.
+[8] D. T. Nguyen, F. Ofli, M. Imran, P. Mitra, Damage assessment from social media imagery data during
+disasters, in: Proceedings of the 2017 IEEE/ACM international conference on advances in social
+networks analysis and mining 2017, 2017, pp. 569–576.
+[9] K. Ahmad, K. Pogorelov, M. Riegler, O. Ostroukhova, P. Halvorsen, N. Conci, R. Dahyot, Automatic
+detection of passable roads after floods in remote sensed and social media data, Signal Processing:
+Image Communication 74 (2019) 110–118.
+[10] L. Palen, A. L. Hughes, Social media in disaster communication, Handbook of disaster research (2018)
+497–518.
+[11] K. Ahmad, A. Sohail, N. Conci, F. De Natale, A comparative study of global and deep features for
+the analysis of user-generated natural disaster related images, in: 2018 IEEE 13th image, video, and
+multidimensional signal processing workshop (IVMSP), IEEE, 2018, pp. 1–5.
+
+[12] B. Bischke, P. Helber, C. Schulze, V. Srinivasan, A. Dengel, D. Borth, The multimedia satellite task at
+mediaeval 2017., in: MediaEval, 2017.
+[13] B. Benjamin, H. Patrick, Z. Zhengyu, B. Damian, et al., The multimedia satellite task at mediaeval
+2018: Emergency response for flooding events (2018).
+[14] J. Devlin, M.-W. Chang, K. Lee, K. Toutanova, Bert: Pre-training of deep bidirectional transformers for
+language understanding, arXiv preprint arXiv:1810.04805 (2018).
+[15] Y. Liu, M. Ott, N. Goyal, J. Du, M. Joshi, D. Chen, O. Levy, M. Lewis, L. Zettlemoyer, V. Stoyanov,
+Roberta: A robustly optimized bert pretraining approach, arXiv preprint arXiv:1907.11692 (2019).
+[16] Z. Yang, Z. Dai, Y. Yang, J. Carbonell, R. R. Salakhutdinov, Q. V. Le, Xlnet: Generalized autoregressive
+pretraining for language understanding, Advances in neural information processing systems 32 (2019).
+[17] K. Ahmad, M. Ayub, J. Khan, N. Ahmad, A. Al-Fuqaha, Social media as an instant source of feedback
+on water quality, IEEE Transactions on Technology and Society (2022).
+

CtAyT4oBgHgl3EQfefgG/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf,len=270
+page_content='Relevance Classification of Flood-related Twitter Posts  via Multiple Transformers  Wisal Mukhtiar1,†, Waliiya Rizwan1,†, Aneela Habib1,†, Yasir Saleem Afridi1,  Laiq Hasan1 and Kashif Ahmad2  1Department of Computer Systems Engineering, University of Engineering and Technology, Peshawar, Pakistan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  2Department of Computer Science, Munsters Technological University, Cork, Ireland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  Abstract  In recent years, social media has been widely explored as a potential source of communication and informa-  tion in disasters and emergency situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Several interesting works and case studies of disaster analytics  exploring different aspects of natural disasters have been already conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Along with the great potential,  disaster analytics comes with several challenges mainly due to the nature of social media content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' In this  paper, we explore one such challenge and propose a text classification framework to deal with Twitter noisy  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' More specifically, we employed several transformers both individually and in combination, so as to  differentiate between relevant and non-relevant Twitter posts, achieving the highest F1-score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Introduction  Natural disasters, which are hazardous events and occur frequently in different parts of the world,  can have devastating effects on society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Depending on the severity of the disaster, it may result in  significant damage to the infrastructure and human lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Rapid response to natural disasters may  help in mitigating their adverse impact on society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' In disasters and emergency situations, access  to relevant and timely information is key to a rapid and effective response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' However, the literature  reports several situations where access to relevant and timely information may not be possible  due to several factors [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  In recent years, social media outlets, such as Twitter, Facebook, and Instagram, have been  explored as a source of communication and information dissemination in emergency situations  [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' The literature already reports the feasibility and effectiveness of social media for a diversified  list of tasks in disaster analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' For instance, Ahmad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' [3] explored social media outlets as a  source of information collection and dissemination during natural disasters by proposing a system  that is able to collect and analyze disaster-related multimedia content from social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Similarly,  social media content has also been utilized for disaster severity and damage assessment [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  Despite being very effective in disaster analytics, social media data also come with several  limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' For instance, social media content contains a lot of noise and irrelevant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  This paper targets one of such challenges by proposing several solutions for the Relevance Classi-  fication of Twitter Posts (RCTP), sub-task introduced in DisasterMM challenge of MediaEval 2022  MediaEval’22: Multimedia Evaluation Workshop, January 13–15, 2023, Bergen, Norwa,y and Online  Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  †These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  � kashif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='ahmad@mtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='ie (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Ahmad)  © 2022 Copyright for this paper by its authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Use permitted under Creative Commons License Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='0 International (CC BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  CEUR  Workshop  Proceedings  http://ceur-ws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='org  ISSN 1613-0073  CEUR Workshop Proceedings (CEUR-WS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='org)  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='00320v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='CL]  1 Jan 2023  [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' The task aims at automatically analyzing and classifying flood-related tweets into relevant  and non-relevant tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Related Work  Disaster analysis in social media content has been one of the active topics of research in the  domain over the last few years [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' During this time, different aspects and applications of disaster  analytics in social media content have been explored [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Some key applications include com-  munication/information dissemination, damage assessment, response management, sentiment  analysis, and identification of the needs of affected individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' The literature already reports  several interesting works on these applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' For instance, Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' [8] utilized social  media content for damage assessment by analyzing disaster-related visual media posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Ahmad  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' [9] analyzed social media imagery for monitoring road conditions after floods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Moreover,  a vast majority of the literature demonstrates how social media outlets can be used as means of  communication in disasters and emergency situations [10, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  In the literature, different types of disasters including natural disasters, such as earthquakes,  landslides, droughts, wildfires, and floods, as well as man-made disasters, such as accidents, have  been explored [1, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' However, the majority of the works have targeted floods, being one of  the most common natural disasters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' The literature reports several interesting works on flood  analysis in social media content for different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' For instance, Ahmad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' [9] proposed a  late fusion-based framework for the automatic detection of passable roads after a flood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' For this  purpose, several deep learning models are trained on flood-related images from social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Alam  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' [4], on the other hand, employed social media imagery for post floods damage severity  assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  Flood detection and analysis in social content have also been a part of the MediaEval benchmark  initiative as a shared task for several years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Each time a separate aspect of flood analysis has  been explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' For instance, in MediaEval 2017 the task aimed at the retrieval of flood-related  images from social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' The task mainly involved analyzing the water level in different areas to  differentiate between floods and regular water reservoirs, such as lakes [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' In MediaEval 2018,  the task was slightly modified by asking the participants to propose multi-modal classification  frameworks for flood-related multimedia content [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' In MediaEval 2019 and 2020, the tasks  aimed at analyzing flood severity and flood events recognition in social media posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Approach  Figure 1 provides the block diagram of the proposed framework for the RCTP task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' The framework  is composed of three main components namely (i) Pre-processing, (ii) Training and Classification,  and (iii) Fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' In the first step, different pre-processing techniques are employed to clean the  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Three different transformers are then trained on the data to obtain classification scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  In the final step, the classification scores of the individual models are combined in a late fusion  scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' The details of these steps are provided below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  Figure 1: Block diagram of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Pre-processing  In the pre-processing step, we employed different techniques for cleaning the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' More  specifically, we removed unnecessary information, such as user names, URLs, emojis, punctuation  marks, stop words, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Besides this, we also performed the necessary pre-possessing tasks that  are required to transform the raw text into a form that is suitable for the transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' To achieve  this, we used the TF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='text library1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Classification via Transformers  After cleaning and pre-processing the data, we trained three different models, namely BERT [14],  RoBERTa [15], and XLNet [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' The selection of these models for the task is motivated by their  proven performance on similar tasks [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' A brief overview of these models is provided below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  BERT: Bidirectional Encoder Representations from Transformers (BERT) is one of the state-  of-the-art NLP algorithms for text processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' The model is pre-trained on a large collection  of unlabeled text and can be fine-tuned for different text-analysis applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' The key  attributes of the model include its bi-directional nature, pre-training with Masked Language  Modeling (MLM), and Next Structure Prediction (NSP) objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' In the experiments with  BERT, we used the Adam optimizer with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='001 and a batch size of 8 for 3  epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  RoBERTa: Robustly Optimized BERT is a modified version of the BERT model with an  improved training mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' More specifically, in RoBERTa the NSP capabilities are  removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Moreover, dynamic masking is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' In addition, a larger batch size and a  larger amount of training data were used in the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' In this work, we used a  learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='001, batch size of 20, and 10 epochs during the fine-tuning of the model  for the desired task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  XLNet: XLNet is another state-of-the-art NLP algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Similar to BERT, XLNet is also  a bidirectional transformer and uses an improved training approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' In contrast to BERT  and traditional NLP algorithms, XLNet relies on Permutation Language Modeling (PLM) by  predicting all the tokens in random order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' This allows XLNet to handle dependencies and  bidirectional relationships in a better way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' In this work, we used a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='002, a  batch size of 32, and 4 epochs during the fine-tuning of the model for the desired task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='tensorflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='org/text/guide/bert_preprocessing_guide#text_preprocessing_with_tftext#  Input Data  Data Pre-processing  Classification  Late Fusion  Model 1  F = S1+S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='.Sn  Score obtained with M2  TextStreams  Pre-processing  Model 2  Mn  Final Score  Score  Model NWe obtained the results in the form of posterior probabilities from these models, which are then  used in the fusion scheme to obtain the final predicted labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' The fusion method used in this work  is described in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Fusion  Our fusion method is based on late fusion, where we combined the classification scores obtained  with the individual models for the final classification decision as shown in Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' In the equation,  𝑆𝑓𝑖𝑛𝑎𝑙 represents the final classification score while 𝑠𝑛 is the score obtained with the nth model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  We note that in the current implementation, we used a simple fusion method by treating all the  models equally (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=', simple aggregation of the individual scores).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  𝑆𝑓𝑖𝑛𝑎𝑙 = 𝑆1 + 𝑆2 + 𝑠3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='. + 𝑆𝑛  (1)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Results and Analysis  Table 1 provides the experimental results of the proposed solutions on the development set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' As  can be been in the table, overall better results are obtained with the BERT model, and surprisingly,  a lower F1-score is observed for RoBERTa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' In the future, we will further investigate the potential  causes of the lower performance of RoBERTa by exploring different implementations and hyper-  parameter settings for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' As far as the performance of the fusion methods is concerned, overall  better results are obtained with the pair of XLNet and BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' One of the potential reasons for the  lower performance of the fusion of all the models is the less accurate prediction of RoBERTa, as  also evident from the performance of the individual models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  Table 1  Experimental results of the proposed solutions on the development set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  Method  F1-Score  BERT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='94  RoBERTa  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='78  XLNet  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='93  Fusion 1 (RoBERTa, BERT, XLNet)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='75  Fusion 2 (BERT, XLNet)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='93  Fusion 3 (RoBERTa, XLNet)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='92  Table 2 provides the official results of the proposed solutions on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' In total, three  different runs were submitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' The first run is based on the fusion of all three models used in this  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' The remaining two runs are based on the fusion of the models in pairs of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' In run 2,  BERT and XLNet are combined while in run 3 RoBERTa and XLNet are jointly used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' As can be  seen in the table, better results are obtained for the fusion of the models in pairs of two where the  best performing pair of two models obtained an improvement of 20% over the fusion of all three  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  Table 2  Experimental results of the proposed solutions on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  Run  Precision  Recall  F1-Score  1 (Fusion of BERT, RoBERTa, XLNet)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='6738  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='5431  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='6014  2 (Fusion of BERT and XLNet)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='8044  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='6948  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='7456  3 (Fusion of RoBERTa and XLNet)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='8977  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='8598  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='8784  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Conclusions  In this paper, we presented our solutions for the RCTP subtask of DisasterMM challenge posted  in MediaEval 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' We proposed a late fusion framework incorporating several state-of-the-art  transformers for the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' In the current implementation, all the models are treated equally by  assigning them equal weights (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=', 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' In the future, we aim to employ merit-based fusion methods  to further improve the final classification score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  References  [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Ahmad, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Pogorelov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Riegler, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Conci, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Halvorsen, Social media and satellites, Multimedia  Tools and Applications 78 (2019) 2837–2875.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  [2] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Said, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Ahmad, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Riegler, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Pogorelov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Hassan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Ahmad, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Conci, Natural disasters  detection in social media and satellite imagery: a survey, Multimedia Tools and Applications 78 (2019)  31267–31302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content='  [3] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Ahmad, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
+page_content=' Riegler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'}
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DNE1T4oBgHgl3EQfqAUl/content/tmp_files/2301.03337v1.pdf.txt

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+Adiabatic theory of one-dimensional curved polariton waveguides
+D. A. Zezyulin∗1 and I. A. Shelykh2, 1
+1Department of Physics, ITMO University, Saint Petersburg 197101, Russia
+2Science Institute, University of Iceland, Dunhagi 3, IS-107, Reykjavik, Iceland
+(Dated: January 10, 2023)
+We construct a general theory of adiabatic propagation of spinor exciton-polaritons in waveguides
+of arbitrary shape, accounting for the effects of TE-TM splitting in linear polarizations and Zeeman
+splitting in circular polarizations. The developed theory is applied for the description of waveguides
+of periodically curved shape. We show that in this geometry the periodic rotation of the effective
+in-plane magnetic field produced by TE-TM interaction results in a nontrivial band-gap structure,
+which can be additionally tuned by application of an external magnetic field. It is also demonstrated,
+that spin-dependent interactions between polaritons lead to the formation of stable gap solitons.
+Introduction.
+Exciton-polaritons are composite half-
+light half-matter quasiparticles emerging in the regime
+of the strong coupling between a photonic mode of a
+planar semiconductor microcavity and an exciton in a
+quantum well (QW) brought in resonance with it. They
+possess a set of remarkable properties, which allow po-
+laritonic systems to serve as a convenient playground for
+study of collective nonlinear phenomena at elevated tem-
+peratures [1]. From their photonic component polaritons
+get extremely small effective mass (about 10−5 of the
+mass of free electrons) and macroscopically large coher-
+ence length [2], while the presence of an excitonic com-
+ponent enables efficient polariton-polariton interactions
+[3–5] and leads to the sensitivity of the polariton systems
+to external electric [6–8] and magnetic [9–11] fields.
+An important property of cavity polaritons is their spin
+(or pseudo-spin) [12], inherited from the spins of QW ex-
+citons and cavity photons. Similar to photons, polari-
+tons have two possible spin projections on the structure
+growth axis corresponding to the two opposite circular
+polarizations which can be mixed by effective magnetic
+fields of various origin. Real magnetic field applied along
+the structure growth axis and acting on the excitonic
+component splits in energy the polariton states with op-
+posite circular polarizations, while TE-TM splitting of
+the photonic modes of a planar resonator couples these
+states to each other via a k-dependent term, thus playing
+a role of an effective spin-orbit interaction [12]. Impor-
+tantly, polariton-polariton interactions are also spin de-
+pendent, as they stem from the interactions of excitonic
+components which are dominated by the exchange term
+[13]. This leads to the fact that polaritons of the same cir-
+cular polarization interact orders of magnitude stronger
+than polaritons with opposite circular polarizations [3].
+Remarkable tunability of cavity polaritons allows to
+engineer their spatial confinement in a variety of ex-
+perimental geometries, ranging from individual micropil-
+lars [14–17] to systems of several coupled pillars form-
+∗email: d.zezyulin@gmail.com
+ing so-called polariton molecules [18, 19] or periodically
+arranged arrays of the pillars forming polariton super-
+lattices [20–24].
+Realization of quasi one-dimensional
+(1D) geometries, where the motion of the polaritons is
+restricted to individual waveguides [7, 25], rings [26–28]
+or systems of coupled waveguides [29, 30], represents par-
+ticular interest from the point of view of the applications
+of polaritonics, as they can form basis for classical [31–33]
+and quantum [34, 35] polaritonic circuits.
+Current state of technology allows routine production
+of quasi 1D polariton waveguides of arbitrary shape, in-
+cluding ones with periodically modulated curvature. Cre-
+ation of the general theory of the polariton propagation
+in these structures, which includes polarization dynam-
+ics and polariton-polariton interactions, is the goal of the
+present Letter.
+The model. The presence of the in-plane spatial con-
+finement results in the strong nonequivalency of the
+states polarized normally and tangentially to a waveg-
+uide, which leads to the appearance of a local effective
+magnetic field, acting on a polariton pseudospin and di-
+rected tangentially to the waveguide. Although one can
+safely assume that in the case of a narrow waveguide of
+a constant width the absolute value of this field remains
+constant (see Supplementary material [36] for further de-
+tails), its direction changes along the curved waveguide,
+and, as we demonstrate below, this has crucial effect on
+polariton dynamics.
+Let us suppose that the shape of a waveguide in (x, y)-
+plane is given parametrically as x = x(ξ), y = y(ξ). The
+components of the effective magnetic field Ωx,y produced
+by TE-TM interaction are proportional to the compo-
+nents of the unit vector tangential to a waveguide τx,y
+and thus read
+Ωx = Ω0τx =
+Ω0x��(ξ)
+�
+x′(ξ)2 + y′(ξ)2 ,
+(1)
+Ωy = Ω0τy =
+Ω0y′(ξ)
+�
+x′(ξ)2 + y′(ξ)2 ,
+(2)
+arXiv:2301.03337v1  [cond-mat.mes-hall]  9 Jan 2023
+
+2
+FIG. 1: (a) Schematic representation of the considered geom-
+etry of a 1D polariton waveguide etched in planar semicon-
+ductor microcavity. The arc length ℓ measures the distance
+along the waveguide. Direction of the in-plane tangential unit
+vector ⃗τ = (τx, τy) changes along the waveguide and leads to
+emergence of an effective space-dependent field for the spinor
+polariton wavefunction.
+(b,c) Real and imaginary parts of
+the L-periodic effective potentials Ω(ℓ) for a waveguide com-
+posed of a chain of touching halfcircles (b) and a sine-shaped
+waveguide (c).
+where primes correspond to derivatives, and
+Ω0 ≈ ℏ2
+4d2
+� 1
+ml
+− 1
+mt
+�
+.
+(3)
+In the above equation, ml and mt stand for the effective
+longitudinal and transverse masses of 2D polaritons, and
+d is an effective width of a polariton channel [37]. As it
+was already mentioned, the presence of the field Ω splits
+in energy the modes polarized normally and tangentially
+to a waveguide.
+Additional splitting in circular polar-
+izations, denoted by ∆z, can be induced by application
+of an external magnetic field perpendicular to a cavity
+interface.
+Let us introduce the coordinate ℓ along the waveguide,
+ℓ =
+� ξ
+0
+�
+x′(η)2 + y′(η)2dη.
+In the adiabatic approxi-
+mation, the effective 1D Hamiltonian governing the dy-
+namics of the spinor wavefunction of polaritons can be
+then represented in the following form (see Supplemen-
+tary material [36] for corresponding derivation):
+ˆH =
+�
+�
+�
+�
+−
+ℏ2
+2meff
+d2
+dℓ2 + ∆z
+2
+Ω−
+Ω+
+−
+ℏ2
+2meff
+d2
+dℓ2 − ∆z
+2
+�
+�
+�
+� , (4)
+where
+Ω± = Ω(ℓ) = Ω0(τx ± iτy)2,
+(5)
+and meff is the effective mass.
+The physical meaning of the above Hamiltonian is
+pretty clear: it describes a motion of a one-dimensional
+spinor particle affected by a constant z-directed magnetic
+field and in-plane magnetic field whose direction changes
+along the way, being always tangential to the waveguide.
+In what follows, we will work with the effective Hamil-
+tonian rewritten in the dimensionless form. To this end,
+we introduce the unit length λ0 and the unit energy
+ε0 ≡ ℏ2/(2meffλ2
+0), and then rescale the variables of
+(22) as ℓ → λ0ℓ and ∆z → ε0∆z.
+Additionally, we
+rescale time as t → (ℏ/ε0)t.
+Assuming, for instance,
+that the unit length λ0 corresponds to 5 µm and meff
+is about 10−5 of the free electron mass, we obtain that
+the unit energy ε0 is about 0.2 meV, and the time unit
+ℏ/ε0 is equivalent to few picoseconds. Supplementing the
+obtained dimensionless Hamiltonian with the interaction
+terms [38], we obtain the following nonlinear evolution
+problem that governs the dynamics of the spinor wave-
+function (Ψ1, Ψ2):
+i∂Ψ1
+∂t
+= −∂2Ψ1
+∂ℓ2 + ∆z
+2 Ψ1 + Ω−(ℓ)Ψ2
++(|Ψ1|2 + σ|Ψ2|2)Ψ1,
+(6)
+i∂Ψ2
+∂t
+= −∂2Ψ2
+∂ℓ2 − ∆z
+2 Ψ2 + Ω+(ℓ)Ψ1
++(|Ψ2|2 + σ|Ψ1|2)Ψ2.
+(7)
+Small negative coefficient σ takes into account weak at-
+traction between polaritons of opposite polarizations (in
+our numerical calculations the value σ = −0.05 was
+used).
+Examples: The chain of halfcircles and the sine-shaped
+waveguide.
+In what follows, we focus on the situation
+when the shape of the curved waveguide can be de-
+scribed by function y(x), see Fig. 1(a) for a schemat-
+ics of the assumed geometry.
+Then the effective field,
+as a function of the arc length ℓ, can be computed as
+Ω±(ℓ) = Ω0 exp{±2i arctan(dy/dx)}, where the deriva-
+tive dy/dx should be expressed as a function of ℓ. In our
+further consideration we focus on the case of periodically
+curved waveguides.
+As a first analytically tractable example we consider
+the situation when the waveguide is composed of a peri-
+odic chain of touching halfcircles of a radius R. In terms
+
+a
+yRe
+[m
+Re, Im (2/20)
+(°/) I 
+0
+Re,
+0.25
+0.5
+0
+0.75
+1
+0
+0.25
+0.5
+0.75
+L3
+FIG. 2: Transformation of the band-gap structure for the sine-shaped waveguide under the fixed TE-TM splitting coefficient
+Ω0 = 0.45 and increasing strength of the external magnetic field ∆z. Here the Bloch quasimomentum k varies within the reduced
+Brillouin zone [−π/L, π/L), where L is the spatial period of the structure. The periodic curvature results in a nontrivial band-
+gap structure. Finite bandgaps are present even in the absence of the external magnetic field (∆z = 0). The increase of ∆z
+leads to the anticrossings of the bands touching at k = 0 and related shift of the band minima and maxima to k ̸= 0.
+of coordinates x and y, the unit cell of the resulting
+periodic structure is given as y(x) =
+�
+R2 − (x − R)2
+for x
+∈
+[0, 2R] (the upper halfcircle) and y(x)
+=
+−
+�
+R2 − (x − 3R)2 for x ∈ [2R, 4R] (the lower halfcir-
+cle).
+In terms of the arc length ℓ, the unit cell cor-
+responds to the interval ℓ ∈ [0, L] where L = 2πR
+is the period of the structure.
+The first halfperiod
+ℓ ∈ [0, πR] corresponds to the first halfcircle, where
+x(ℓ) = R[1 − cos(ℓ/R)] and y(ℓ) = R sin(ℓ/R), and the
+second halfperiod ℓ ∈ [πR, 2πR] corresponds to the sec-
+ond halfcircle, where we have parametrization x(ℓ) =
+R[3 + cos(ℓ/R)] and y(ℓ) = R sin(ℓ/R), and the rest of
+waveguide is obtained by the periodic repetition of the
+unit cell. Performing straightforward calculations, we ob-
+tain that within the unit cell the resulting potential reads
+Ω±(ℓ) = −Ω0 exp{∓2iℓ sign (πR − ℓ)/R}.
+The shape
+of the resulting dependency is illustrated in Fig. 1(b).
+While the obtained dependence is rather simple, its imag-
+inary part is not a smooth function: it has a cusp exactly
+at the center of the unit cell ℓ = πR, where the two half-
+circles touch.
+As a second example, which results in a smooth peri-
+odic potential (which is therefore better suited for the
+numerical analysis), we consider a sine-shaped waveg-
+uide y(x) = V0 sin x.
+Then the arc length along the
+waveguide is given by the incomplete elliptic integral of
+the second kind [39]: ℓ(x) =
+�
+1 + V 2
+0 E(sin x, m), where
+m = V 2
+0 /(1 + V 2
+0 ). To the best of our knowledge, there
+is neither a commonly used special function nor a closed-
+form expression that allows to invert the incomplete el-
+liptic integral of the second kind, i.e., to express x and
+y through ℓ in our case. In the meantime, there exists a
+simple iterative numerical procedure for inversion of the
+incomplete elliptic integral of the second kind [40]. Us-
+ing this procedure, one can easily obtain the dependence
+Ω(ℓ), see Fig. 1(c) for a representative example.
+The
+resulting 1D Hamiltonian ˆH defined by (22) becomes ef-
+fectively periodic with the spatial period in ℓ given as
+L = 4E(m), where E(m) is the complete elliptic integral
+of the second kind.
+Band structure. Periodic nature of the resulting sys-
+tem suggests to look at the band structure which can
+be presented in the form of the dependencies of the en-
+ergy E versus Bloch quasimomentum k, which, without
+loss of generality, can be assumed to belong to the Bril-
+louin zone [−π/L, π/L), where L is the period. For sinu-
+soidal waveguide the result computed for system (6)–(7)
+with omitted nonlinear terms (|Ψ1,2|2 + σ|Ψ2,1|2)Ψ1,2 is
+shown in Fig. 2.
+We have focused on the transforma-
+tion of the spectral structure subject the the increase
+of the external magnetic field, which is characterized by
+the Zeeman splitting coefficient ∆z. As one can see, the
+periodic curvature of a waveguide results in a nontriv-
+ial band-gap structure as the effective periodic potential
+Ω(ℓ) opens finite gaps even in the absence of the external
+magnetic field (∆z = 0). The increase of ∆z leads to
+a transformation of the band-gap structure. In particu-
+lar, it leads to the anticrossing of the bands touching at
+k = 0 and related shift of the band minima and max-
+ima to k ̸= 0. Dispersion curves having two degenerate
+extrema at k = ±k0 ̸= 0 can be, in particular, relevant
+for the observation of the so-called stripe phase charac-
+terized by spinor wavefunctions carrying a more complex
+internal structure, see e.g. [41–45] and [46] for discussion
+of stripe phase and stripe solitons in spin-orbit coupled
+atomic and polariton condensates, respectively.
+Gap solitons. The presence of finite gaps in the band-
+gap structure suggests that when the repulsive interac-
+tions between the polaritons of the same circular po-
+larization are taken into account, the waveguide can
+support formation of polariton gap solitons [22, 46–51].
+These localized states can be found using the substitu-
+tion Ψ1,2(t, ℓ) = e−iµtψ1,2(ℓ), where stationary wavefunc-
+tions ψ1,2(ℓ) satisfy zero boundary conditions at ℓ → ∞
+and ℓ → −∞, and µ characterizes the chemical poten-
+tial of the polariton condensate.
+The numerical study
+
+=0
+△= 0.4
+△= 1.0
+△z = 1.4
+△= 2.0
+8
+8
+8
+6
+6
+6
+6
+E
+2
+2
+0
+0
+0
+0
+kL/π
+kL/π
+kL/π
+kL/π
+kL/π4
+indicates that the system supports a variety of solitons
+which form continuous families, i.e., can be parameter-
+ized by the continuous change of the chemical potential
+µ within the energy spectrum bandgap. To describe the
+found solitons, we introduce the polariton density inte-
+gral N =
+� ∞
+−∞(|ψ1|2 + |ψ2|2)dℓ which characterizes the
+squared norm of the solution. In Fig. 3(a) we illustrate
+the family of fundamental (simplest) gap solitons as a de-
+pendence N on µ. The soliton family detaches from the
+left edge of the bandgap, where the soliton norm van-
+ishes: N → 0.
+In this limit, small-amplitude solitons
+transform to a linear Bloch wave. As the chemical po-
+tential increases towards the right gap edge, the total
+norm N grows monotonously. To quantify the degree of
+the soliton localization, we introduce an additional char-
+acteristics n99 which amounts to the number of spatial
+periods where 99% of quasiparticles are confined. The de-
+pendence n99 on µ is also plotted in Fig. 3(a). It demon-
+strates nonmonotonic behavior approaching its minimal
+values in the center of the gap. In this regime the soli-
+tons are most localized, and almost all energy can be
+trapped in the segment of waveguide composed of ap-
+proximately from five to ten unit cells. At the same time,
+the quantity n99 becomes extremely large near the edges
+of the gap, which means that the corresponding solitons
+are very broad and relatively poorly localized. Examples
+of spatial profiles of solitons having different amplitudes
+and degrees of localization are shown in Fig. 3(b).
+It is known that gap solitons and, in particular, those
+in systems dominated by repulsive nonlinearities, can be
+be prone to dynamical instabilities [52–55]. In the mean-
+time, using the dynamical simulations, we found that the
+family of fundamental gap solitons presented in Fig. 3(a)
+contains stable solutions which can robustly preserve the
+steady shape for the indefinite simulation time (much
+larger than typical polariton lifetimes), even if the ini-
+tial profiles are perturbed by a small-amplitude random
+noise. Example of such stable dynamics is presented in
+Fig. 3(c,d). At the same time, more complex solitons can
+develop dynamical instabilities which eventually lead to
+their delocalization. The corresponding example is shown
+in Fig. 3(e,f).
+Conclusion. In conclusion, we constructed a theory of
+the propagation of cavity polaritons in narrow quasi-1D
+waveguides of arbitrary shape and applied it to the case of
+periodically curved waveguides. We demonstrated that
+the periodic rotation of an effective in-plane magnetic
+field produced by TE-TM splitting in linear polarizations
+leads to the formation of nontrivial band structure. The
+shape of the bands, the bandgaps and the positions of
+the band extrema can be tuned by application of an ex-
+ternal magnetic field. In the nonlinear regime the system
+supports formation of dynamically stable gap solitons.
+Acknowledgements.
+The research was supported by
+Priority 2030 Federal Academic Leadership Program.
+IAS acknowledges support from Icelandic Research Fund
+FIG. 3: (a) Gap solitons norm N and the localization measure
+n99 as functions of chemical potential µ for a family of funda-
+mental gap solitons in the first finite gap. Here the coefficient
+of TE-TM splitting Ω0 = 0.4 and amplitude of the Zeeman
+splitting ∆z = 0.3. Shaded regions correspond to the values of
+µ that belong to spectral bands. (b) Example of a broad soli-
+ton near the left edge of the gap (specifically, at µ = 0.24) and
+a strongly localized soliton in the center of the gap at µ = 0.5.
+(c,d) Stable dynamics of the gap soliton with chemical poten-
+tial µ = 0.29. Initial conditions correspond to the stationary
+wavefunctions perturbed with a random noise whose ampli-
+tude is about 2% of the soliton’s amplitude. (e,f) Example
+of unstable evolution of a gap soliton of more complex shape
+corresponding to Ω0 = 0.4, µ = 0.4, and ∆z = 0.009.
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+
+7
+SUPPLEMENTAL MATERIAL: DERIVATION OF
+THE 1D ADIABATIC HAMILTONIAN
+The two-dimensional Hamiltonian of a polariton mov-
+ing inside a waveguide defined by a confining potential
+U(x, y) is [38]:
+ˆH2D =
+�
+�
+�
+�
+−
+ℏ2
+2meff
+� ∂2
+∂x2 + ∂2
+∂y2
+�
++ ∆z
+2 + U(x, y)
+β
+�
+∂
+∂y + i ∂
+∂x
+�2
+β
+�
+∂
+∂y − i ∂
+∂x
+�2
+−
+ℏ2
+2meff
+� ∂2
+∂x2 + ∂2
+∂y2
+�
+− ∆z
+2 + U(x, y)
+�
+�
+�
+� ,
+(8)
+where
+β = ℏ2
+4
+� 1
+ml
+− 1
+mt
+�
+.
+(9)
+Let us introduce in each point of a waveguide local
+coordinate system with axis ℓ directed tangential to it
+and n normal to it. The elementary lengths dℓ and dn
+read:
+dℓ = τx(ℓ)dx + τy(ℓ)dy,
+(10)
+dn = −τy(ℓ)dx + τx(ℓ)dy
+(11)
+where τx,y are components of the unit vector tangential
+to the waveguide at a given point characterized by coor-
+dinate ℓ along the waveguide.
+We can now right down:
+∂
+∂x = ∂ℓ
+∂x
+∂
+∂ℓ + ∂n
+∂x
+∂
+∂n = τx
+∂
+∂ℓ − τy
+∂
+∂n,
+(12)
+∂
+∂y = ∂ℓ
+∂y
+∂
+∂ℓ + ∂n
+∂y
+∂
+∂n = τy
+∂
+∂ℓ + τx
+∂
+∂n,
+(13)
+∂
+∂y ± i ∂
+���x = ±iτ∓
+∂
+∂ℓ + τ∓
+∂
+∂n,
+(14)
+where
+τ± = τx ± iτy.
+(15)
+We thus have:
+∂2
+∂x2 + ∂2
+∂y2 = ∂2
+∂ℓ2 + ∂2
+∂n2 +
+�
+τy
+∂τx
+∂ℓ − τx
+∂τy
+∂ℓ
+� ∂
+∂n,(16)
+where we used that
+τ 2
+x + τ 2
+y = 1.
+(17)
+Similarly
+� ∂
+∂y ± i ∂
+∂x
+�2
+= (18)
+= τ 2
+∓
+∂2
+∂n2 − τ∓
+∂
+∂ℓτ∓
+∂
+∂ℓ ± iτ∓
+�
+τ∓
+∂
+∂ℓ + ∂
+∂ℓτ∓
+� ∂
+∂n.
+Let us now suggest that the confining potential locally
+depends on the transverse coordinate n only, and use adi-
+abatic approximation for the spinor wavefunction Ψ(x, y)
+representing it as:
+Ψ(x, y) = ψ(ℓ)φ(n),
+(19)
+where the part ψ(ℓ) describes the propagation of the po-
+laritons along the waveguide, and φ(n) corresponds to
+their 1D lateral confinement and can be taken real. This
+approximation holds if an effective thickness of a waveg-
+uide d is much less then its local curvature R, which for
+a parametrically given curve is given by
+R =
+�
+x′(ξ)2 + y′(ξ)2�3/2
+|x′(ξ)y′′(ξ) − y′(ξ)x′′(ξ)|.
+(20)
+Multiplying the Schr¨odinger equation ˆH2DΨ = EΨ by
+φ(n) and integrating by n from −∞ to +∞, one gets for
+the dynamics of the propagation along the channel the
+following 1D Schr¨odinger equation:
+ˆHψ(ℓ) = Eψ(ℓ),
+(21)
+where
+
+8
+ˆH =
+�
+�
+�
+�
+�
+�
+E0 −
+ℏ2
+2meff
+d2
+dℓ2 + ∆z
+2
+Ω− − βτ−
+d
+dℓτ−
+d
+dℓ
+Ω+ − βτ+
+d
+dℓτ+
+d
+dℓ
+E0 −
+ℏ2
+2meff
+d2
+dℓ2 − ∆z
+2
+�
+�
+�
+�
+�
+�
+,
+(22)
+and we have used that
+� +∞
+−∞
+φ(n)dφ
+dn dn = 0,
+(23)
+and
+E0 =
+� +∞
+−∞
+φ(n)
+�
+−
+ℏ2
+2meff
+d2
+dn2 + U(n)
+�
+φ(n)dn
+(24)
+is the energy of the confinement, and
+Ω± = βτ 2
+±
+� +∞
+−∞
+φ(n) d2φ
+∂n2 dn ≈ β
+d2 τ 2
+± = Ω0τ 2
+±,
+(25)
+where d is an effective width of the confining channel, and
+we used Gaussion approximation, φ(n) = d√πe−n2/(2d2)
+Note, that E0 is just a constant, which can be safely
+dropped. As for the off-diagonal terms βτ± d
+dℓτ± d
+dℓ, one
+can note, that by the order of magnitude d/dℓ ∼ k, where
+k is a wavenumber, describing the propagation of the
+polaritons along the waveguide. Therefore, for narrow
+waveguides and small k, when k ≪ d−1, these terms
+can be neglected as compared to Ω±, and one gets the
+Hamiltonian (4) of the main text.
+

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+8th Brunei International Conference on Engineering and Technology (BICET 2021), Universiti Teknologi Brunei 
+Internet of Things: Digital Footprints Carry A Device 
+Identity 
+Rajarshi Roy Chowdhury1, 2, a), Azam Che Idris1 and Pg Emeroylariffion Abas1 
+1Faculty of Integrated Technologies, Universiti Brunei Darussalam, Jalan Tungku Link, Gadong BE1410, Brunei 
+Darussalam  
+2Department of Computer Science and Engineering, Sylhet International University, Shamimabad Road, Sylhet 
+3100, Bangladesh 
+ 
+Corresponding author: a) 19h0901@ubd.edu.bn or rajarshiry@gmail.com 
+ 
+ABSTRACT. The usage of technologically advanced devices has seen a boom in many domains, including education, 
+automation, and healthcare; with most of the services requiring Internet-connectivity. To secure a network, device 
+identification plays key role. In this paper, a device fingerprinting (DFP) model, which is able to distinguish between 
+Internet of Things (IoT) and non-IoT devices, as well as uniquely identify individual devices, has been proposed. Four 
+statistical features have been extracted from the consecutive five device-originated packets, to generate individual device 
+fingerprints. The method has been evaluated using the Random Forest (RF) classifier and different datasets. Experimental 
+results have shown that the proposed method achieves up to 99.8% accuracy in distinguishing between IoT and non-IoT 
+devices and over 97.6% in classifying individual devices. These signify that the proposed method is useful in assisting 
+operators in making their networks more secure and robust to security breaches and unauthorised access. 
+Keywords : digital footprint; network traffic traces; machine learning algorithm; internet of things; device 
+fingerprinting 
+ 
+INTRODUCTION 
+ 
+It has been predicted that the number of network-connected Internet of Things (IoT) and non-IoT devices 
+worldwide will reach approximately 30.9 billion and 10.3 billion, respectively, by the year 2025 [1]⁠. Proliferated 
+growth of these devices with their heterogeneous functionalities, has imposed new challenges to network 
+administrators and operators, in providing, managing, and controlling the operations and security of the network 
+services [2]⁠. Accurate device identification is one key aspect that needs to be seriously considered in securing 
+network-connected devices. Conventionally, internet protocol (IP) enabled devices have been using user-defined 
+identifiers, such as IP and media access control (MAC) addresses, as a form of identifications. However, these 
+identifiers have been proven to be vulnerable [3]⁠ to various attacks, such as spoofing [4]⁠ and device mobility, due to 
+the availability of malicious software [5]⁠, for performing such attacks. Device fingerprinting (DFP) [3]⁠ represents 
+one technique that may be used to identify devices based on their communication traffic traces (or digital footprints) 
+without using explicit identifiers, and it can be performed, either actively or passively, from different layers of the 
+communication model [6]⁠.  
+ 
+Due to the prominent characteristics of network traffic features, many researchers [2, 7]⁠ have used packet-level 
+features for different purposes [8]⁠, including for device identification [9]⁠. Sivanathan et al. [10]⁠ have described a 
+DFP scheme based on the analysis of passively observed network traffic traces. A total of 11 statistical features are 
+used as device fingerprints, from packet traffic-flows over a period of one day, by looking at the devices’ sleeping 
+time, average packet size and traffic rate, active time, number of servers and protocols used in a flow, number of 
+
+8th Brunei International Conference on Engineering and Technology (BICET 2021), Universiti Teknologi Brunei 
+unique domain name system (DNS) request, and intervals of DNS and network time protocol (NTP) requests. 
+Subsequently, these features are used to train an ML model for classification. It has been shown that the DFP 
+scheme is able to distinguish between IoT and non-IoT devices with high accuracy and achieve over 95% accuracy 
+in identifying individual IoT devices. The same researchers [9]⁠ have also presented another device fingerprinting 
+scheme, by utilizing statistical characteristics of hourly network traffic traces, to generate 8 device-specific 
+fingerprints. Experimental result has shown over 99% accuracy using the UNSW dataset. Charyyev et al. [11]⁠ have 
+utilized Nilsimsa hash value of packet flows (n packets) for device-specific fingerprints, to classify individual IoT 
+devices, to achieve 93% precision. 
+ 
+Researchers in [2, 12]⁠ have used 12 packets information, to generate device signatures for classifying IoT 
+devices, with 81.5% global accuracy and 76.15% accuracy using an aggregated model, whilst Aksoy and Gunes [13]⁠ 
+have presented a DFP approach, known as SysID, which utilizes features from a single packet, for identifying smart 
+home IoT devices with 82% average classification accuracy. Bezawada et al. [14]⁠ have utilized 5 consecutive 
+packets information, including protocols headers and payload (20 features), for classifying IoT devices uniquely 
+with mean identification accuracy of 93% to 100% using a laboratory dataset of 14 IoT devices. In [15]⁠, the authors 
+have used a one second window to group packets, for generating statistical fingerprinting features. These features 
+are then used to train a binary classifier for categorizing IoT and non-IoT devices with high accuracy of 99%, whilst 
+a multi-class classifier has been used to uniquely identify IoT devices with about 96% accuracy. All these existing 
+DFP models, however, require either a large number of features set from different layers of the communication 
+model, or a large number of network packets information for generating fingerprints. Consequently, these models 
+consume a long period of time, and require complex computation. As such, a more efficient DFP model is required 
+for classifying devices with high accuracy, but with less computation cost. 
+ 
+In this paper, a supervised machine learning (ML) based DFP model, which generates device-specific signatures 
+by computing four statistical features from consecutive five packets of the network traffic, has been proposed. An 
+intuition that these features carry device-specific characteristics in terms of device memory and processing speed. 
+Experimental results have shown that over 97.0% accuracy is achievable in classifying individual non-IoT devices 
+from traffic collected in a laboratory environment, and 97.3% accuracy on the non-IoT traffic traces from the 
+UNSW dataset. The proposed DFP model is also capable of distinguishing between IoT and non-IoT devices with 
+up to 99.8% accuracy on the UNSW dataset. The key contributions of this research work are: 
+ 
+• 
+Identifying device-specific features from the device-originated communication traffic traces, to generate 
+device signatures for classification.  
+• 
+Instrument an experimental testbed of non-IoT devices in a laboratory environment for data collection.  
+• 
+Evaluate the proposed DFP scheme performance based on a supervised ML algorithm, to distinguish between 
+IoT and non-IoT devices and identify individual devices. 
+ 
+The rest of the paper is organized as follows. The proposed ML-based device fingerprinting method, as well as 
+the datasets, data collection procedure, and an ML classifier are described in Section II. Section III describes 
+experimental results on various datasets, and finally, conclusion is given in Section IV. 
+ 
+METHODOLOGY 
+ 
+The proposed DFP method is used to extract unique device features from network traffic traces. These features 
+are used to train an ML classifier, and subsequently, used to test the performance of the proposed DFP method on 
+different datasets. This section describes the proposed DFP method, the datasets used for training and testing, as 
+well as the classification method used to test the model. 
+ 
+Datasets: IoT and Non-IoT 
+ 
+The proposed device fingerprinting model performance has been evaluated by utilizing a publicly available 
+dataset: UNSW [9]⁠, and a testbed dataset of non-IoT devices, which has been collected from a laboratory 
+environment. Summary of the datasets are listed in Table 1. The UNSW dataset comprises network traffic traces 
+from both IoT and non-IoT devices, including TP-Link camera, smart bulb, Belkin camera, smart doorbell, printer, 
+
+8th Brunei International Conference on Engineering and Technology (BICET 2021), Universiti Teknologi Brunei 
+smart photo frame, laptop, smartphone, and tablet devices, with these heterogeneous devices coming from different 
+manufacturers: Belkin, Philips Hue, Netatmo, TP-Link, Withings, HP, Apple. On the other hand, the laboratory 
+dataset comprises 7 non-IoT devices, including laptops, smartphones, and desktops, from different manufacturers. 
+The data collection procedure from the 7 non-IoT devices is described in the following section. 
+ 
+TABLE 1. List of IoT and non-IoT Datasets. 
+Dataset 
+Devices 
+Total Packets 
+Source 
+IoT 
+Non-IoT 
+UNSW 
+22 
+-- 
+6,844,740 
+[9]⁠ 
+-- 
+7 
+3,515,705 
+Lab Dataset 
+-- 
+7 
+442,970 
+-- 
+ 
+TABLE 2. List of non-IoT devices for experimental set up. 
+No. 
+Device Category 
+Device Name/Model 
+Operating System 
+Connectivity 
+MAC Address 
+1 
+Laptop 
+Aspire-S7 
+Windows 
+WiFi 
+34:23:87:b7:56:17 
+2 
+ProBook-4410s  
+WiFi/Ethernet 
+00:25:b3:47:da:6f 
+3 
+Desktop 
+Asus 
+Ethernet 
+08:60:6e:c1:79:c2 
+4 
+HP-EliteDesk 
+Ethernet 
+80:e8:2c:d6:9e:49 
+5 
+Smart Phone 
+MYA-U29  
+Android 
+WiFi 
+d0:ff:98:95:57:af 
+6 
+MLXP2ZA-A 
+iOS 
+WiFi 
+e0:c7:67:45:a3:62 
+7 
+MWC22KH-A 
+WiFi 
+06:44:b7:aa:20:98 
+ 
+Dataset Collection Methodology 
+ 
+An experimental design, consisting of local area network (LAN) and wireless local area network (WLAN) with 
+non-IoT devices, was set up in a laboratory environment at Universiti Brunei Darussalam (UBD). Design of the 
+testbed is depicted in Figure 1, with the seven non-IoT devices from different manufacturers and of different types, as 
+listed in Table 2. These devices were configured, to connect with an access point (AP) either using ethernet or wireless 
+fidelity (WiFi) interfaces. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+FIGURE 1. An experimental testbed of non-IoT devices network (LAN/WLAN). 
+
+DNS
+NTP
+Server
+Connectivity:
+Server
+Server
+N
+Ethernet
+WiFi
+Other
+a
+Internet
+WiFi
+Hotspot
+Gateway
+ubuntu?
+(UBD Network)
+Hub
+Ethernet
+USB Ethernet
+Port
+Port
+Monitoring Station
+(Capture Network Traffic)8th Brunei International Conference on Engineering and Technology (BICET 2021), Universiti Teknologi Brunei 
+A laptop was used to configure an access point (AP), which was used to provide network services to the non-IoT 
+devices, as well as to monitor and capture communication footprints from the devices. The Dell Inspiron 15 5000 
+Series laptop runs Ubuntu 18.04 as an operating system (OS), and was connected to the UBD network via its built-in 
+Ethernet interface, to provide the Internet connections. The built-in WiFi interface was configured as a WiFi Hotspot, 
+providing wireless connectivity to the WiFi-enabled (IEEE 802.11 standard) devices. Additionally, a TU3-ETG USB 
+Ethernet adapter was connected to the laptop, and used to set up a LAN network using the D-Link Switch Hub DES-
+1005A hub for providing network services to the connected non-IoT devices. On the Ubuntu OS, the network 
+connection editor tool, i.e. nm-connection-editor, was been utilised for connection establishment. 
+ 
+Devices generally generate two types of traffic [9]⁠: autonomous traffic, including traffic generated for 
+connection establishment, application and system synchronizations, and activity traffic, which is generated due to 
+human or object interactions. These inbound and outbound communication traffic traces, flowing over both 
+interfaces (external Ethernet and built-in WiFi interfaces) were captured using tcpdump 4.9.3 utility, and stored into 
+.pcap (packet capture) files format, similar to [16]⁠. Device-originated traffic traces were then extracted using TShark 
+utility and stored in .csv (comma-separated values) files format, along with labelling of individual devices names. 
+Finally, the recorded dataset was cleaned for further processing, by eliminating inconsistent instances, including 
+empty rows, and duplicate values.  
+ 
+Device Fingerprinting Model 
+ 
+The proposed DFP scheme architecture is depicted in Figure 2, which uses device-originated communication 
+traffic traces to generate device fingerprints for classification. Device-originated traffic traces are filtered according 
+to individual device MAC addresses, with tcp.window_size and ip.len values extracted from each packet from the 
+available captured data. These two values of a network packet carry significant device-specific information. 
+tcp.window_size value depends on a device buffer size and computation speed [14]⁠ whilst ip.len value specifies the 
+total length of a packet to represent unique characteristics of a devices communication pattern [15]⁠. tcp.window_size 
+and ip.len values from five consecutive packets (as one instance) are utilized, to compute mean (µ) and standard 
+deviation (σ), for constructing device-specific fingerprints, i.e. iplen_µ, iplen_σ, tcpwinsiz_µ, and tcpwinsiz_σ. 
+These 4 statistical fingerprints have been used for training a machine learning (ML) model, and subsequently, to 
+evaluate the performance, of the model in classifying devices using datasets, which have been randomly split into 
+training (80% instances) and testing (20% instances) datasets. 
+ 
+FIGURE 2. The proposed device fingerprinting scheme. 
+ 
+Random Forest Classifier 
+ 
+Random Forest (RF) classifier is a supervised machine learning (ML) algorithm, that can be used for both 
+classification [9]⁠ and regression [17]⁠ problems. The algorithm randomly generates a group of trees, with majority 
+voting used to make a decision from the ensemble of decision trees [18, 19]⁠, for the classification task, as presented 
+in Figure 3. This assists in avoiding over-fitting problem. Researchers in different domains have utilized RF 
+classifier for different classification tasks. In [9]⁠, the RF algorithm has been used for classifying IoT devices with 
+high accuracy. Primartha et al. [20] have performed anomaly detection using the algorithm, and it has also been used 
+
+Testing Dataset
+(20%)
+Training Dataset
+(80%)
+Capture
+Filter and Extract
+Fingerprint Generation
+Training Model
+Test Model
+Classification
+Network Traffic
+Traffie Traces
+(Mean, Standard Deviation)
+ML Algorithm
+ML Algorithm
+IoT and Non-IoT
+# Inbound and outbound
+# Outbound traffic traces
+# Statistical analysis
+# Train a machine
+# Test model performance
+# Category: IoT and non-loT
+traffic traces
+# Packet header features
+# Device fingerprint/Signature
+Learning (ML) model
+# Device identification
+(Store in pcap files fomat)
+# Label instances
+# Tune hyperparameter
+# Train and test datasets (csv files)8th Brunei International Conference on Engineering and Technology (BICET 2021), Universiti Teknologi Brunei 
+for disease identification in medical science [21]⁠. In this paper, RF classifier is used to appraise the performance of 
+the proposed DFP method, by using the extracted features for training the RF classifier, and subsequently, using the 
+trained RF classifier to determine classification performance. Some of the significant tunable hyper-parameters are 
+set experimentally, including the number of iterations (or number of trees) = 100, seed = 1, and batch size (number 
+of instances) = 100, to improve classification accuracy and reduce the root mean squared error (RMSE) [22]⁠. 
+ 
+ 
+FIGURE 3. An abstract representation of a RF classifier. 
+ 
+RESULTS AND DISCUSSION 
+ 
+The proposed DFP method has been evaluated using waikato environment for knowledge analysis (Weka) tool 
+[23]⁠. An online dataset: UNSW [9]⁠ dataset, and an experimental dataset, as presented in Table 3, have been utilized 
+to evaluate the classification performance based on the RF classifier. The UNSW dataset consists of network traffic 
+traces from IoT and non-IoT devices, which are referred to as the U-IoT and U-NonIoT datasets, respectively. On 
+the other hand, the experimental dataset contains only network traffic traces from non-IoT devices, and it is referred 
+to as the L-NonIoT dataset. 
+ 
+TABLE 3. Total number of instances used for evaluating the proposed DFP model. 
+Dataset 
+Devices 
+Training Dataset 
+(80%) 
+Test Dataset  
+(20%) 
+Total Instances 
+(100%) 
+IoT 
+Non-IoT 
+UNSW  (U-IoT) 
+* 
+--- 
+1,095,158 
+273,790 
+1,368,948 
+UNSW  (U-NonIoT) 
+--- 
+* 
+562,513 
+140,628 
+703,141 
+Lab  
+(L-NonIoT) 
+--- 
+* 
+70,875 
+17,719 
+88,594 
+ 
+ 
+The proposed DFP method utilises 5 network traffic packets as one instance to generate fingerprint. As such, a 
+total of 1,368,948 (6,844,740 / 5) and 703,141 (3,515,705 / 5) instances have been used from the U-IoT and U-
+NonIoT datasets, respectively, whilst a total of 88,594 (442,970 / 5) instances have been used from the L-NonIoT 
+dataset. 80% of the datasets have been used for training and the remainder for testing. The performance of the 
+trained RF classifier has been measured with respect to its ability to a) distinguish between IoT and non-IoT devices, 
+and b) classify individual devices. 
+ 
+Device Category: IoT and Non-IoT Devices 
+ 
+Classification performances of the proposed DFP model in distinguishing between IoT and non-IoT devices are 
+presented in Figure 4, on combined U-IoT and U-NonIoT datasets (i.e. UNSW dataset), and combined U-IoT and L-
+
+Dataset
+Data
+Subset of Data
+Subset of Data
+Subset of Data
+Random Samples
+1
+2
+n
+Decision Trees
+Selected Class
+Selected Class
+Class
+Selected Class
+(Vote)
+(Vote)
+(Vote)
+Majority Voting
+Final Decision
+(Class)8th Brunei International Conference on Engineering and Technology (BICET 2021), Universiti Teknologi Brunei 
+NonIoT datasets. The figure shows that device categorization accuracy reaches up to 99.9% using the RF classifier 
+on the combined U-IoT and L-NonIoT datasets. On the UNSW dataset [9]⁠, which consists of instances from 22 IoT 
+and 7 non-IoT devices, the proposed DFP method achieves 99.8% accuracy. 
+ 
+ 
+FIGURE 4. Categorize IoT and non-IoT devices: UNSW and Lab datasets. 
+ 
+ 
+FIGURE 5. Classification performance of the non-IoT devices: UNSW and Lab datasets. 
+ 
+Individual Device Classification 
+ 
+The performances of the proposed DFP method in classifying individual IoT and non-IoT devices on different 
+datasets, are depicted in Figure 5 and Figure 6. In Figure 5, the proposed DFP model achieves over 97.0% accuracy 
+in classifying non-IoT devices from the L-NonIoT and U-NonIoT datasets, with accuracy a little bit higher on the U-
+NonIoT dataset. Individual IoT devices classification performance of the proposed DFP model, on the U-IoT dataset 
+with 22 IoT devices, is given in Figure 6. Most of the IoT devices in the dataset can be classified with over 97.6% 
+accuracy, with the exception of the BlipcareBPmeter, the BelkinWemoSensor and BelkinWemoSwitch devices, 
+which give classification accuracies of about 75.0%, 96.5% and 91.4%, respectively. The lowest accuracy for the 
+BlipcareBPmeter device is due to the limited number of instances available from this device for training and testing.  
+ 
+CONCLUSION  
+ 
+A large number of heterogeneous IoT and non-IoT devices from different manufacturers are being connected to 
+the Internet, to obtain network-based services. In terms of network security, it is challenging for network 
+administrators and operators to identify the connected devices using conventional identifiers, as they are prone to 
+security breaches. In this paper, a DFP model based on the analysis of network traffic traces has been proposed, 
+which is capable of distinguishing between IoT and non-IoT devices as well as classifying individual IoT and non-
+IoT devices. As opposed to other methods in the literature, which require relatively large number of features and 
+
+loTvs NonloT
+U-loT: UNSW-loT, U-NonloT: UNSW-NonloT, L-NonloT: Lab-NonloT Datasets
+U-loT vs
+U-NonloT
+0.998
+Datasets
+U-loT vs
+L-NonloT
+0.999
+0.00
+0.25
+0.50
+0.75
+1.00
+AccuracyNonloTDevices
+L-NonloT: Lab-NonloT, U-NonloT: UNSW-NonloT Datasets
+L-NonloT
+0.970
+Datasets
+U-NonloT
+0.973
+0.00
+0.25
+0.50
+0.75
+1.00
+Accuracy8th Brunei International Conference on Engineering and Technology (BICET 2021), Universiti Teknologi Brunei 
+requiring longer sequence of packet network traffics to construct their DFP features, only 4 statistical features from 
+5 consecutive packet network traffics are required to construct the DFP features. These are used for training and 
+testing an ML classifier. Evaluations on the UNSW dataset have shown that the proposed DFP method is able to 
+distinguish between IoT and non-IoT devices with up to 99.8% accuracy, and individually classify most of the IoT 
+and non-IoT devices with over 97.6% accuracy. On the laboratory collected network traces, the proposed DFP 
+model is able to classify individual devices with 97.0% accuracy. The research outcomes signify that the proposed 
+DFP model is useful for device identification and may assist network administrators in providing a more secure 
+network.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+FIGURE 6. Individual IoT device classification performance: U-IoT dataset. 
+ 
+ACKNOWLEDGEMENTS 
+ 
+The authors are profoundly grateful to the Faculty of Integrated Technologies (FIT), Universiti Brunei 
+Darussalam (UBD), for supporting this research work, as well as to UBD for awarding the UBD Graduate 
+Scholarship (UGS) to the first author. 
+ 
+REFERENCES 
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+ 
+

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+page_content=' Universiti Brunei Darussalam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Jalan Tungku Link,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Gadong BE1410,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Brunei Darussalam 2Department of Computer Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Sylhet International University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Shamimabad Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Sylhet 3100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Bangladesh  Corresponding author: a) 19h0901@ubd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='bn or rajarshiry@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='com  ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' The usage of technologically advanced devices has seen a boom in many domains, including education, automation, and healthcare;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' with most of the services requiring Internet-connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' To secure a network, device identification plays key role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' In this paper, a device fingerprinting (DFP) model, which is able to distinguish between Internet of Things (IoT) and non-IoT devices, as well as uniquely identify individual devices, has been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Four statistical features have been extracted from the consecutive five device-originated packets, to generate individual device fingerprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' The method has been evaluated using the Random Forest (RF) classifier and different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Experimental results have shown that the proposed method achieves up to 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='8% accuracy in distinguishing between IoT and non-IoT devices and over 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='6% in classifying individual devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' These signify that the proposed method is useful in assisting operators in making their networks more secure and robust to security breaches and unauthorised access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Keywords : digital footprint;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' network traffic traces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' machine learning algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' internet of things;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' device fingerprinting  INTRODUCTION  It has been predicted that the number of network-connected Internet of Things (IoT) and non-IoT devices worldwide will reach approximately 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='9 billion and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='3 billion, respectively, by the year 2025 [1]\u2060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Proliferated growth of these devices with their heterogeneous functionalities, has imposed new challenges to network administrators and operators, in providing, managing, and controlling the operations and security of the network services [2]\u2060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Accurate device identification is one key aspect that needs to be seriously considered in securing network-connected devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Conventionally, internet protocol (IP) enabled devices have been using user-defined identifiers, such as IP and media access control (MAC) addresses, as a form of identifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' However, these identifiers have been proven to be vulnerable [3]\u2060 to various attacks, such as spoofing [4]\u2060 and device mobility, due to the availability of malicious software [5]\u2060, for performing such attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Device fingerprinting (DFP) [3]\u2060 represents one technique that may be used to identify devices based on their communication traffic traces (or digital footprints) without using explicit identifiers, and it can be performed, either actively or passively, from different layers of the communication model [6]\u2060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='  Due to the prominent characteristics of network traffic features, many researchers [2, 7]\u2060 have used packet-level features for different purposes [8]\u2060, including for device identification [9]\u2060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Sivanathan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' [10]\u2060 have described a DFP scheme based on the analysis of passively observed network traffic traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' A total of 11 statistical features are used as device fingerprints, from packet traffic-flows over a period of one day, by looking at the devices’ sleeping time, average packet size and traffic rate, active time, number of servers and protocols used in a flow, number of  8th Brunei International Conference on Engineering and Technology (BICET 2021), Universiti Teknologi Brunei unique domain name system (DNS) request, and intervals of DNS and network time protocol (NTP) requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Subsequently, these features are used to train an ML model for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' It has been shown that the DFP scheme is able to distinguish between IoT and non-IoT devices with high accuracy and achieve over 95% accuracy in identifying individual IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' The same researchers [9]\u2060 have also presented another device fingerprinting scheme, by utilizing statistical characteristics of hourly network traffic traces, to generate 8 device-specific fingerprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Experimental result has shown over 99% accuracy using the UNSW dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Charyyev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' [11]\u2060 have utilized Nilsimsa hash value of packet flows (n packets) for device-specific fingerprints, to classify individual IoT devices, to achieve 93% precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='  Researchers in [2, 12]\u2060 have used 12 packets information, to generate device signatures for classifying IoT devices, with 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='5% global accuracy and 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='15% accuracy using an aggregated model, whilst Aksoy and Gunes [13]\u2060 have presented a DFP approach, known as SysID, which utilizes features from a single packet, for identifying smart home IoT devices with 82% average classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Bezawada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' [14]\u2060 have utilized 5 consecutive packets information, including protocols headers and payload (20 features), for classifying IoT devices uniquely with mean identification accuracy of 93% to 100% using a laboratory dataset of 14 IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' In [15]\u2060, the authors have used a one second window to group packets, for generating statistical fingerprinting features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' These features are then used to train a binary classifier for categorizing IoT and non-IoT devices with high accuracy of 99%, whilst a multi-class classifier has been used to uniquely identify IoT devices with about 96% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' All these existing DFP models, however, require either a large number of features set from different layers of the communication model, or a large number of network packets information for generating fingerprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Consequently, these models consume a long period of time, and require complex computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' As such, a more efficient DFP model is required for classifying devices with high accuracy, but with less computation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='  In this paper, a supervised machine learning (ML) based DFP model, which generates device-specific signatures by computing four statistical features from consecutive five packets of the network traffic, has been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' An intuition that these features carry device-specific characteristics in terms of device memory and processing speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Experimental results have shown that over 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='0% accuracy is achievable in classifying individual non-IoT devices from traffic collected in a laboratory environment, and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='3% accuracy on the non-IoT traffic traces from the UNSW dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' The proposed DFP model is also capable of distinguishing between IoT and non-IoT devices with up to 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='8% accuracy on the UNSW dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' The key contributions of this research work are:  Identifying device  specific features from the device  originated communication traffic traces, to generate device signatures for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='  Instrument an experimental testbed of non  IoT devices in a laboratory environment for data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='  Evaluate the proposed DFP scheme performance based on a supervised ML algorithm, to distinguish between IoT and non  IoT devices and identify individual devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='  The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' The proposed ML-based device fingerprinting method, as well as the datasets, data collection procedure, and an ML classifier are described in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Section III describes experimental results on various datasets, and finally, conclusion is given in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='  METHODOLOGY  The proposed DFP method is used to extract unique device features from network traffic traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' These features are used to train an ML classifier, and subsequently, used to test the performance of the proposed DFP method on different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' This section describes the proposed DFP method, the datasets used for training and testing, as well as the classification method used to test the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='  Datasets: IoT and Non IoT  The proposed device fingerprinting model performance has been evaluated by utilizing a publicly available dataset: UNSW [9]\u2060, and a testbed dataset of non-IoT devices, which has been collected from a laboratory environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Summary of the datasets are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' The UNSW dataset comprises network traffic traces from both IoT and non-IoT devices, including TP-Link camera, smart bulb, Belkin camera, smart doorbell, printer,  8th Brunei International Conference on Engineering and Technology (BICET 2021), Universiti Teknologi Brunei smart photo frame, laptop, smartphone, and tablet devices, with these heterogeneous devices coming from different manufacturers: Belkin, Philips Hue, Netatmo, TP-Link, Withings, HP, Apple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' On the other hand, the laboratory dataset comprises 7 non-IoT devices, including laptops, smartphones, and desktops, from different manufacturers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' The data collection procedure from the 7 non-IoT devices is described in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='  TABLE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' List of IoT and non-IoT Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Dataset Devices Total Packets Source IoT Non-IoT UNSW 22 -- 6,844,740 [9]\u2060 -- 7 3,515,705 Lab Dataset -- 7 442,970 --  TABLE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' List of non-IoT devices for experimental set up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Device Category Device Name/Model Operating System Connectivity MAC Address 1 Laptop Aspire-S7 Windows WiFi 34:23:87:b7:56:17 2 ProBook-4410s WiFi/Ethernet 00:25:b3:47:da:6f 3 Desktop Asus Ethernet 08:60:6e:c1:79:c2 4 HP-EliteDesk Ethernet 80:e8:2c:d6:9e:49 5 Smart Phone MYA-U29 Android WiFi d0:ff:98:95:57:af 6 MLXP2ZA-A iOS WiFi e0:c7:67:45:a3:62 7 MWC22KH-A WiFi 06:44:b7:aa:20:98  Dataset Collection Methodology  An experimental design,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' consisting of local area network (LAN) and wireless local area network (WLAN) with non-IoT devices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' was set up in a laboratory environment at Universiti Brunei Darussalam (UBD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Design of the testbed is depicted in Figure 1, with the seven non-IoT devices from different manufacturers and of different types, as listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' These devices were configured, to connect with an access point (AP) either using ethernet or wireless fidelity (WiFi) interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='  FIGURE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' An experimental testbed of non-IoT devices network (LAN/WLAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='  DNS NTP Server Connectivity: Server Server N Ethernet WiFi Other a Internet WiFi Hotspot Gateway ubuntu?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' (UBD Network) Hub Ethernet USB Ethernet Port Port Monitoring Station (Capture Network Traffic)8th Brunei International Conference on Engineering and Technology (BICET 2021), Universiti Teknologi Brunei A laptop was used to configure an access point (AP), which was used to provide network services to the non-IoT devices, as well as to monitor and capture communication footprints from the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' The Dell Inspiron 15 5000 Series laptop runs Ubuntu 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='04 as an operating system (OS), and was connected to the UBD network via its built-in Ethernet interface, to provide the Internet connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' The built-in WiFi interface was configured as a WiFi Hotspot, providing wireless connectivity to the WiFi-enabled (IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='11 standard) devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Additionally, a TU3-ETG USB Ethernet adapter was connected to the laptop, and used to set up a LAN network using the D-Link Switch Hub DES- 1005A hub for providing network services to the connected non-IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' On the Ubuntu OS, the network connection editor tool, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' nm-connection-editor, was been utilised for connection establishment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='  Devices generally generate two types of traffic [9]\u2060: autonomous traffic, including traffic generated for connection establishment, application and system synchronizations, and activity traffic, which is generated due to human or object interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' These inbound and outbound communication traffic traces, flowing over both interfaces (external Ethernet and built-in WiFi interfaces) were captured using tcpdump 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='3 utility, and stored into .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='pcap (packet capture) files format, similar to [16]\u2060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Device-originated traffic traces were then extracted using TShark utility and stored in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='csv (comma-separated values) files format, along with labelling of individual devices names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Finally, the recorded dataset was cleaned for further processing, by eliminating inconsistent instances, including empty rows, and duplicate values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='  Device Fingerprinting Model  The proposed DFP scheme architecture is depicted in Figure 2, which uses device-originated communication traffic traces to generate device fingerprints for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Device-originated traffic traces are filtered according to individual device MAC addresses, with tcp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='window_size and ip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='len values extracted from each packet from the available captured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' These two values of a network packet carry significant device-specific information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' tcp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='window_size value depends on a device buffer size and computation speed [14]\u2060 whilst ip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='len value specifies the total length of a packet to represent unique characteristics of a devices communication pattern [15]\u2060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' tcp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='window_size and ip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='len values from five consecutive packets (as one instance) are utilized, to compute mean (µ) and standard deviation (σ), for constructing device-specific fingerprints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' iplen_µ, iplen_σ, tcpwinsiz_µ, and tcpwinsiz_σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' These 4 statistical fingerprints have been used for training a machine learning (ML) model, and subsequently, to evaluate the performance, of the model in classifying devices using datasets, which have been randomly split into training (80% instances) and testing (20% instances) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='  FIGURE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' The proposed device fingerprinting scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='  Random Forest Classifier  Random Forest (RF) classifier is a supervised machine learning (ML) algorithm, that can be used for both classification [9]\u2060 and regression [17]\u2060 problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' The algorithm randomly generates a group of trees, with majority voting used to make a decision from the ensemble of decision trees [18, 19]\u2060, for the classification task, as presented in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' This assists in avoiding over-fitting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Researchers in different domains have utilized RF classifier for different classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' In [9]\u2060, the RF algorithm has been used for classifying IoT devices with high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Primartha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' [20] have performed anomaly detection using the algorithm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' and it has also been used  Testing Dataset (20%) Training Dataset (80%) Capture Filter and Extract Fingerprint Generation Training Model Test Model Classification Network Traffic Traffie Traces (Mean,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='Standard ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='Deviation) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='ML ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='Algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='ML ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='Algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
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+page_content='on ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='Engineering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='Technology ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='(BICET ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='2021),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Universiti Teknologi Brunei for disease identification in medical science [21]\u2060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' In this paper, RF classifier is used to appraise the performance of the proposed DFP method, by using the extracted features for training the RF classifier, and subsequently, using the trained RF classifier to determine classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Some of the significant tunable hyper-parameters are set experimentally, including the number of iterations (or number of trees) = 100, seed = 1, and batch size (number of instances) = 100, to improve classification accuracy and reduce the root mean squared error (RMSE) [22]\u2060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='  FIGURE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' An abstract representation of a RF classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='  RESULTS AND DISCUSSION  The proposed DFP method has been evaluated using waikato environment for knowledge analysis (Weka) tool [23]\u2060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' An online dataset: UNSW [9]\u2060 dataset, and an experimental dataset, as presented in Table 3, have been utilized to evaluate the classification performance based on the RF classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' The UNSW dataset consists of network traffic traces from IoT and non-IoT devices, which are referred to as the U-IoT and U-NonIoT datasets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' On the other hand, the experimental dataset contains only network traffic traces from non-IoT devices, and it is referred to as the L-NonIoT dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='  TABLE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Total number of instances used for evaluating the proposed DFP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Dataset Devices Training Dataset (80%) Test Dataset (20%) Total Instances (100%) IoT Non-IoT UNSW  (U-IoT) * --- 1,095,158 273,790 1,368,948 UNSW  (U-NonIoT) --- * 562,513 140,628 703,141 Lab (L-NonIoT) --- * 70,875 17,719 88,594  The proposed DFP method utilises 5 network traffic packets as one instance to generate fingerprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' As such, a total of 1,368,948 (6,844,740 / 5) and 703,141 (3,515,705 / 5) instances have been used from the U-IoT and U- NonIoT datasets, respectively, whilst a total of 88,594 (442,970 / 5) instances have been used from the L-NonIoT dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' 80% of the datasets have been used for training and the remainder for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' The performance of the trained RF classifier has been measured with respect to its ability to a) distinguish between IoT and non-IoT devices, and b) classify individual devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='  Device Category: IoT and Non-IoT Devices  Classification performances of the proposed DFP model in distinguishing between IoT and non-IoT devices are presented in Figure 4, on combined U-IoT and U-NonIoT datasets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' UNSW dataset), and combined U-IoT and L-  Dataset Data Subset of Data Subset of Data Subset of Data Random Samples 1 2 n Decision Trees Selected Class Selected Class Class Selected Class (Vote) (Vote) (Vote) Majority Voting Final Decision (Class)8th Brunei International Conference on Engineering and Technology (BICET 2021), Universiti Teknologi Brunei NonIoT datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' The figure shows that device categorization accuracy reaches up to 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='9% using the RF classifier on the combined U-IoT and L-NonIoT datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' On the UNSW dataset [9]\u2060, which consists of instances from 22 IoT and 7 non-IoT devices, the proposed DFP method achieves 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='8% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='  FIGURE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Categorize IoT and non-IoT devices: UNSW and Lab datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='  FIGURE 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Classification performance of the non-IoT devices: UNSW and Lab datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='  Individual Device Classification  The performances of the proposed DFP method in classifying individual IoT and non-IoT devices on different datasets, are depicted in Figure 5 and Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' In Figure 5, the proposed DFP model achieves over 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='0% accuracy in classifying non-IoT devices from the L-NonIoT and U-NonIoT datasets, with accuracy a little bit higher on the U- NonIoT dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Individual IoT devices classification performance of the proposed DFP model, on the U-IoT dataset with 22 IoT devices, is given in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Most of the IoT devices in the dataset can be classified with over 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='6% accuracy, with the exception of the BlipcareBPmeter, the BelkinWemoSensor and BelkinWemoSwitch devices, which give classification accuracies of about 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='0%, 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='5% and 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='4%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' The lowest accuracy for the BlipcareBPmeter device is due to the limited number of instances available from this device for training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='  CONCLUSION  A large number of heterogeneous IoT and non-IoT devices from different manufacturers are being connected to the Internet, to obtain network-based services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' In terms of network security, it is challenging for network administrators and operators to identify the connected devices using conventional identifiers, as they are prone to security breaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' In this paper, a DFP model based on the analysis of network traffic traces has been proposed, which is capable of distinguishing between IoT and non-IoT devices as well as classifying individual IoT and non- IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' As opposed to other methods in the literature, which require relatively large number of features and  loTvs NonloT U-loT: UNSW-loT, U-NonloT: UNSW-NonloT, L-NonloT: Lab-NonloT Datasets U-loT vs U-NonloT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='998 Datasets U-loT vs L-NonloT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
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+page_content='00 Accuracy8th Brunei International Conference on Engineering and Technology (BICET 2021), Universiti Teknologi Brunei requiring longer sequence of packet network traffics to construct their DFP features, only 4 statistical features from 5 consecutive packet network traffics are required to construct the DFP features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' These are used for training and testing an ML classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Evaluations on the UNSW dataset have shown that the proposed DFP method is able to distinguish between IoT and non-IoT devices with up to 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='8% accuracy, and individually classify most of the IoT and non-IoT devices with over 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='6% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' On the laboratory collected network traces, the proposed DFP model is able to classify individual devices with 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='0% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' The research outcomes signify that the proposed DFP model is useful for device identification and may assist network administrators in providing a more secure network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='  FIGURE 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Individual IoT device classification performance: U-IoT dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  The authors are profoundly grateful to the Faculty of Integrated Technologies (FIT), Universiti Brunei Darussalam (UBD), for supporting this research work, as well as to UBD for awarding the UBD Graduate Scholarship (UGS) to the first author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
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+page_content=', Study of cardiovascular disease prediction model based on random forest in eastern China, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' 10 (2020) 1–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='1038/s41598-020-62133-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Breiman, Random forests, Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' 45 (2001) 5–32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content='1023/A:1010933404324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Frank, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Hall, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Witten, The WEKA Workbench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Online Appendix for Data Mining: Practical Machine Learning Tools and Techniques, 4th ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}
+page_content=' Morgan Kaufmann (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'}

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+J/ψ polarization in large-PT semi-inclusive deep-inelastic scattering at the EIC
+Umberto D’Alesio,1, 2, ∗ Luca Maxia,1, 2, † Francesco Murgia,2, ‡ Cristian Pisano,1, 2, § and Sangem Rajesh3, 4, ¶
+1Dipartimento di Fisica, Universit`a di Cagliari, Cittadella Universitaria, I-09042 Monserrato (CA), Italy
+2INFN, Sezione di Cagliari, Cittadella Universitaria, I-09042 Monserrato (CA), Italy
+3Department of Physics, School of Advanced Sciences,
+Vellore Institute of Technology, Vellore, Tamil Nadu 632014, India
+4INFN, Sezione di Perugia, via A. Pascoli snc, 06123, Perugia, Italy
+(Dated: January 31, 2023)
+We present a detailed phenomenological study of J/ψ polarization in semi-inclusive deep inelastic
+scattering processes, focusing on the kinematics accessible at the future Electron-Ion Collider. We
+show theoretical estimates for the standard polarization parameters for different frames usually
+adopted in the literature, in the large PT region, namely PT ≫ ΛQCD, where collinear factorization
+is expected to hold. We adopt both the Color Singlet Model and the Nonrelativistic QCD approach,
+paying special attention to the role of different sets of Long Distance Matrix Elements. Finally we
+present a preliminary analysis of some frame independent polarization invariants.
+I.
+INTRODUCTION
+Our understanding of the J/ψ production mechanism at high energies has improved significantly since its discovery
+almost 50 years ago [1, 2], thanks to the combined efforts from both the theoretical and experimental communities.
+However, there are still major problems in the theoretical analyses of the available data, such as the long-standing
+J/ψ polarization puzzle. Namely, J/ψ polarization measurements cannot yet be explained in a way entirely consistent
+with the world experimental results for the unpolarized J/ψ yields.
+The present theoretical frameworks all agree in providing a perturbative description of the creation of the charm
+quark-antiquark (c¯c) pair. The charm mass mc plays the role of the hard scale, since it is much larger than the
+asymptotic scale parameter of QCD, ΛQCD. These approaches nonetheless differ in the treatment of the subsequent
+nonperturbative transition to the hadronic bound state. For instance, in the traditional Color-Singlet Model (CSM) [3]
+the c¯c pair is produced at short distances directly with the quantum numbers of the J/ψ meson, i.e. in a color-singlet
+(CS) state with spin one and no orbital angular momentum. This is possible by the emission of an additional hard
+gluon, which implies the suppression of the cross section by one power of the strong coupling constant αs. However,
+the CSM cannot be considered as a complete theory, since at the next-to-leading order (NLO) P-wave quarkonia are
+affected by uncanceled infrared singularities.
+These singularities are properly removed in the effective field theory approach of nonrelativistic QCD (NRQCD),
+based on a rigorous factorization theorem, which was assumed in the original paper by Bodwin, Braaten, and Lep-
+age [4], and later explicitly proven to next-to-next-to-leading order (NNLO) [5]. NRQCD therefore implies a sep-
+aration of process-dependent short-distance coefficients, to be calculated perturbatively as expansions in αs, from
+long-distance matrix elements (LDMEs), which are expected to be universal and have to be extracted from experi-
+ments. Scaling rules [6] predict each of the LDMEs to scale with a definite power of the relative velocity v of the heavy
+quark-antiquark pair in the quarkonium rest frame in the limit v ≪ 1. Observables are hence evaluated by means of
+a double expansion in αs and in v, with αs ≃ 0.2 and v2 ≃ 0.3 for charmonium states. An essential feature of this
+approach is that the c¯c pair at short distance can be produced in any Fock state n = 2S+1L[c]
+J with definite orbital
+angular momentum L, spin S, total angular momentum J and color configuration c = 1, 8. NRQCD hence predicts
+the existence of intermediate color-octet (CO) states, which subsequently evolve into physical, CS quarkonia by the
+emission of soft gluons. For S-wave quarkonia, the CSM is recovered in the limit v → 0. In the specific case of J/ψ
+production, the CSM prediction is based only on the 3S[1]
+1
+CS state, while NRQCD includes the leading relativistic
+corrections as well, which at the relative order O(v4) are given by the CO states 1S[8]
+0 , 3S[8]
+1 , and 3P [8]
+J
+with J = 0, 1, 2.
+The values of the CO LDMEs extracted from different fits to data on J/ψ and Υ yields [7–11] are not compatible
+with each other, even within the large uncertainties [12–14]. Therefore, any new method to determine them with
+better precision is worth exploring [15–17]. In this paper we propose to look at the J/ψ polarization parameters in
+∗ umberto.dalesio@ca.infn.it
+† luca.maxia@ca.infn.it
+‡ francesco.murgia@ca.infn.it
+§ cristian.pisano@unica.it
+¶ sangem.rajesh@vit.ac.in
+arXiv:2301.11987v1  [hep-ph]  27 Jan 2023
+
+2
+semi-inclusive deep-inelastic scattering (SIDIS), e p → e′ J/ψ X, in a kinematic region where the transverse momentum
+of the J/ψ meson PT is large, namely PT ≫ ΛQCD, and collinear factorization is expected to hold. Analysing SIDIS at
+finite values of the exchanged photon virtuality Q2 has certain experimental and theoretical advantages as compared to
+photoproduction. Namely, as Q2 increases theoretical uncertainties in the different frameworks decrease and resolved
+photon contributions are expected to be negligible. Moreover, background from diffractive J/ψ production is expected
+to decrease with Q2 faster than the SIDIS cross section. The distinct signature of the scattered lepton makes the
+process particularly easy to detect. Clearly, cross sections are smaller than those expected in the photoproduction
+case, however, considering the achievable high luminosities, this study should be feasible at the future Electron-Ion
+Collider (EIC) planned in the United States [18–20].
+So far, only a single experimental study of J/ψ polarization in SIDIS has been performed, by the H1 Collaboration
+at HERA [21]. Such a measurement is limited to the polarization parameter λ in the helicity frame. This result turns
+out to be compatible with the predictions provided in Refs. [22, 23], but it can hardly discriminate among the different
+models. In analogy with Refs. [22, 23], our phenomenological analysis has been carried out at the perturbative order
+α2
+s, which has to be considered as the state of the art for these observables. Higher-order effects have been calculated
+very recently only for the unpolarized cross section within the CSM [24]. Anyway, we expect these effects (at least
+in the large Q2 region) to be small for the observables we are investigating, because they are ratios of cross sections.
+We point out that our estimates include also the polarization parameters µ and ν, not addressed in Refs. [22, 23],
+which are studied in different reference frames. Furthermore, we perform a preliminary study of rotational invariant
+combinations of these parameters.
+The remainder of the paper is organized as follows. In section II we recall the standard SIDIS variables and collect
+the expressions of the differential cross section for quarkonium production and its leptonic decay in terms of the helicity
+structure functions and the polarization parameters. In section III we discuss the three polarization parameters λ, µ,
+ν, showing their estimates in two reference frames and paying special attention to their energy, z and PT dependences
+as well as to the impact of the LDME set adopted. To overcome the intrinsic frame dependence of the polarization
+parameters, in section IV we present two classes of the so-called rotational invariant quantities, and show, as a case
+of study, some results for one of them. Finally in section V we gather our conclusions.
+II.
+KINEMATICS AND FORMALISM
+In this section we provide the main analytic expressions needed to carry out the phenomenological analysis. For
+more details and the complete formalism we refer the reader to Ref. [25]. We consider the SIDIS process
+e(k) + p(P) → e′(k′) + J/ψ(Pψ) + X(PX) ,
+(1)
+with the subsequent J/ψ decay into a lepton pair
+J/ψ(Pψ) → l+(l) + l−(l′) ,
+(2)
+where, in brackets, we have shown the four-momenta of each particle. The J/ψ meson is produced via the partonic
+subprocess
+γ∗(q) + a(pa) → c¯c[n](Pψ) + a(p′
+a) ,
+(3)
+with q2 = −Q2 and P 2
+ψ = M 2
+ψ = (2mc)2. The initial parton momentum, pa, is related to the parent proton one, P, as
+pa = ξP .
+(4)
+We adopt the following three standard invariant quantities, defined in terms of the photon and hadron momenta
+xB =
+Q2
+2P · q ,
+y = P · q
+P · k ,
+z = P · Pψ
+P · q ,
+(5)
+where xB is the Bjorken variable, y is the inelasticity and z is the energy fraction carried out by the J/ψ (in the
+proton rest frame). All these variables are constrained in the region 0 ≤ xB, y, z ≤ 1 and they are connected to other
+kinematical quantities of the system, like the total center-of-mass (cm) energy √s and the virtual photon-proton cm
+energy, W.
+The cross section that describes the J/ψ formation and its decay into a lepton pair can be written as
+1
+Bll
+dσ
+dxB dy dz d2PT dΩ =
+α
+8 y z Q2
+3
+8π
+�
+WT (1 + cos2 θ) + WL(1 − cos2 θ) + W∆ sin 2θ cos φ + W∆∆ sin2 θ cos 2φ
+�
+,
+(6)
+
+3
+where PT is the J/ψ transverse momentum in the cm frame of the virtual photon and the proton, Bll is the branching
+ratio for the decay process J/ψ → ℓ+ℓ− and Ω(θ, φ) refers to the solid angle spanned by the lepton ℓ+ in a reference
+frame where the system formed by ℓ+ and ℓ− is at rest. Moreover, we have introduced the following helicity structure
+functions
+WT ≡ W11 = W−1,−1 ,
+WL ≡ W00 ,
+W∆ ≡
+1
+√
+2 (W10 + W01) =
+√
+2 Re [W10] ,
+W∆∆ ≡ W1,−1 = W−1,1 ,
+(7)
+where the subscripts refer to the J/ψ polarization states. More specifically, WT and WL are respectively the structure
+functions for transversely and longitudinally polarized J/ψ mesons, W∆ is the single-helicity flip structure function,
+and W∆∆ is the double-helicity flip one. Notice that in Eq. (6) we have introduced a proper overall constant factor
+w.r.t. Eq. (2.35) of Ref. [25] to ensure the normalization when integrated over the solid angle, see Eq. (8) below.
+This does not affect any conclusion of Ref. [25], where all relevant quantities are defined as ratios of helicity structure
+functions.
+As shown in Ref. [25], the structure functions in Eq. (7) can be further decomposed in terms of the contributions
+coming from the longitudinal ( ) and transverse (⊥) polarizations of the virtual photon. Moreover, within a collinear
+factorization scheme, they are given as convolutions of collinear parton distribution functions (PDFs) with partonic
+helicity structure functions (weighted by proper LDMEs). These, in turn, can be expressed as functions of the partonic
+Mandelstam invariants.
+The unpolarized cross section is obtained by integrating Eq. (6) over the solid angle Ω,
+1
+Bll
+dσ
+dxB dy dz d2PT
+=
+α
+8 y z Q2 (2WT + WL) .
+(8)
+It is then useful to introduce the ratio of polarized and unpolarized cross sections
+dN
+dΩ ≡
+dσ
+dxB dy dz d2PT dΩ
+�
+dσ
+dxB dy dz d2PT
+�−1
+,
+(9)
+which can be expressed as follows
+dN
+dΩ = 3
+4π
+1
+λ + 3
+�
+1 + λ cos2 θ + µ sin 2θ cos ϕ + 1
+2 ν sin2 θ cos 2ϕ
+�
+,
+(10)
+where we have defined the polarization parameters
+λ = W11 − W00
+W11 + W00
+,
+µ =
+√
+2 Re [W10]
+W11 + W00
+,
+ν =
+W1, −1
+W11 + W00
+,
+(11)
+or alternatively adopting Eq. (7),
+λ = WT − WL
+WT + WL
+,
+µ =
+W∆
+WT + WL
+,
+ν =
+2 W∆∆
+WT + WL
+.
+(12)
+The parameterizations shown in Eqs. (6) and (10) are standard for the study of the angular distribution of a spin-one
+particle decay into a lepton pair and, indeed, they are commonly adopted in Drell-Yan processes [26] and in J/ψ
+photoproduction [27].
+Among the polarization coefficients, λ, µ and ν, the most investigated experimentally is λ.
+Moreover, from
+the phenomenological point of view it has a very intuitive interpretation, with λ = +1(−1) describing a trans-
+verse(longitudinal) polarization state for the J/ψ (i.e. a J/ψ helicity equal to ±1 or 0), while λ = 0 for an unpolarized
+one.
+The main goal of our study is to present estimates for these polarization quantities, within both the CSM and the
+NRQCD frameworks, focusing on the kinematic region accessible at the future EIC. As we will show in the following,
+such a detailed phenomenological study could help in disentangling among the production mechanisms.
+
+4
+LDME Set
+⟨O1[ 3S1]⟩
+�
+GeV3� ⟨O8[ 1S0]⟩
+�
+GeV3� ⟨O8[ 3S1]⟩
+�
+GeV3� ⟨O8[ 3P0]⟩
+�
+GeV5�
+C12
+1.16
+0.089
+0.003
+0.0126
+G13
+1.16
+0.097
+−0.0046
+−0.0214
+BK11
+1.32
+0.0304
+0.00168
+−0.00908
+Table I. LDME set (central) values for the J/ψ state: C12 [8], G13 [28] and BK11 [29]. For the other 3PJ states we use the
+standard spin-symmetry relation ⟨O8[ 3PJ]⟩ = (2J + 1) ⟨O8[ 3P0]⟩.
+III.
+ANGULAR DISTRIBUTIONS
+In this section we analyze the polarization parameters defined in Eq. (11) showing both their z and PT distributions.
+The explicit analytic expressions of the underlying partonic structure functions, calculated at the perturbative order
+α2
+s, are presented in Ref. [25] for the so-called Gottfried-Jackson frame, together with all prescriptions needed to
+transform them in the other relevant frames. For the predictions based on the NRQCD approach, in addition to the
+CS contribution, given by a pure gluon fusion channel, we consider the CO channels up to the order v4, which involve
+both gluon and quark final states. The CTEQ6L1 set [30] is used for the unpolarized parton distribution functions.
+Moreover, in order to assess the stability of our results against higher order corrections, we produce uncertainty bands
+by varying the factorization scale µF in the range µ0/2 < µF < 2µ0, around the central value µ0 =
+�
+Q2 + M 2
+ψ.
+Concerning the CO LDME values, three different sets are adopted, see Table I. Here we only recall their main
+features: the C12 set [8] has been extracted simultaneously from both polarized and unpolarized J/ψ production
+data in pp collision at PT > 7 GeV, measured by the CDF (Run II) Collaboration; the G13 set [28] is obtained
+including only PT > 7 GeV unpolarized data from the CDF and LHCb Collaborations and then used to predict
+the J/ψ polarization in pp collisions; it is in agreement with the C12 set if feed-down contribution is negligible; the
+BK11 set [29] is based on a fit without any polarization data, but starting from a lower PT value, around 3 GeV, and
+including both photoproduction and hadroproduction data.
+The high cm energy kinematical set-ups expected at the EIC are an ideal environment to study J/ψ polarization in
+electroproduction. Moreover, they will allow to better explore high photon virtualities (Q), avoiding the competing
+contributions from photoproduction. Furthermore, since we are interested in the region where collinear factorization
+holds, our results will be shown only for PT values above PT min = 1 GeV. Notice that around this value we actually
+enter the region where the transverse momentum dependent (TMD) factorization could be applied and therefore our
+estimates are pushed down to the overlapping region of validity of the two factorization schemes.
+A.
+The λ parameter
+In Fig. 1 we present our predictions for λ at √s = 140 GeV, as a function of both the J/ψ energy fraction z
+(left panels) and its transverse momentum PT (right panels). Two quarkonium rest frames are explicitly considered:
+the Gottfried-Jackson (upper panels) and the Helicity (lower panels) ones. In this and in the following figures, the
+kinematical ranges explored are indicated in the legend boxes. For completeness we report here the corresponding
+regions explored in xB and y at √s = 140 GeV, 10−3 ≲ xB ≲ 0.2 and y ≲ 0.5 respectively, even if the effectively
+probed maximum value in xB is around 0.07.
+Concerning other typical frames, like the Target and Collins-Soper ones, we only notice that the first one give
+estimates very close to those in the Helicity frame, while predictions obtained in the second one, at least for the
+kinematics considered, are in general much smaller than those in the Gottfried-Jackson frame or even close to zero.
+Notice that for such observable, defined as a ratio of cross sections, the dependence on the scale µF in the range
+[µ0/2, 2µ0] is barely appreciable and therefore is not shown.
+The study of the λ parameter as a function of z presents very interesting features from the phenomenological
+point of view. The reasons are manifold: first of all its expected relative large size as compared to the µ and ν
+parameters. Moreover, it is experimentally under more active investigation. On the other hand, theoretical estimates
+for λ as a function of z (for small and moderate values) do not vary significantly adopting different frameworks
+(Fig. 1, left panels), which implies that, in order to get information on the quarkonium formation mechanism, one
+would need highly precise measurements. The same problem was found in different analyses performed by the HERA
+Collaborations, Refs. [21, 23].
+The situation changes considerably at z > 0.6, which represents a very interesting region from the phenomenological
+point of view. As is well known, NRQCD estimates for the unpolarized cross section manifest a divergent behavior as
+
+5
+0.50
+0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+Gottfried-Jackson
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.2
+0.4
+0.6
+0.8
+z
+0.50
+0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+Helicity
+CSM
+NRQCD (C12)
+NRQCD (BK11)
+NRQCD (G13)
+2
+4
+6
+8
+10
+PT [GeV]
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+s = 140 GeV
+9 GeV2 < Q2 < 100 GeV2
+20 GeV < W < 100 GeV
+0.2 < z < 0.9 or PT > 1 GeV
+Figure 1. Estimates for λ at √s = 140 GeV as a function of z (left panels) and PT (right panels) for different models and
+LDME sets and two reference frames: Gottfried-Jackson (upper panels) and Helicity (lower panels) frames. Integration ranges
+are given in the light-blue legend box.
+z → 1, due to the corresponding ˆt → 0 singularities. This can potentially spoil the validity of NRQCD factorization.
+As shown in Ref. [31], in order to extend the region of applicability of NRQCD up to 1 − z ∼ v2, one can introduce
+a new set of functions, the so-called shape functions [32], that allow to improve noticeably the convergence for
+photoproduction. We expect such quantities to be relevant also for the SIDIS process, together with their TMD
+extensions, which have been adopted in the study of pp collisions in Refs. [33, 34] and whose perturbative tails have
+been derived in Ref. [35] for unpolarized and in Ref. [25] for polarized J/ψ SIDIS. On the other hand, the impact of
+the shape functions on λ is expected to be strongly reduced since λ is a ratio of cross sections. This can be tested
+with future available data.
+A much more powerful tool to assess the relevance of the CO contributions is the study of the PT distribution
+(Fig. 1, right panels). In the Gottfried-Jackson frame (upper panel) we see a clear separation as well as a different
+behavior between the CSM and NRQCD curves, in particular in the region 4 < PT < 7 GeV; similarly in the Helicity
+frame there is a wide separation between the CSM and the NRQCD curves, while different LDME sets give predictions
+much closer to each other and closer to λ = 0. It is worth noticing that, even if the unpolarized cross section decreases
+as PT increases, a good separation can be found already around PT ≃ 5 GeV, which is also far away from the TMD
+region.
+Before concluding the analysis of λ at large cm energies, a comment on the contributions from different partonic
+channels and/or different NRQCD waves can be useful. Concerning the z distribution, we find that the main con-
+tribution to the numerator of λ comes from the (gluon) CS wave, while the differences among NRQCD predictions,
+especially around z → 0.9, are due to the gluon P-wave, modulated by the corresponding LDME parameter. For the
+PT distribution we find, similarly, that the CS term is on the whole the most relevant contribution, followed again by
+the gluon P-wave one. In particular at PT → 1 GeV the size of the gluon P-wave contribution becomes comparable
+to (or even larger than) the CS one; moreover, since the low-PT region dominates the integration over PT , one can
+also understand why the gluon P-wave is so relevant in our estimates vs. z, with the most visible effects for z → 0.9.
+At medium PT values the quark P-wave starts becoming important and at even higher PT values it is similar in
+size to the gluon one; this means that in this region, the full P-wave contribution (gluon+quark) dominates over the
+CS one.
+Another interesting possibility given by the future EIC facility is the corresponding analysis at smaller energies:
+in the following we will adopt √s = 45 GeV. In this case, different integration ranges have been considered for W
+and Q2, as reported in the legend box of Fig. 2. These, in turn, correspond to 10−3 ≲ xB ≲ 0.5 (with an effective
+
+6
+0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+Gottfried-Jackson
+0.2
+0.0
+0.2
+0.4
+0.2
+0.4
+0.6
+0.8
+z
+0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+Helicity
+s = 45 GeV
+2.5 GeV2 < Q2 < 100 GeV2
+10 GeV < W < 40 GeV
+0.2 < z < 0.9 or PT > 1 GeV
+2
+4
+6
+8
+10
+PT [GeV]
+0.2
+0.0
+0.2
+0.4
+CSM
+NRQCD (C12)
+NRQCD (BK11)
+NRQCD (G13)
+Figure 2.
+Estimates for λ at cm energy √s = 45 GeV. The integration region, different with respect to the higher-energy case,
+is given in the red legend box, while curves and panels have the same meaning as in Fig. 1. The scale error bands are sizable
+and explicitly shown only for the CSM prediction as a function of PT .
+upper limit around xB ≃ 0.2) and y ≲ 0.8, a more valence-like region w.r.t. the previous case. Moreover, since at
+lower energies it is more difficult to reach high photon virtualities, we get contributions mostly from moderately low
+Q2. Consistently we adopt a lower limit, Qmin ≃ 1.6 GeV, in the integration. Notice that in this kinematic region, at
+least for the high PT dependence of λ within the CSM, the scale error bands are once again sizeable enough.
+From Fig. 2 (left panels) we can see that the z distribution does not depend significantly on the energy for z ≤ 0.6,
+while at higher z values the estimates are closer to zero, at variance with those at higher cm energy. As said, a
+polarization study pushed up to this regime can suffer from factorization breaking effects in NRQCD even if data in
+this region could be relevant from the phenomenological point of view. We also observe a rapid variation of all curves
+in the Helicity frame at z ∼ 0.1. This is due to geometrical factors which are energy dependent (see also Eq. (A16)
+of Ref. [25]). The same variation is also present at higher cm energy, but for z < 0.1 (outside the range shown in the
+lower-left panel of Fig. 1).
+Concerning the PT dependence, Fig. 2 (right panels), we notice that the CSM results are very different with respect
+to the corresponding ones in Fig. 1, while the same is not true for the NRQCD cases. This is related to the different
+virtualities explored, on which the CSM estimates depend heavily. This difference can be considered as an extra tool
+in the quest of discerning among different frameworks.
+Finally, we briefly comment on how the parton and/or wave contributions vary with the energy.
+While the z
+distribution manifests almost no energy dependence, the PT spectrum presents interesting features in the two frames
+considered. For the Gottfried-Jackson one the relative contribution from the quark P-wave is widely increased at this
+lower energy, making it the leading term in the numerator at medium/high PT . Regarding the Helicity frame the
+situation is, potentially, even more interesting, since the CSM and P-wave (both gluon and quark) contributions are
+highly suppressed at this energy, especially at large PT . The main role is then played by the 3S(8)
+1
+quark wave, which
+is responsible for the difference among the predictions based on the LDME sets considered. Even if in this region it
+is quite hard to expect precise enough data to discriminate between models, it is nevertheless worth stressing that it
+could be very useful in constraining the nonperturbative physics.
+
+7
+0.8
+0.6
+0.4
+0.2
+0.0
+0.2
+Gottfried-Jackson
+0.75
+0.50
+0.25
+0.00
+0.25
+0.50
+0.2
+0.4
+0.6
+0.8
+z
+0.8
+0.6
+0.4
+0.2
+0.0
+0.2
+Helicity
+s = 140 GeV
+9 GeV2 < Q2 < 100 GeV2
+20 GeV < W < 100 GeV
+0.2 < z < 0.9 or PT > 1 GeV
+2
+4
+6
+8
+10
+PT [GeV]
+0.75
+0.50
+0.25
+0.00
+0.25
+0.50
+CSM
+NRQCD (C12)
+NRQCD (BK11)
+NRQCD (G13)
+Figure 3. Estimates for the parameter µ at √s = 140 GeV. Paneling order is the same as in Fig. 1. Integration ranges are
+given in the blue legend box.
+B.
+The µ parameter
+Estimates for the µ parameter are again provided both in the Gottfried-Jackson and in the Helicity frames, as a
+function of z and PT at √s = 140 GeV, Fig. 3, and √s = 45 GeV, Fig. 4.
+From these figures we see that the Gottfried-Jackson frame is the best choice to discern among the CSM and
+NRQCD approach. A similar conclusion holds for the parameter ν as well, see the discussion in Sec. III C. Indeed,
+in Fig. 3 the separation between the CSM estimates and the corresponding NRQCD ones are remarkably sizeable for
+z ≳ 0.5 and PT ≳ 5 GeV. On the contrary, estimates in the Helicity frame both with respect to z and PT are so close
+to each other that one cannot draw any conclusion.
+The wave/parton decomposition of the W∆ helicity function, that is directly related to the µ numerator, allows us
+to get some further insights. The main CO contribution comes from the P-wave term. In particular, differences in
+NRQCD predictions as a function of z (left panels of Fig. 3) are driven by the gluon P-wave LDMEs. Moreover, the
+gluon P-wave dominates the numerator behavior with respect to PT too (right panels of Fig. 3). In addition, we find
+that the NRQCD predictions in the Gottfried-Jackson frame receive a significant contribution from the gluon P-wave
+also at low-PT , namely PT ≲ 3 GeV. At variance with the behavior in z, here the quark P-wave channel is relevant
+at high PT , especially when considering the Helicity frame.
+Moving to the lower cm energy, we see that the CSM µ estimates in the Gottfried-Jackson frame, Fig. 4 (upper
+panels), vary significantly for z ≳ 0.5 and PT ≳ 5 GeV, as compared with what happens at √s = 140 GeV. We
+remark that this variation can also appear via a proper Q-binning in the higher cm energy case (√s = 140 GeV).
+In contrast, estimates within the Helicity frame at lower energies (lower panels of Fig. 4) do not present the same
+energy/Q-binning dependence. The only remarkable exception resides in the PT distribution, where CSM predictions
+increase up to ∼ 40%, to be compared with the √s = 140 GeV case where the CSM result is at most ∼ 25%. Despite
+this, µ estimates in the Helicity frame do not differ enough to discern among different models.
+Looking at the wave/parton decomposition, we confirm that also for the µ numerator the role of quarks is enhanced
+at lower energies. This is particularly true for the PT dependence. Here we find that NRQCD predictions at the
+higher PT values, namely PT ≳ 6 GeV, are mostly driven by the quark P-wave; moreover, in the same PT region we
+observe that the 3S[8]
+1
+quark wave is non-negligible.
+
+8
+0.75
+0.50
+0.25
+0.00
+0.25
+0.50
+Gottfried-Jackson
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.2
+0.4
+0.6
+0.8
+z
+0.75
+0.50
+0.25
+0.00
+0.25
+0.50
+Helicity
+s = 45 GeV
+2.5 GeV2 < Q2 < 100 GeV2
+10 GeV < W < 40 GeV
+0.2 < z < 0.9 or PT > 1 GeV
+2
+4
+6
+8
+10
+PT [GeV]
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+CSM
+NRQCD (C12)
+NRQCD (BK11)
+NRQCD (G13)
+Figure 4. Estimates for the parameter µ at √s = 45 GeV. Paneling order is the same as in Fig. 1. Integration ranges are given
+in the red legend box.
+C.
+The ν parameter
+We now discuss the parameter ν, which is particularly important in the TMD framework, since it is directly related
+to the TMD distribution of linearly polarized gluons inside an unpolarized proton, h⊥g
+1 . This could play a role in the
+region of moderately low PT , where the two factorization schemes overlap.
+Again, we focus initially on the higher cm energy (√s = 140 GeV), Fig. 5, and then we describe the main differences
+with respect to the smaller cm energy (√s = 45 GeV), Fig. 6.
+Starting from the z-dependent distribution in Fig. 5 (left panels), we see once again that even if the estimated
+ν values are potentially sizeable, at least in the Helicity frame, the separation among the different approaches is in
+general very poor. Nevertheless, it is worth remarking that at high z we find more sensitivity to the LDME sets in
+the NRQCD framework. The situation is slightly different for the PT case (right panels): if the Helicity frame does
+not show a promising scenario, in the Gottfried-Jackson case the differences in the medium/high-PT region between
+the two approaches are sizeable.
+As said, results at high z and/or small PT are in general promising for future analyses regarding the h⊥g
+1
+gluon
+distribution in the TMD region. Nevertheless, it is important to remark that for the ν parameter the shape functions
+and their TMD extensions enter, potentially, in a different way in the numerator and the denominator, and their role
+could be important. This requires further investigation, together with a full higher-order description in αs, which is
+not available at present.
+It is once again interesting to look into the parton and wave decomposition. The z-dependent W∆∆ is dominated,
+for almost all z values, by the CS wave; only for z → 0.9 the CS contribution becomes negligible, and the results
+are driven by the CO P-wave, in particular by the gluon term. Moving to the PT dependence, we find again some
+similarities with the λ case: the CS term is the relevant contribution to the numerator over the whole PT spectrum,
+together with the gluon P-wave. At variance with the λ parameter case, the quark contribution to the P-wave term
+starts becoming important already at small-PT values.
+Moving to the lower cm energy, from Fig. 6 we see that the z distribution is sensitive to the energy change in the
+whole spectrum, at variance with the λ case. The differences, particularly noticeable in the Gottfried-Jackson frame,
+are mostly in size and not in the general behavior, implying that even in this case it would be difficult to extract any
+information. Again, we remark that the rapid variation of ν estimates at low-z values is due to a geometrical factor
+(Eq. (A16) of Ref. [25]). The PT -dependent distributions, instead, have a quite different behavior for the two frames
+
+9
+0.2
+0.0
+0.2
+0.4
+Gottfried-Jackson
+s = 140 GeV
+9 GeV2 < Q2 < 100 GeV2
+20 GeV < W < 100 GeV
+0.2 < z < 0.9 or PT > 1 GeV
+0.6
+0.4
+0.2
+0.0
+0.2
+0.2
+0.4
+0.6
+0.8
+z
+0.2
+0.0
+0.2
+0.4
+Helicity
+2
+4
+6
+8
+10
+PT [GeV]
+0.6
+0.4
+0.2
+0.0
+0.2
+CSM
+NRQCD (C12)
+NRQCD (BK11)
+NRQCD (G13)
+Figure 5. Estimates for the parameter ν at √s = 140 GeV. Paneling order is the same as in Fig. 1. Integration ranges are
+given in the blue legend box.
+displayed. The Gottfried-Jackson estimates vary significantly in size, especially if one considers the CSM; moreover all
+the LDME sets give similar predictions, compatible with zero, for PT > 5 GeV, while predictions, in both approaches,
+are sizeable (up to ∼ 20%) at low-PT values. This could be very promising for further extensions to the TMD region.
+The curves in the Helicity frame, instead, do not show the same dependence on the energy. In general, we conclude
+that the study of the ν parameter, at least in this frame, is not very effective. Nevertheless it becomes more interesting
+when its information is combined with other parameters, as done in the study of the invariant quantities in the next
+section, Sec. IV.
+Concerning the wave decomposition, we find that both quark and gluon P-wave contributions to the PT and z
+distributions are enhanced at lower energies, even if for the latter this is true only at large z values. Notice that
+the different (larger) size of the ν parameter at z → 0.9 could also affect the TMD region, increasing the possibility
+of extracting information on the linearly polarized gluon distribution.
+The main source of this enhancement at
+√s = 45 GeV is related once again to the lower photon virtualities explored. In this sense, very similar predictions
+might be expected at higher cm energy via a binned analysis with 1.6 GeV < Q < Mψ.
+IV.
+ROTATIONAL INVARIANTS
+The polarization parameters λ, µ and ν, as widely discussed in the previous sections, are frame dependent by
+definition, since they are expressed with respect to the solid angle Ω spanned by the l+ particle in the J/ψ decay and
+in its rest frame. As already pointed out, the frame choice is not unique and the results appear different from frame
+to frame. On the other hand, the relations among the most used reference frames are computable, since they differ
+only in the Z-axis direction.
+A complementary and powerful tool to study J/ψ polarization, both from the experimental and the phenomeno-
+logical points of view, is the use of rotational invariant parameters, that are rest-frame independent by construction.
+These can be defined taking into account what follows.
+For all the most common choices, the Z- and X-axes, lying in the J/ψ production plane, are defined in terms of
+physical momenta in the quarkonium rest frame (see Appendix A of Ref. [25]), with the Y -axis always perpendicular
+with respect to this plane and always pointing in the same direction. This implies that two frames (F, F ′) can be
+connected by a simple rotation of an angle ψ around the Y -axis, and the corresponding polarization parameters can
+
+10
+0.1
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Gottfried-Jackson
+s = 45 GeV
+2.5 GeV2 < Q2 < 100 GeV2
+10 GeV < W < 40 GeV
+0.2 < z < 0.9 or PT > 1 GeV
+0.2
+0.1
+0.0
+0.1
+0.2
+0.3
+0.4
+CSM
+NRQCD (C12)
+NRQCD (BK11)
+NRQCD (G13)
+0.2
+0.4
+0.6
+0.8
+z
+0.1
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Helicity
+2
+4
+6
+8
+10
+PT [GeV]
+0.2
+0.1
+0.0
+0.1
+0.2
+0.3
+0.4
+Figure 6. Estimates for the parameter ν at √s = 45 GeV. Paneling order is the same as in Fig. 1. Integration ranges are given
+in the red legend box.
+be directly related as1
+�
+�
+λ
+µ
+ν
+�
+�
+F ′
+=
+1
+1 + ρ
+�
+�
+1 − 3
+2 sin2 ψ
+3
+2 sin 2ψ
+3
+4 sin2 ψ
+− 1
+2 sin 2ψ
+cos 2ψ
+1
+4 sin 2ψ
+sin2 ψ
+− sin 2ψ 1 − 1
+2 sin2 ψ
+�
+�
+�
+�
+λ
+µ
+ν
+�
+�
+F
+,
+(13)
+with
+ρ = sin2 ψ
+2
+�
+λF − νF
+2
+�
+− sin 2ψ µF
+2 ,
+(14)
+as given in Eqs. (A.18) and (A.19) of Ref. [25], where we have changed the rotation angle from θ to ψ to avoid any
+confusion with the polar angle of the final lepton l+. Notice that the quantity ρ depends on the kinematics, since the
+rotation angle itself depends on the partonic Mandelstam variables (see Eqs. (A.14)-(A.16) of Ref. [25] for details).
+From Eq. (13), one can construct several quantities which do not change upon rotation around the Y direction.
+The following relations are extremely useful in this respect:
+3 + λF ′ =
+1
+1 + ρ (3 + λF ) ,
+1 − νF ′
+2
+=
+1
+1 + ρ
+�
+1 − νF
+2
+�
+.
+(15)
+A group of rotational invariants, as initially proposed in Ref. [36], can be defined in terms of two polarization
+parameters, namely λ and ν,
+F(ci) = c0(3 + λ) + c1(1 − ν/2)
+c2(3 + λ) + c3(1 − ν/2) ,
+(16)
+where ci are suitable free constants.
+1 Here µF stands for the µ parameter in a specific frame F, not to be confused with the factorization scale µF defined in the previous
+sections.
+
+11
+0.20
+0.25
+0.30
+0.35
+0.40
+0.45
+0.50
+s = 140 GeV
+s = 140 GeV
+9 GeV2 < Q2 < 100 GeV2
+20 GeV < W < 100 GeV
+0.2 < z < 0.9 or PT > 1 GeV
+0.25
+0.30
+0.35
+0.40
+0.2
+0.4
+0.6
+0.8
+z
+0.20
+0.25
+0.30
+0.35
+0.40
+0.45
+0.50
+s = 45 GeV
+CSM
+NRQCD (C12)
+NRQCD (BK11)
+NRQCD (G13)
+2
+4
+6
+8
+10
+PT [GeV]
+0.25
+0.30
+0.35
+0.40
+s = 45 GeV
+2.5 GeV2 < Q2 < 100 GeV2
+10 GeV < W < 40 GeV
+0.2 < z < 0.9 or PT > 1 GeV
+Figure 7. Estimates for the invariant F, Eq. (17), as a function of z (left panels) and PT (right panels) at two cm energies,
+√s = 140 GeV (upper panels) and √s = 45 GeV (lower panels), for different approaches and LDME sets. Kinematic ranges
+are given in the legend boxes.
+Among all possible combinations, two of them play an important role and have received special attention [37–41]
+F ≡ F(1,−2,1,0) = 1 + λ + ν
+3 + λ
+(17)
+and
+˜λ ≡ F(1,−3,0,1) = 2 λ + 3 ν
+2 − ν
+.
+(18)
+These invariants have been widely studied in pp and heavy-ion processes [42, 43].
+It is worth noticing that both invariants can be similarly defined for Drell-Yan processes, where they acquire a
+constant value if the Lam-Tung relation (1 − λ = 2ν) holds [26]: FDY = 1/2 and ˜λDY = +1, as pointed out in
+Refs. [38, 41].
+Another interesting feature is that ˜λ = +1(−1) is related to a natural transverse (longitudinal)
+polarization [36]. It is important to stress that the constant behavior is purely dynamical, and in particular for the
+Drell-Yan case is a consequence of rotational invariance and helicity conservation [44]. Since J/ψ couples differently
+in SIDIS processes, the Lam-Tung relation is expected to be broken in this case.
+Not all the invariants belong to the previous family. Indeed, one can exploit another relation that involves all
+polarization parameters in two frames and that, upon rotation around the Y -axis, reads
+(λF ′ − νF ′/2)2 + 4µ2
+F ′ = (λF − νF /2)2 + 4µ2
+F
+(1 + ρ)2
+.
+(19)
+From this, one can construct an invariant quantity involving the polarization parameters squared, as first pointed
+out in Ref. [45]. As an example, we recall
+˜λ′ = (λ − ν/2)2 + 4µ2
+(3 + λ)2
+,
+(20)
+as introduced in Ref. [41].
+
+12
+The study of rotational invariants has not only a theoretical interest, but it is relevant also from the experimental
+point of view, since their expected equality among different frames is an important check of experimental acceptances
+and systematics as shown, for instance, by the ATLAS Collaboration [46].
+For these reasons, we consider, as a case of study, one of these quantities at the kinematics explored by the EIC.
+In Fig. 7 we show the theoretical estimates in the collinear framework, for the invariant F, Eq. (17), as a function of
+z (left panels) and PT (right panels). Once again we compute this quantity at two energies, √s = 140 GeV (upper
+panels) and √s = 45 GeV (lower panels) for different approaches and LDME sets.
+From Fig. 7 we clearly see that F is not equal to 1/2, as expected from the Lam-Tung relation. Moreover, it is
+neither a constant, since its value depends on both z and PT variables. In principle, for some LDME sets a constant
+behavior could accidentally appear, but this would be limited to a specific kinematic region.
+Another interesting remark is that, while the denominator of F is proportional to the unpolarized cross section, its
+numerator is controlled by the relative size of the λ and ν parameters. This can vary significantly, depending on the
+frames and approaches adopted, as discussed in the previous Section.
+From this preliminary study we can conclude that, even if not easily accessible from the experimental point of view,
+these invariant quantities could represent an invaluable tool to learn on the J/ψ polarization mechanism.
+V.
+CONCLUSIONS
+The study of quarkonium polarization, interesting by itself, is also a powerful tool to explore the still challenging
+issue of its formation mechanism within QCD. In this spirit, we have presented a phenomenological analysis of J/ψ
+polarization in SIDIS at large PT .
+More specifically, we have looked at the dilepton angular distribution in the
+J/ψ → ℓ+ℓ− decay in terms of the associated polarization parameters, that could be accessed at the future EIC. By
+exploiting the theoretical results of Ref. [25], we have computed the parameters, λ, µ and ν, in different frames, trying
+to emphasize whether one can use these observables to discriminate among two well consolidated frameworks, still
+under investigation: the Color Singlet Model and the NRQCD approach. Moreover, for the latter we have employed
+three different LDME sets, based on different extractions and assumptions, highlighting their impact on quarkonium
+polarization estimates.
+We have shown results both as a function of z and PT , adopting two quite different cm energies, for standard
+kinematics at the EIC, together with a detailed analysis in terms of parton and NRQCD wave contributions.
+The main findings of our study can be summarized as follows: i) concerning the λ parameter, the large-z region,
+both in the Gottfried-Jackson and the Helicity frame, turns out to be very promising, with the only caveat of possible
+contributions from (TMD) shape functions (even if expected to be reduced being λ a ratio of helicity structure
+functions); similarly its PT distribution, at medium-large values, could be an ideal ground to disentangle the formation
+mechanisms, both at high and low energies. ii) The µ parameter displays some interesting features when studied in
+the Gottfried-Jackson frame, namely: a clear separation among the estimates in different frameworks at medium-large
+z or as a function of PT in the high-energy set-up; a different behavior with respect to the corresponding lower-energy
+estimates at medium-large z or at moderate PT . Moreover, in the Helicity frame at low energies one could extract
+important information by looking in the large PT region. iii) Similarly, for the ν parameter, relevant also in the
+context of the TMD framework, medium-large PT values in the Gottfried-Jackson frame are certainly worth to be
+explored.
+Finally, we have discussed a selection of frame-independent (rotational invariant) polarization parameters, relevant
+not only from the theory point of view, but extremely useful as an important check of experimental acceptances and
+systematics. In particular, we have focused on the invariant F, controlled by the relative weight of the λ and ν
+parameters, that strongly depend on the frames and frameworks adopted. As shown, this observable could clearly
+help in getting information on the J/ψ formation mechanism, both at large z (high- and low-energy set-ups) and as
+a function of PT (at large energy).
+We can certainly conclude that a study of the dilepton angular distribution in J/ψ decay in SIDIS at the EIC could
+be an invaluable tool to shed light on the J/ψ polarization as well as on its formation mechanism.
+ACKNOWLEDGMENTS
+We thank P. Faccioli, T. Stebel and R. Venugopalan for clarifying some aspects concerning the rotational invariants.
+This project has received funding from the European Union’s Horizon 2020 research and innovation programme under
+grant agreement STRONG 2020—No 824093. U.D. and C.P. also acknowledge financial support by Fondazione di
+Sardegna under the project “Proton tomography at the LHC”, project number F72F20000220007 (University of
+
+13
+Cagliari).
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+

FtE3T4oBgHgl3EQfVwq1/content/tmp_files/2301.04463v1.pdf.txt

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+Astronomy & Astrophysics manuscript no. main
+© ESO 2023
+January 12, 2023
+New members of the Lupus I cloud based on Gaia astrometry
+⋆
+Physical and accretion properties from X-Shooter spectra
+F. Z. Majidi1,2, J. M. Alcal´a3, A. Frasca4, S. Desidera2, C. F. Manara5, G. Beccari5, V. D’Orazi2,6, A.
+Bayo5,7, K. Biazzo8, R. Claudi2, E. Covino3, G. Mantovan1,2, M. Montalto4, D. Nardiello2,9, G. Piotto1, and
+E. Rigliaco2
+1 Dipartimento di Fisica e Astronomia, Universit´a degli Studi di Padova, Vicolo dell’Osservatorio 3, 35122 Padova,
+Italy
+2 INAF-Osservatorio Astronomico di Padova, vicolo dell’Osservatorio 5, 35122 Padova, Italy
+3 INAF-Osservatorio Astronomico di Capodimonte, via Moiariello 16, 80131 Napoli, Italy
+4 INAF-Osservatorio Astrofisico di Catania, via S. Sofia, 78, 95123 Catania, Italy
+5 European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching bei M¨unchen, Germany
+6 Department of Physics, University of Rome Tor Vergata, via della ricerca scientifica 1, 00133, Rome, Italy
+7 Instituto de F´ısica y Astronom´ıa, Facultad de Ciencias, Universidad de Valpara´ıso, Av. Gran Breta˜na 1111, Valpara´ıso,
+Chile
+8 INAF - Rome Astronomical Observatory, Via di Frascati, 33, I-00044, Monte Porzio Catone, Italy
+9 Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France
+Received
+ABSTRACT
+We characterize twelve young stellar objects (YSOs) located in the Lupus I region, spatially overlapping with the Upper
+Centaurus Lupus (UCL) sub-stellar association. The aim of this study is to understand whether the Lupus I cloud has
+more members than what has been claimed so far in the literature and gain a deeper insight into the global properties
+of the region. We selected our targets using Gaia DR2 catalog, based on their consistent kinematic properties with the
+Lupus I bona fide members. In our sample of twelve YSOs observed by X-Shooter, we identified ten Lupus I members.
+We could not determine the membership status of two of our targets, namely Gaia DR2 6014269268967059840 and
+2MASS J15361110-3444473 due to technical issues. We found out that four of our targets are accretors, among them
+2MASS J15551027-3455045, with a mass of ∼0.03 M⊙, is one of the least massive accretors in the Lupus complex to
+date. Several of our targets (including accretors) are formed in-situ and off-cloud with respect to the main filaments of
+Lupus I, hence, our study may hint that there are diffused populations of M-dwarfs around Lupus I main filaments. In
+this context, we would like to emphasize that our kinematic analysis with Gaia catalogs played a key role in identifying
+the new members of the Lupus I cloud.
+Key words. Accretion, Accretion Disks – Stars: activity, atmospheres, chromospheres, low-mass, pre-main sequence
+1. Introduction
+Observation of young stellar populations in nearby star-
+forming regions and comparison of their properties with
+more massive and distant ones is a key to understanding
+the impact of the environment on the star formation process
+and the properties of protoplanetary disks.
+The Lupus dark cloud complex is one of the main low-
+mass star-forming regions (SFRs) within 200 pc of the Sun.
+It consists of a loosely connected group of dark clouds
+and low-mass pre-main sequence (PMS) stars. The complex
+hosts four active SFRs plus five other looser dark clouds
+with signs of moderate star-formation activity (Comer´on
+2008). Infrared (IR) and optical surveys (Evans et al. 2009;
+Rygl et al. 2012) have shown that objects in all evolution-
+ary phases, from embedded Class I objects to evolved Class
+III stars, are found majorly concentrated in the Lupus I, II
+and III clouds with Lupus III being the richest in YSOs.
+⋆ Based on observations collected at the European Southern
+Observatory at Paranal, under program 105.20P9.001
+Different distances to the Lupus stellar sub-groups have
+been claimed in the past from Hipparcos parallaxes and
+extinction star counts (Comer´on 2008), but recent investi-
+gations based on Gaia DR2 showed that the vast majority
+of YSOs in all Lupus clouds are at a distance of ∼160 pc
+(see the Appendix in Alcal´a et al. 2019). Out of the three
+main clouds, Lupus III has been recognized as the most
+massive and active star-forming region in Lupus by far,
+with a great number of young low-mass and very-low mass
+stars (Comer´on 2008), while Lupus I, II and IV represent
+regions of low star-formation activity, with Lupus V and
+VI lacking star-formation (Spezzi et al. 2011; Manara et al.
+2018).
+In this paper we investigate the Lupus I cloud. This
+cloud has less than thirty bona fide members, which from
+now on we refer to as Lupus I core members. The main
+motivation for selecting this cloud over the others with a
+low star-forming activity was the recent discovery of the
+star GQ Lup C (Alcal´a et al. 2020; Lazzoni et al. 2020),
+which is located on the main filament.
+1
+arXiv:2301.04463v1  [astro-ph.SR]  11 Jan 2023
+
+Majidi et al.: New members of the Lupus I cloud
+This target was specifically selected by our team for
+discovering possible wide companions to SPHERE-GTO
+targets on Gaia DR2 with a high specific interest in the
+presence of planets, brown dwarfs, or spatially resolved cir-
+cumstellar disks (Alcal´a et al. 2020; Majidi et al. 2020). GQ
+Lup C was proved to be a strong accretor that surprisingly
+had escaped detection in previous IR and Hα surveys, sug-
+gesting the possibility that many YSOs in the region are
+yet to be discovered. This discovery hence motivated us to
+conduct a more extended search in Gaia DR2 to select new
+YSO candidates in the same region. In this work, we present
+the spectroscopic characterization of 12 YSOs in the Lupus
+I cloud.
+The outline of this paper is as follows: in Sect. 2, we
+discuss the target selection criteria, as well as compiling
+a complete list of the bona fide Lupus I members, in ad-
+dition to the observation and data reduction methods; in
+Sect. 3, we discuss the data analysis methods employed for
+analyzing the X-Shooter spectra, the membership criteria,
+and accreting objects; in Sect. 4, we discuss the results of
+our analysis; in Sect. 5, we introduce additional qualities of
+our targets in Lupus I, present their spectral energy distri-
+butions (SEDs), and evaluate them as potential wide com-
+panion candidates; and eventually, Sect. 6 will present our
+conclusions.
+2. Target selection, observations, and data
+reduction
+2.1. Target selection
+The Gaia astrometric catalog (Gaia Collaboration 2018)
+has been recently used to efficiently identify young clus-
+ters and associations within 1.5 kpc from the Sun (see
+Prisinzano et al. 2022, and references therein). We selected
+our sample of YSO candidates based on a statistical anal-
+ysis using the Gaia DR2 catalog detailed in the following.
+As a first step, we identified the genuine population (core
+members) of Lupus I. These core members were gathered
+from the catalogs existing in the literature (Hughes et al.
+1994; Mer´ın et al. 2008; Mortier et al. 2011; Galli et al.
+2013; Alcal´a et al. 2014; Frasca et al. 2017; Benedettini et
+al. 2018; Dzib et al. 2018; Comer´on et al. 2013; Galli et al.
+2020), and are listed in Table 1. We calculated the member-
+ship probability of these targets to Upper Centaurus Lupus
+(UCL) with BANYAN Σ (Gagn´e et al. 2018) which are also
+quoted in Table 1. It should be noted that the catalog does
+not evaluate the Lupus membership.
+We then extracted the kinematic properties (i.e., par-
+allaxes, ϖ, and proper motions µα∗ and µδ) of these core
+members from Gaia DR2, and constrained a range over
+these parameters (see Appendix B of Alcal´a et al. 2020).
+Using this constrained range, we searched for the objects
+with similar kinematic properties to Lupus I core members
+in Gaia DR2 in a radius of 3 degrees from the center of
+the Lupus I cloud. At this stage, we found 247 objects. We
+placed these objects on a color-magnitude diagram (CMD)
+with Main Sequence (MS) stars (Pecaut & Mamajek 2013)
+and we removed those that were close to the limiting magni-
+tude of Gaia (with photometric errors preventing a reliable
+classification according to their position on CMD) and we
+ended up with 186 targets. For generating this CMD, we
+used G magnitudes and Bp − Rp colors. This sample was
+then restricted to objects with a parallax within 5.5 to 7.5
+mas (140-170 pc), within the < ϖ > ±4·σϖ parallax range
+of Lupus I core members, but we kept both sources lying
+close and far from the main filaments of the Lupus I to
+be inclusive both with the kinematic properties and spatial
+location of the selected targets. We also excluded those ob-
+jects which were too faint for X-Shooter to observe (J > 15
+mag) or older than typical YSOs in Lupus I (inconsistent
+with the Lupus I core members on our generated CMD).
+Taking into account all these constraints, we identi-
+fied 43 candidates as potential members of Lupus I. As
+shown in the CMD in Fig. 1, all of our eventual candidates
+lie above the MS stars identified by Pecaut & Mamajek
+(2013) and possess magnitudes and colors very similar to
+those of Lupus I members. Among these 43 objects, there
+are targets that i) have never been recognized as poten-
+tial members of Lupus I (17 objects), ii) were introduced
+as candidate members of Lupus I according to their con-
+sistent kinematic and/or photometric properties, but need
+spectroscopic confirmation (23 objects), iii) were known as
+members of Lupus I, but were poorly characterized in the
+literature, and, were never observed with X-Shooter (3 ob-
+jects). We chose to include all these categories of objects
+to be followed up by X-Shooter, and the main reason for
+keeping the third category was that with X-Shooter spec-
+troscopy we can determine their radial velocity (RV) and
+projected radial velocity (v sin i), or further explore their
+chromospheric and accretion properties in a more detailed
+fashion than previously done.
+Targets in this category are Sz 70 (Hughes et al. 1994),
+2MASS J15383733-3422022 (Comer´on et al. 2013), and
+2MASS J15464664-3210006 (Eisner et al. 2007). Among the
+eight objects selected in Lupus I in the unbiased photomet-
+ric survey by Comer´on et al. (2013, see their Table 2), only
+three were selected by our criteria and are those for which
+these authors provide stellar parameters, qualifying them
+as genuine YSOs. The other five were suspected to be fore-
+ground objects. Indeed, we confirmed that the astrometric
+parameters of the latter are out of range of our selection
+criteria.
+As a final step, we cross-matched our full sample of
+43 objects with the OmegaCAM Hα survey in Lupus (see
+Beccari et al. 2018, for details of this survey), with only 4
+being recognized as Hα emitters. This confirms that many
+potential YSOs may have escaped detection in Hα imag-
+ing surveys and motivated us to spectroscopically charac-
+terize our full sample, giving a high priority to the four
+OmegaCAM Hα emitters as potentially strong accretors.
+2.2. Observations
+The observations were done with the X-Shooter spectro-
+graph (Vernet et al. 2011) at the VLT, within a filler pro-
+gram, and terminated at the end of the observing period,
+when only ∼28% of the proposed sample was observed.
+Hence, of the 43 proposed targets, only 12 were eventu-
+ally observed which are fully characterized in this paper,
+and are listed in Table 2. The list of the targets that were
+not observed is reported in Appendix A. These 12 targets
+were selected by ESO staff from the list of our proposed
+43 targets, and include all of the Hα emitters. Although
+the observed sample is small, all the 12 observed targets
+were confirmed to be YSOs whose physical and chromo-
+spheric/accretion properties are worth to be investigated.
+For two stars the OBs were not validated by ESO observing
+2
+
+Majidi et al.: New members of the Lupus I cloud
+Table 1: Lupus I core members known from the literature (measurement errors are displayed in parenthesis). The column
+under Prob stands for the UCL membership probability percentage of the targets calculated by BANYAN Σ (Gagn´e et
+al. 2018).
+Name
+α (J2000)
+δ (J2000)
+ϖ
+µα∗
+µδ
+RV
+Prob
+age
+(h:m:s)
+(d:m:s)
+(mas)
+(mas/yr)
+(mas/yr)
+(km/s)
+%
+Myr
+RX J1529.7-3628
+15 29 47.26
+–36 28 37.41
+6.04(0.09)
+–14.69(0.10)
+–19.66(0.08)
+0.90(0.27)a
+98.6
+-
+IRAS 15334-3411
+15 36 39.92
+–34 21 42.17
+6.89(0.13)
+–11.80(0.19)
+–19.84(0.12)
+-
+91.6
+-
+Sz 65/V∗ IK Lup
+15 39 27.77
+–34 46 17.21
+6.44(0.05)
+–13.27(0.12)
+–22.24(0.07)
+–2.70(2.00)
+98.6
+1.9b
+Sz 66
+15 39 28.28
+–34 46 18.09
+6.36(0.09)
+–13.60(0.19)
+–21.56(0.12)
+2.40(1.80)
+99.5
+3.9b
+RX J1539.7-3450A
+15 39 46.38
+–34 51 02.54
+6.40(0.04)
+–15.25(0.09)
+–22.33(0.05)
+7.17(1.28)a
+99.6
+-
+UCAC4 274-081081
+15 48 06.26
+–35 15 48.13
+6.61(0.09)
+–12.12(0.19)
+–22.33(0.13)
+-
+97.4
+-
+RX J1539.7-3450B
+15 39 46.37
+–34 51 03.66
+6.40(0.13)
+–13.52(0.26)
+–20.85(0.13)
+-
+98.2
+-
+2MASS J15440096-3531056
+15 44 00.96
+–35 31 05.72
+6.45(0.14)
+–11.49(0.26)
+–24.07(0.19)
+-
+89.3
+-
+AKC2006 18
+15 41 40.81
+–33 45 18.86
+6.69(0.35)
+–18.84(0.33)
+–22.06(0.27)
+9.10(2.30)
+95.3
+8.3
+AKC2006 19
+15 44 57.89
+–34 23 39.36
+6.54(0.14)
+–18.94(0.089)
+–22.75(0.06)
+9.60(2.10)
+97.0
+8.0
+Sz 68/HT LUP A-B
+15 45 12.87
+–34 17 30.65
+6.49(0.06)
+–13.63(0.13)
+–21.60(0.08)
+–4.3(1.8)
+99.1
+0.5b
+HT Lup C
+15 45 12.67
+–34 17 29.37
+6.55(0.19)
+–15.43(0.22)
+–20.27(0.15)
+1.2(3.9)d
+97.8
+-
+Sz 69
+15 45 17.41
+–34 18 28.29
+6.47(0.08)
+–15.05(0.15)
+–22.15(0.11)
+5.40(2.90)
+99.6
+2.6b
+2MASS J15451851-3421246
+15 45 18.52
+–34 21 24.56
+6.59(0.18)
+–15.14(0.34)
+–21.77(0.22)
+4.40(2.90)
+99.7
+0.5b
+IRAS 15422-3414
+15 45 29.78
+–34 23 38.81
+6.46(0.17)
+–15.25(0.31)
+–22.52(0.24)
+-
+99.1
+-
+RX J1546.6-3618
+15 46 41.20
+–36 18 47.44
+6.69(0.07)
+–17.38(0.12)
+–24.29(0.08)
+7.20(0.10)c
+99.8
+-
+Sz 71/GW LUP
+15 46 44.73
+–34 30 35.68
+6.41(0.06)
+–14.03(0.10)
+–23.36(0.07)
+–3.30(1.90)
+99.0
+2.0b
+Sz 72/HM LUP
+15 47 50.63
+–35 28 35.40
+6.41(0.05)
+–14.26(0.09)
+–23.16(0.06)
+6.90(2.40)
+99.6
+2.9b
+Sz 73/THA 15-5
+15 47 56.94
+–35 14 34.79
+6.38(0.06)
+–14.20(0.11)
+–22.26(0.07)
+5.00(2.20)
+99.7
+3.7b
+GQ LUP/CD-3510525
+15 49 12.11
+–35 39 05.05
+6.59(0.05)
+–14.26(0.09)
+–23.59(0.07)
+–3.60(1.30)
+99.4
+0.9b
+Sz 76
+15 49 30.74
+–35 49 51.42
+6.27(0.05)
+–12.77(0.11)
+–23.37(0.08)
+1.40(1.00)
+99.4
+2.3b
+Sz 77
+15 51 46.96
+–35 56 44.11
+6.46(0.05)
+–12.42(0.09)
+–24.16(0.06)
+2.40(1.50)
+99.3
+3.0b
+RX J1556.0-3655
+15 56 02.09
+–36 55 28.27
+6.33(0.04)
+–11.66(0.07)
+–22.50(0.05)
+2.60(1.20)
+99.3
+7.8b
+2MASS J15443392-3352540d
+15 44 33.92
+–33 52 54.11
+7.48(0.24)
+–22.03(0.27)
+–24.92(0.16)
+0.9(3.8)
+96.3
+4.5e
+2MASS J15392180-3400195d
+15 39 21.81
+–34 00 19.56
+6.39(0.19)
+–17.23(0.2)
+–20.18(0.15)
+1.1(3.8)
+97.8
+7.1e
+a Gaia Collaboration (2018)
+b Both RV and age are obtained by Frasca et al. (2017)
+c Torres et al. (2006)
+d RV for this YSO candidate is the optimal RV determined by BANYAN Σ as a member of UCL.
+e Age obtained by Comer´on et al. (2013).
+Fig. 1: CMD of all the potential members of Lupus I in our
+original sample of 43 objects (blue dots), with the MS stars
+(Pecaut & Mamajek 2013) (orange dots) and the Lupus I
+core members (red triangles) included in Table 1.
+staff (due to not fulfilling some of our requirements). But
+the spectra are nevertheless useful for classification pur-
+poses and are used in this work.
+X-Shooter spectra are divided into three arms (Vernet
+et al. 2011), the UVB (λ ∼ 300–500 nm), VIS (λ ∼ 500-
+1050 nm), and NIR (λ ∼ 1000–2500 nm). We decided to
+observe all our targets with 1.′′0, 0.′′9, and 0.′′9 slit widths
+(for UVB, VIS, and NIR arms respectively) for one or two
+cycles based on their J band magnitudes. For our faintest
+objects with J > 14 mag, we considered two cycles of ABBA
+nodding mode. Among our observed targets, only 2MASS
+J15551027-3455045 belongs to this category, and due to its
+faintness, the final signal-to-noise ratio (SNR) of its spec-
+tra was lower than expected. The exposure time for each
+arm and the total execution time taking into account the
+overheads are reported for each target in Table 3. For our
+brightest target, TYC7335-550-1 with J = 9.65 mag, we
+decided that only one cycle of ABBA nodding would be
+sufficient for our scientific aims.
+For some targets with a higher scientific significance to
+our program or because of their faintness, we decided to also
+observe telluric standard stars. Only a few of our targets
+(analyzed in this work) did not have a telluric star observa-
+tion included in their observation block (OB) and these are
+UCAC4 273-083363, 2MASS J15414827-3501458 (with J =
+11.55 mag and 11.05 mag respectively), UCAC4 269-083981
+(J = 10.72 mag), and Gaia DR2 6014269268967059840 (J
+= 13.64 mag) which had a lower scientific priority for our
+program – either were not lying on the main filament, were
+not strong candidates for membership in Lupus I, were not
+3
+
+OurLupusICandidates
+Pecaut and Mamajek Objects
+Lupus ICore Members
+G
+10
+15
+20
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+Bp-RpMajidi et al.: New members of the Lupus I cloud
+Table 2: Objects observed with X-Shooter (measurement errors are displayed in parenthesis). The column under Prob
+stands for the UCL membership probability percentage of the targets calculated by BANYAN Σ (Gagn´e et al. 2018).
+The four candidates detected in the OmegaCAM Hα imaging survey are flagged with ( Hα) right to their names (See
+Sect. 2.1).
+Name
+α (J2000)
+δ (J2000)
+ϖ
+µα∗
+µδ
+Prob
+G
+(h:m:s)
+(d:m:s)
+(mas)
+(mas/yr)
+(mas/yr)
+%
+(mag)
+Partially known targets:
+2MASS J15383733-3422022
+15 38 37.34
+–34 22 02.26
+6.79(0.15)
+–18.25(0.26)
+–24.15(0.19)
+99.4
+16.78
+Sz 70
+15 46 42.99
+–34 30 11.55
+6.09(0.21)
+–12.58(0.39)
+–22.16(0.25)
+95.7
+14.50
+Candidates:
+TYC 7335-550-1a
+15 36 11.55
+–34 45 20.54
+6.26(0.07)
+–13.93(2.43)
+–19.51(1.01)
+99.2
+11.31
+2MASS J15361110-3444473b ( Hα)
+15 36 11.09
+–34 44 47.82
+5.83(0.29)
+–13.56(0.29)
+–20.21(0.23)
+94.8
+18.92
+2MASS J15523574-3344288c ( Hα)
+15 52 35.74
+–33 44 28.87
+5.98(0.17)
+–20.06(0.37)
+–22.17(0.23)
+50.2
+17.06
+2MASS J15551027-3455045d ( Hα)
+15 55 10.28
+–34 55 04.67
+6.78(0.26)
+–11.09(0.54)
+–23.94(0.31)
+93.8
+18.23
+2MASS J16011870-3437332e ( Hα)
+16 01 18.70
+–34 37 33.20
+7.35(0.07)
+–16.59(0.07)
+–24.97(0.05)
+98.5
+16.46
+UCAC4 269-083981f
+15 56 19.06
+–36 13 25.15
+6.095(0.04)
+–13.77(0.09)
+–22.29(0.06)
+98.7
+13.02
+Gaia DR2 6010590577947703936
+15 56 55.36
+–36 11 10.73
+6.83(0.11)
+–15.64(0.24)
+–25.82(0.15)
+98.7
+16.37
+2MASS J15414827-3501458g
+15 41 48.28
+–35 01 45.84
+6.74(0.13)
+–17.99(0.25)
+–25.39(0.18)
+99.5
+13.98
+UCAC4 273-083363
+15 46 46.15
+–35 24 11.40
+6.99(0.06)
+–18.14(0.11)
+–25.04(0.08)
+99.6
+14.46
+Gaia DR2 6014269268967059840
+15 36 55.30
+–33 45 22.19
+6.68(0.24)
+–16.23(0.37)
+–22.29(0.27)
+95.3
+17.39
+a Proposed candidate member of Lupus I by Zari et al. (2018).
+b aka Gaia DR1 6014141205925321984.
+c aka Gaia DR2 6012155767105823616.
+d aka Gaia DR2 6011827867821601792, candidate Lupus I member also proposed by Galli et al. (2020).
+e Gaia DR3 6011165313293141760.
+f Dipper, candidate member of Lupus I also proposed by Nardiello et al. (2020).
+g aka SSTc2dJ154148.3-350145, a candidate Lupus I member previously proposed by Comer´on et al. (2009).
+Table 3: Observing log of the new candidate members of Lupus I.
+Name
+Date
+Exposure time
+Seeing
+Ttot
+airmass
+SNR
+J
+Grade
+(yyyy-mm-dd)
+(sec)
+(′′)
+(hour)
+(mag)
+2MASS J15383733-3422022
+2021-08-03
+1920/1800/1920 1.72/1.72/1.72
+0.67
+1.04
+5.4/47.1/68.6
+13.39
+A
+Sz 70
+2021-07-06
+600/500/600
+0.55/0.52/0.52
+0.33
+1.03
+6.9/67.8/132.4
+10.85
+A
+TYC7335-550-1
+2021-06-27
+300/200/300
+0.72/0.77/0.77
+0.33
+1.36
+71.1/117.0/245.6
+9.65
+A
+2MASS J15361110-3444473
+2021-06-27
+3600/3400/3840 0.73/0.69/0.70
+1.25
+1.15
+0.1/4.9/21.3
+14.91
+A
+2MASS J15523574-3344288
+2021-06-27
+1800/1700/1920 0.72/0.72/0.69
+0.7
+1.43
+0.4/12.2/33.3
+13.49
+A
+2MASS J15551027-3455045
+2021-08-01
+1800/1700/1920 1.73/1.79/1.79
+0.62
+1.11
+0.7/15.0/41.2
+13.76
+A
+2MASS J16011870-3437332
+2021-08-08
+1800/1700/1920 1.49/1.49/1.49
+0.72
+1.35
+5.6/48.9/76.8
+13.07
+A
+UCAC4 269-083981
+2021-08-01
+600/500/600
+2.27/2.27/2.27
+0.33
+1.19
+39.5/108.4/123.2 10.72
+Ca
+Gaia DR2 6010590577947703936
+2021-08-06
+1920/1820/1920 2.04/1.92/1.92
+0.67
+1.14
+5.9/51.0/78.9
+13.08
+A
+2MASS J15414827-3501458
+2021-07-14
+600/500/600
+1.13/1.13/1.13
+0.33
+1.12
+25.4/100.2/232.3 11.05
+A
+UCAC4 273-083363
+2021-07-14
+600/500/600
+1.33/1.29/1.33
+0.33
+1.08
+18.3/73.6/171.0
+11.55
+A
+Gaia DR2 6014269268967059840
+2021-08-04
+1800/1700/1800 2.49/2.49/2.49
+0.65
+1.13
+1.5/26.1/50.5
+13.64
+Cb
+Notes. Date of observation, exposure time allocated to each arm, mean seeing, and SNR (in order for UVB, VIS, and NIR
+wavelengths) as well as the total execution time, mean airmass, and the observation grades (as provided by the ESO observing
+staff) are reported.
+a UCAC4 269-083981 had an out of constraint seeing (2.′′0 which was exceeded).
+b Gaia DR2 6014269268967059840 was reported to have an out of constraint seeing.
+Hα emitters, or were not faint for X-shooter to necessitate
+the observation of a telluric template. As we will detail
+later, we will also adopt a different approach to remove
+telluric lines for these objects. For the targets containing
+telluric observation in their OBs, the same nodding strat-
+egy as those of the targets was employed to minimize noise
+4
+
+Majidi et al.: New members of the Lupus I cloud
+and cosmetics, with an airmass as close as possible to the
+targets. The airmass and seeing reported in Table 3 are
+averaged over the exposure times for each arm.
+2.3. Data reduction
+The data used in this work have been reduced with the X-
+Shooter pipeline xshoo of version 2.3.12 and higher1, and
+hence they have been de-biased, flat-fielded, wavelength-
+calibrated, order-merged, extracted, sky-subtracted and
+eventually flux-calibrated. The result of this pipeline output
+is an ESO one-dimensional standard binary table and the
+two-dimensional ancillary files ready for scientific analysis.
+Flux calibration based on the photometric data available
+in the literature was done later directly on the available
+spectra, along with the telluric removal process which is
+not done for the distributed spectra reduced by the xshoo
+pipeline.
+We used the Image Reduction and Analysis Facility
+(IRAF, Tody 1986, 1993) to remove the telluric lines from
+the target spectra and to flux calibrate them, as well as
+to derive the stellar parameters from the spectra, which
+we shall discuss in detail in the upcoming sections. Since
+the strategy for arranging our observation blocks did not
+include wide slit observations, the flux calibration of our
+targets totally relies on the photometric data available in
+the literature, which have been collected in various surveys
+(with the corresponding flux errors of e-16 W.m−2 for the
+UVB arm, e-16 W.m−2 for the VIS arm, and 2.5e-15 W.m−2
+for the NIR arm). For some of our faint objects, we only
+had access to very limited photometric data and had to cal-
+ibrate the UVB portion of the spectra in accordance with
+the available photometric data in the VIS range.
+For the objects with observations of telluric standard
+stars, we removed the telluric lines and molecular bands
+using the IRAF task Telluric. For the three targets with-
+out telluric star observations in our sample, which namely
+are 2MASS J15414827-3501458, UCAC4 273-083363, and
+Gaia DR2 6014269268967059840, we used the TelFit
+Python code. This code fits the telluric absorption spec-
+trum in the observed spectra (Gullikson et al. 2014) using
+the LBLRTM code which models the line-by-line radiative
+transfer (Clough et al. 2005). Applying TelFit, we cor-
+rected the spectra for oxygen and water molecular bands
+in the visible range (∼550-1000 nm), as well as for water,
+oxygen, and CO2 molecular bands in the NIR (∼1000-2500
+nm) (for the details on the wavelength ranges where these
+molecular bands dominate the spectrum the reader is re-
+ferred to Smette et al. 2015).
+3. Data Analysis
+There are several immediate aims that we planned to fulfill
+through our program. With the X-Shooter spectra, we can
+confirm the youth of the selected candidates through the
+presence of the Li i (6708 ˚A) absorption line, in addition to
+Hα emission, and other lines of the Balmer series as further
+hints. We also determine the spectral type (SpT) classifi-
+cation and the determination of stellar physical parameters
+such as effective temperature (Teff), luminosity (L), mass
+(M) and age. It is also possible that some of our candidates
+1 https://www.eso.org/sci/software/pipelines/
+xshooter/
+may belong to Scorpius-Centaurus Association (with an age
+10-18 Myr, UCL sub-association) rather than Lupus (1-2
+Myr). We can single out these objects once we have fully
+characterized them. The disentanglement between the two
+associations would be useful for clarifying their relation-
+ship. Using spectral lines of the Balmer series, we will also
+measure the accretion luminosity (Lacc) and mass accretion
+rate ( ˙Macc) of those objects that we qualify as accretors. In
+the following, we describe the methods used for achieving
+our immediate goals.
+3.1. Spectroscopic analysis methods
+3.1.1. Spectral typing and line equivalent widths
+To obtain the SpTs of our objects, we first compared the
+spectrum obtained with X-Shooter’s VIS arm with a li-
+brary of visible spectra of already characterized stars and
+brown dwarfs formerly observed by X-Shooter (Manara et
+al. 2013). For the quantitative spectral typing of the stars,
+we then calculated the spectral indices described in Riddick
+et al. (2007) based on the ratios of the average flux of
+molecular absorption bands within narrow wavelength re-
+gions, yielding in all cases an uncertainty of 0.5 subclasses.
+For TYC 7335-550-1 and UCAC4 269-083981, which are
+brighter than the rest of the targets and do not show clear
+molecular bands in their spectra suitable for measuring the
+Riddick’s indices, the SpT is instead estimated through the
+Teff obtained by the ROTFIT code (see Sect. 3.1.2). The
+results can be found in Table 7.
+The EW of the atomic lines reported in Table 5 is mea-
+sured by taking an average over i) the direct integration of
+the line profiles between two marked pixels and ii) fitting
+a Gaussian. The errors associated with these values thus
+report the difference between the measurements made with
+these methods. There are cases for which we could not de-
+tect the Li i line at 6708 ˚A. Hence, for these objects we
+only report an upper limit on the measurement of EWLi i.
+As suggested by Cayrel (1988), a three-sigma upper limit
+on the flux of the lithium line can be calculated as:
+dEW = 3 × 1.06
+�
+(FWHM)dx/(S/N),
+(1)
+in which FWHM is the full width at half maximum, S/N is
+the signal-to-noise ratio, and the bin size (dx) can be fixed
+to 0.2 ˚A for the VIS arm. The values of these measurements
+are reported in Table 5 and Table 6 for TYC7335-550-1.
+3.1.2. ROTFIT
+We used ROTFIT as the basis of our analysis for assessing
+the stellar parameters of our targets. Using ROTFIT, we
+evaluated their RV, v sin i, and surface gravity (log g). The
+version of ROTFIT used for this purpose is the one designed
+for the optimal usage of the X-Shooter spectra (Frasca et al.
+2017). The stellar parameters obtained with ROTFIT can
+be found in Table 4. The fitting process with ROTFIT code
+was carried out within a veiling (the UV excess continuum
+that influences the entire photosphere of the star from UVB
+to NIR) range from 0 to 1. None of our objects showed
+significant veiling, hence the veiling parameter for all our
+studied targets in this paper is equal to zero.
+5
+
+Majidi et al.: New members of the Lupus I cloud
+Table 4: Physical stellar parameters of the targets obtained with the ROTFIT code.
+Name
+Teff
+log g
+vsini
+RV
+Prob
+(K)
+(km/s)
+(km/s)
+%
+2MASS J15383733-3422022
+3111±70
+4.75±0.13
+<8
+4.1±2.7
+99.8
+Sz 70
+3038±76
+4.02±0.11
+14.0±14.0
+1.1±2.6
+84.6
+TYC 7335-550-1
+4488±140
+4.06±0.22
+<8
+2.6±2.0
+99.2
+2MASS J15361110-3444473
+2883±104
+4.41±0.12
+13.0±10.0
+6.9±2.6
+97.9
+2MASS J15523574-3344288
+2981±44
+4.54±0.10
+<8
+2.6±2.7
+75.3
+2MASS J15551027-3455045
+2700±103
+3.60±0.11
+19.0±8.0
+0.1±2.9
+97.9
+2MASS J16011870-3437332
+3121±90
+4.73±0.14
+12.0±8.0
+–0.5±2.3
+98.7
+UCAC4 269-083981
+3846±47
+4.53±0.11
+<8
+0.6±2.7
+99.6
+Gaia DR2 6010590577947703936
+3154±72
+4.77±0.13
+40.8±3.6
+0.5±4.7
+99.2
+2MASS J15414827-3501458
+3213±94
+4.52±0.23
+53.3±5.7
+3.4±4.3
+99.8
+UCAC4 273-083363
+3211±56
+4.51±0.15
+<8
+1.3±2.3
+99.8
+Gaia DR2 6014269268967059840
+3019±108
+4.75±0.14
+44.0±12.0
+1.7±4.6
+98.3
+Notes. The column Prob represents the probability of the target to be member of Lupus I according to BANYAN Σ, which is
+based on the RVs measured with ROTFIT and the kinematic properties reported by Gaia DR2.
+Table 5: EWs of the relevant lines indicating the chromospheric and accretion tracers for our targets. Negative values
+indicate the lines that are in emission.
+Name
+EWLi i
+EWHα
+EWHβ
+EWHγ
+EWHδ
+WHα(10%)
+(˚A)
+(˚A)
+(˚A)
+(˚A)
+(˚A)
+(km/s)
+2MASS J15383733-3422022
+0.74±0.04
+–8.77±0.92
+–7.71±0.04
+–7.99±0.21
+–7.20±0.52
+128±18
+Sz 70
+0.55±0.05
+–43.37±3.97
+–9.97±1.07
+–10.28±1.04
+–11.14±1.51
+366±14
+2MASS J15361110-3444473
+< 0.25a
+–71.4±8.77
+. . .
+. . .
+. . .
+292±14
+2MASS J15523574-3344288
+0.81±0.09
+–13.52±0.76
+–10.9±0.88
+–3.9±1.1
+–2.84±0.49
+146±9
+2MASS J15551027-3455045
+-b
+–88.9±1.17
+–29.7±0.85
+–6.68±0.24
+–5.09±0.49
+229±14
+2MASS J16011870-3437332
+0.67±0.03
+–21.47±1.59
+–21.61±1.28
+–19.41±0.75
+–13.34±2.18
+274±14
+UCAC4 269-083981
+0.56±0.01
+–1.69±0.07
+–1.63±0.08
+–1.56±0.24
+–1.44±0.21
+174±5
+Gaia DR2 6010590577947703936
+0.68±0.06
+–6.53±0.38
+–6.75±0.25
+–6.97±0.09
+–6.69±0.22
+183±5
+2MASS J15414827-3501458
+< 0.012a
+–10.04±0.53
+–9.55±0.61
+–10.64±0.29
+–10.21±0.7
+210±18
+UCAC4 273-083363
+< 0.017a
+–11.4±0.94
+–11.12±0.45
+–11.15±1.35
+–8.59±0.67
+155±9
+Gaia DR2 6014269268967059840
+< 0.047a
+–17.53±2.20
+. . .
+. . .
+. . .
+219±14
+a Three-sigma upper limits on the measurement (read Subsection for further explanation).
+b Li I line was affected by a cosmic ray hit and could not be measured.
+Table 6: EWs of the relevant lines indicating the chromospheric and accretion tracers for TYC 7335-550-1.
+Name
+EWLi i
+EWHα
+EWHϵ
+EW H
+Ca ii
+EW K
+Ca ii
+EW 8498
+Ca ii
+EW 8542
+Ca ii
+EW 8662
+Ca ii
+(˚A)
+(˚A)
+(˚A)
+(˚A)
+(˚A)
+(˚A)
+(˚A)
+(˚A)
+TYC 7335-550-1
+0.39±0.02
+–0.45±0.06
+–0.32±0.16
+–1.07±0.14
+–1.41±0.19
+–0.47±0.03
+–0.78±0.06
+–0.68±0.06
+Notes. The EW of Hα, Hϵ, and Ca ii lines relate to the emission in the cores of these lines obtained by the subtraction of the photospheric
+template.
+3.1.3. Physical parameters
+We used the bolometric correction (BC) relation proposed
+by Pecaut & Mamajek (2013, 2016) for evaluating the lu-
+minosity in both V and J bands and the radius of can-
+didates according to their observed parallaxes and magni-
+tudes. This is possible because none of our targets show
+significant near-IR excess (Fig. 2) nor strong veiling (Sect.
+3.1.2).
+For the objects only resolved in Gaia DR2 catalog, the
+BC relationship introduced by the Gaia DR2 science team2
+is used. In order to have a correct estimation of the lu-
+minosity, we have also taken into account the extinction
+2 https://gea.esac.esa.int/archive/documentation/
+GDR2/Data_analysis/chap_cu8par/sec_cu8par_process/
+ssec_cu8par_process_flame.html
+of the objects which was determined using the grid of X-
+Shooter spectra of zero-extinction non-accreting T Tauri
+stars (Manara et al. 2013), as explained in Sect. 3.2 of
+Alcal´a et al. (2014). It is evident from Fig. 2 that the targets
+have low extinction and little or no NIR excess, probably
+except for the rightmost point in the diagram, which corre-
+sponds to 2MASS J15361110-3444473. The relatively red-
+der H −Ks color of this object in comparison with the oth-
+ers, may be due to the presence of an unresolved very late-
+type companion. This will be further discussed in Appendix
+C.
+Once the Teff (from ROTFIT), luminosity, and ra-
+dius of the targets are derived, their mass, age, and log g
+can be evaluated through various evolutionary tracks and
+isochrones available in the literature. The corresponding
+values of these parameters, which are reported in Table
+6
+
+Majidi et al.: New members of the Lupus I cloud
+Table 7: Physical stellar parameters of the targets.
+Name
+SpT
+AV
+L⋆
+R⋆
+M⋆
+Age
+log g
+(mag)
+(L⊙)
+(R⊙)
+(M⊙)
+(Myr)
+2MASS J15383733-3422022
+M5
+0
+0.012±0.006
+0.39±0.01
+0.09±0.05
+10.7±5
+4.20±0.5
+Sz 70
+M5
+0.5
+0.25±0.11
+1.87±0.05
+0.17±0.05
+0.5±0.3
+3.28±0.2
+TYC 7335-550-1
+K4.5
+0.7
+0.94±0.56
+1.60±0.05
+1.1±0.1
+3.50±1
+4.04±0.2
+2MASS J15361110-3444473
+M5.5
+1.75
+0.006±0.003
+0.32±0.01
+0.05±0.05
+9.77±5
+4.13±0.3
+2MASS J15523574-3344288
+M5.5
+0.5
+0.02±0.01
+0.55±0.01
+0.11±0.03
+6.3±3
+4.04±0.4
+2MASS J15551027-3455045
+M7.5
+0.75
+0.0072±0.0034
+0.39±0.02
+0.03±0.02
+1.7±1.5
+3.71±0.3
+2MASS J16011870-3437332
+M5
+0
+0.013±0.006
+0.41±0.01
+0.09±0.04
+9.55±5
+4.16±0.5
+UCAC4 269-083981
+M0
+0.5
+0.30±0.14
+1.23±0.02
+0.6±0.3
+4.2±1
+4.03±0.5
+Gaia DR2 6010590577947703936
+M4.5
+0
+0.017±0.007
+0.45±0.01
+0.11±0.05
+8.8±4
+4.16±0.3
+2MASS J15414827-3501458
+M4
+0
+0.12±0.06
+1.13±0.03
+0.2±0.08
+1.82±1
+3.64±0.4
+UCAC4 273-083363
+M3.5
+0
+0.069±0.032
+0.83±0.01
+0.2±0.04
+3.63±1.5
+3.88±0.3
+Gaia DR2 6014269268967059840
+M6
+0
+0.01±0.005
+0.41±0.02
+0.05±0.03
+6.46±2
+3.93±0.5
+Notes. The methods used for calculating SpT, AV , L⋆, and R⋆ are described in the text. M⋆, log g, and age of the stars are
+evaluated according to Baraffe et al. (2015) isochrones, except for TYC 7335-550-1, for which we have used the MIST isochrones.
+The SpT for TYC 7335-550-1 and UCAC4 269-083981 (in italic) are obtained using the temperatures derived by the ROTFIT code
+(Table 4) and the SpT–Teff calibration of Pecaut & Mamajek (2013). The errors associated with SpT and AV are 0.5 subclasses
+and 0.4 mag respectively. The errors associated with mass and age are internal to the tracks and isochrones.
+Fig. 2: J − H (mag) vs. H − Ks (mag) diagram of all our
+targets. The red dots show the chromospherically-dominant
+targets, the cyan dots are the accretors, and the blue line
+represents the colors of MS objects, down to spectral type
+M9.5. The normal reddening vector, shown with the black
+arrow, corresponds to AV = 2 mag. The rightmost target is
+2MASS J15361110-3444473 which is suspected to be a bi-
+nary, hence, it might have color contribution from a second
+target.
+7, are derived by the evolutionary models calculated by
+Baraffe et al. (2015). The Hertzsprung-Russel (HR) dia-
+gram of the Lupus I targets, including the previously known
+and the newly discovered members, is displayed Fig. 3. One
+of our targets, namely TYC 7335-550-1, is much brighter
+than the other stars investigated in the present work, and
+falls outside the range covered by the Baraffe et al. (2015)
+models. Therefore, to derive its stellar parameters, we used
+MESA Isochrones and Stellar Tracks (MIST Paxton et al.
+2015; Choi et al. 2016; Dotter 2016). For modeling pur-
+poses, we assumed that all targets have solar metallicity
+(Baratella et al. 2020).
+Some of our objects display strong emission lines which
+is a sign of noticeable chromospheric activity (see the EW of
+some of the chromospheric activity indicators in Table 5) or
+magnetospheric accretion from a circumstellar disk. If the
+magnetic activity is relevant, the position of the star in the
+HR diagram can be significantly affected by photospheric
+starspots and by the changes in the internal structure in-
+duced by the magnetic fields (see Gangi et al. 2022, for in-
+teresting cases in the Taurus SFR). In this case, isochrones
+that do not take into account these effects (such as Baraffe
+et al. 2015) may lead to systematic effects in the estimate
+of mass and age. In particular, they may indicate an age
+half the real age of star (Asensio-Torres et al. 2019; Feiden
+2016). This is crucial for our study which also aims at de-
+termining the membership of the stars in Lupus I or UCL
+associations. Thus, in addition to MIST and the isochrones
+provided by Baraffe et al. (2015), we used other isochrones.
+A set of evolutionary models that considers the mag-
+netic activity of the stars is the Dartmouth magnetic
+isochrones (Feiden 2016), which we also use in this work to
+estimate the ages of all our targets. These isochrones were
+originally developed for estimating the age of the Upper
+Scorpius members (11±2 Myr), almost coeval to the UCL
+(15±3 Myr), and hence are quite useful to fulfill our sci-
+entific aims. In addition to Baraffe et al. (2015) and MIST
+models, we used both Dartmouth std and Dartmouth mag
+(Feiden 2016, and the references therein) models, as well as
+PARSEC + COLIBRI S37 (Bressan et al. 2012; Pastorelli
+et al. 2019, 2020). For all our targets, we obtained over-
+estimated ages using PARSEC + COLIBRI S37 isochrones
+totally inconsistent with the other isochrones, hence, we
+do not report our results obtained with this isochrone to
+avoid confusion. The results of age estimation with all the
+other isochrones are included in Table B.1. For all the mod-
+els, we have assumed our targets have solar metallicity. For
+PARSEC models, extinction is also a free parameter that
+can be fixed and was thus set to the corresponding ex-
+tinction of the targets reported in Table 7. Eventually, we
+would like to point out that it is not straightforward to
+state which targets may have an under-estimated age, par-
+ticularly in the case of objects that are as young as the
+members of Lupus I and UCL considered in this work.
+7
+
+1.5
+1
+J-H
+0.5
+0
+0
+0.5
+1
+H-KsMajidi et al.: New members of the Lupus I cloud
+Fig. 3: log L⋆(L⊙) vs log Teff (K) diagram for all our tar-
+gets (cyan and red dots represent accretors and non-
+accretors, respectively), together with the previously char-
+acterized Lupus members (black dots, Alcal´a et al. 2019,
+sub-luminous objects are not plotted). Blue dashed lines
+represent evolutionary tracks of Baraffe et al. (2015) for
+stars with masses indicated by the number (in M⊙) next
+to the top or bottom of each track. The red lines indicate
+isochrones calculated with the same models at ages of 1, 3,
+30 Myrs, and 10 Gyrs, from the right to the left.
+3.2. Lupus I membership criteria
+According to the works previously done in the Lupus com-
+plex (Alcal´a et al. 2014, and the references therein), in ad-
+dition to the kinematical properties expressed by the Gaia
+parallax and proper motions, membership criteria in this
+star-forming region are:
+i) the presence of lithium in their atmospheres, which
+is the main signature of youth. Despite the obviousness of
+this criterion, there are previously acknowledged members
+of the Lupus cloud that lack lithium. An example is rep-
+resented by Sz 94 in the Lupus III cloud (Manara et al.
+2013; Biazzo et al. 2017; Frasca et al. 2017); ii) an age con-
+sistent with the core members of the cloud. Although the
+estimated age of the Lupus complex is ∼ 1–2 Myr, there are
+previously recognized members of the complex that exceed
+this age range. Examples of such targets are AKC2006 18
+and AKC2006 19 in Lupus I, although their apparent old
+age may be ascribed to disks seen edge-on that obscure
+the central objects making them sub-luminous on the HR
+diagram (see other examples in Sect. 7.4 in Alcal´a et al.
+2014); iii) an RV consistent with the values of the genuine
+members of the Lupus I (Frasca et al. 2017).
+If an object does not match the membership criteria
+defined above, there are two possibilities. Either it is older
+than the UCL (age>20 Myr), and we would hence identify it
+as field star; or it has a consistent age with UCL (∼15 Myr)
+which would confirm its membership to this sub-cloud of
+the Scorpius-Centaurus stellar association. To this aim, we
+have used various isochrones to evaluate the age of our tar-
+gets.
+Fig. 4: |EWHα| vs SpT of our targets with the weak lined T
+Tauri stars studied by Manara et al. (2013, blue dots). The
+cyan dots represent accretors, and the red dots represent
+chromospherically-dominant objects. The horizontal lines
+in red represent the thresholds that separate non-accreting
+and accreting objects considering their SpTs (White &
+Basri 2003).
+3.3. Accreting objects
+There are several criteria for determining whether an object
+is actively accreting matter. Usually, an accreting object is
+characterized by strong emission lines, strong UV and NIR
+continuum excess emission, or structured line profiles (e.g.,
+Manara et al. 2013). Here, to establish whether an object is
+an accretor, we use the criterion proposed by White & Basri
+(2003) which distinguishes the accreting and non-accreting
+objects based on the EW of their Hα emission versus SpT.
+The method used in this paper for calculating the Lacc (ac-
+cretion luminosity) and
+˙Macc (mass accretion rate) of our
+targets involves measuring the line luminosity of the emis-
+sion lines of the accreting targets and using the established
+relationships between the Lline (for each emission line) with
+Lacc (Alcal´a et al. 2017). We quote the eventual accretion
+line luminosity that is obtained this way as log Lacc−line in
+Table 8 and Table 9.
+The whole procedure that we carried out for this task
+can be summarized as follows: we corrected the spectra for
+telluric lines and flux-calibrated them, then measured the
+flux at Earth of the emission lines by integrating their pro-
+file above the local continuum, corrected the flux for ex-
+tinction, calculated the luminosity of each emission line by
+multiplying the flux at Earth for 4πd (adopting a distance
+d = 1000/ϖ pc, with ϖ in mas), and eventually took an
+average over all the values of log Lacc−line. We chose Hα,
+Hβ, and Hγ emission lines to measure the accretion lumi-
+nosity of our targets. After deducing the log Lacc for each
+target, we obtained their
+˙Macc accordingly (Alcal´a et al.
+2017). The results of our measurements are presented in
+Table 8.
+Among all our targets, only TYC 7335-550-1 does not
+show Hydrogen emission lines above the continuum, and
+its Hα line is instead in absorption. For this target, we
+used ROTFIT to subtract the photospheric template in or-
+der to measure the flux of the emission components that
+fill the cores of Hydrogen and Ca ii lines. This method has
+been successfully used to emphasize chromospheric emis-
+sion or a moderate accretion whenever the photospheric
+8
+
+1.0
+0
+(o)
+logL 
+0.5
+0.05
+0.4
+2
+0.3
+0.2
+0.02
+Y
+3.8
+3.7
+3.6
+3.5
+3.4
+logTeff (K)100
+10
+IEWHαl
+1
+0.1
+K3
+K4
+K5
+K6
+K7
+K8
+K9
+MO
+M1M2
+M3
+M4
+M5
+M6
+M7
+M8M9M10
+SpTMajidi et al.: New members of the Lupus I cloud
+flux is large and the emission is only detectable as a filling of
+the line core or an emission bump within the photospheric
+line wings that do not emerge above the continuum (e.g.,
+Frasca et al. 2015, 2017, and references therein). The spec-
+tral subtraction allows us to recognize and measure the EW
+of the emission that fills in the Hα line (Fig. 5). Adopting
+the same method, we measured the fluxes of the H&K lines
+of the Ca ii and in the cores of the three infrared lines of
+the Ca ii IRT at λ =849.8, 854.2, and 866.2 nm (Fig. 6).
+We were also able to separate the contribution of the Hϵ
+emission from the nearby Ca ii H line.
+Fig. 5: X-Shooter spectrum of TYC 7335-550-1 in the Hα
+region, normalized to the local continuum (black solid line)
+along with the inactive photospheric template (red dotted
+line). The latter is produced by ROTFIT with the BT-
+Settl synthetic spectrum at the Teff and log g of this target
+that is degraded to the resolution of X-Shooter, rotationally
+broadened, and wavelength shifted according to the target
+RV. The difference target − template is displayed at the
+bottom of the box and emphasizes the Hα emission that
+fills in the line core (green hatched area), which has been
+integrated to obtain the Hα line flux.
+4. Results
+4.1. Stellar parameters and membership
+The physical stellar parameters that we obtained from
+the spectral analysis and the HR diagram as described in
+Sects. 3.1.1 and 3.1.3 are reported in Table 7. The stellar pa-
+rameters obtained with ROTFIT are presented in Table 4,
+where the membership probability was recalculated with
+the BANYAN Σ using the values of RVs measured with
+ROTFIT. Both Teff and log g found with ROTFIT are in
+good agreement with those derived from SpT and the HR
+diagram and reported in Table 7.
+We note that, at the resolution of the X-Shooter VIS
+spectra, the minimum value of v sin i that can be measured
+is 8 km/s (see, e.g., Frasca et al. 2017) and hence this value
+should be considered as an upper limit. With this knowl-
+edge, we can classify targets with v sin i < 8 km/s as slow
+rotators, and those with v sin i > 40 km/s as fast rotators.
+Moreover, the large RV range of the bona fide members of
+Lupus I (∼ –5-12 km/s, according to Table 1) denies us to
+Fig. 6: a) X-Shooter UVB spectrum of TYC 7335-550-1 in
+the Ca ii H&K region (black solid line) along with the in-
+active photospheric template (red dotted line). b) and c)
+Residual (target − template) spectrum around the Ca ii K
+and Ca ii H line, respectively. The hatched green areas mark
+the residual H and K emissions that have been integrated to
+obtain the EWs and fluxes. The purple-filled area relates to
+Hϵ. d) and e) Observed Ca ii IRT line profiles (black solid
+lines) with the photospheric template overlaid with red dot-
+ted lines. The residual spectra are shown at the bottom of
+each panel shifted downward by 0.2 in relative flux units
+for clarity.
+put a strict constraint on the Lupus I membership of our
+targets (Fig. 7). The RVs of the Lupus I members confirmed
+in this work, however, are within a smaller range with re-
+spect to the previously confirmed core members of the same
+region, except for 2MASS J15361110-3444473 which may or
+may not be a Lupus I member.
+According
+to
+our
+full
+characterization,
+besides
+TYC 7335-550-1 which is a K4.5 type star, all the
+others have M spectral types. Three-quarters of our
+targets, have spectral types between M4 and M6, which
+is in accordance with the previously identified members
+of the Lupus complex (Alcal´a et al. 2014; Frasca et al.
+2017; Krautter et al. 1997; Herczeg & Hillenbrand 2014;
+Comer´on et al. 2013; Galli et al. 2020). The ages of these
+targets cover a large range of 0.7-11 Myrs, with masses in
+the range of 0.02 to 1.1 M⊙ (as also indicated in Fig. 3).
+As discussed in Sect. 2.1, Sz 70 and 2MASS J15383733-
+3422022 were partially known in the literature. The phys-
+ical parameters that we report here for Sz 70 are in excel-
+lent agreement with the results of Hughes et al. (1994). For
+9
+
+Tyc7335-550-
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+LAW
+0.2
+6520
+6540
+6560
+6580
+6600
+x (A)Tyc7335-550-1
+2.0
+1.5
+1.0
+0.5
+3920
+3940
+3960
+3980
+(A)
+6
+1.5
+1.5
+0
+Call K
+Call H
+1.0
+1.0
+0.5
+0.5
+He
+0.0
+0.0
+3926
+3929
+39.32
+3935
+3938
+3941
+3962
+3965
+3968
+3971
+3974
+3977
+^ (A)
+> (A)1.0
+0.5
+0.5
+0.0
+0'0
+8480
+8500
+B520
+8540
+8560
+8640 8650 8660 8670 8680 8690
+^ (A)
+A (A)Majidi et al.: New members of the Lupus I cloud
+Table 8: Accretion luminosity of the accretors derived from the line luminosities. The mass accretion rates are derived
+from the average of these values (Lacc−average).
+Name
+log Lacc−Hα
+log Lacc−Hβ
+log Lacc−Hγ
+log Lacc−average
+log
+˙Macc
+(L⊙)
+(L⊙)
+(L⊙)
+(L⊙)
+(M⊙yr−1)
+Accretors:
+Sz 70
+–2.73
+–2.95
+–2.91
+–2.85
+–9.22
+2MASS J15361110-3444473
+–3.62
+. . .
+. . .
+–3.62
+–10.21
+2MASS J15551027-3455045
+–3.85
+–3.95
+–3.96
+–3.92
+–10.20
+2MASS J16011870-3437332
+–4.04
+–4.29
+–4.20
+–4.16
+–10.91
+Active stars:
+2MASS J15383733-3422022
+–5.41
+–5.43
+–5.52
+–5.45
+–12.21
+2MASS J15523574-3344288
+–4.62
+–4.87
+–4.80
+–4.75
+-11.46
+UCAC4 269-083981
+–4.07
+–4.09
+–4.24
+–4.13
+-11.22
+Gaia DR2 6010590577947703936
+–5.12
+–5.09
+–5.03
+–5.08
+–11.86
+2MASS J15414827-3501458
+–3.97
+–3.93
+–4.07
+-3.99
+-10.63
+UCAC4 273-083363
+–4.01
+–4.14
+–4.19
+–4.11
+–10.89
+Gaia DR2 6014269268967059840
+–5.22
+. . .
+. . .
+–5.22
+–11.07
+Table 9: Accretion luminosity of TYC 7335-550-1 derived from its line luminosities. Its mass accretion rate is derived
+from the average of these values (Lacc−average).
+Name
+log Lacc log Lacc
+log Lacc
+log Lacc
+log Lacc
+log Lacc
+log Lacc
+log Lacc
+log
+˙
+Macc
+Hα
+Hϵ
+Ca II (H) Ca II (K) Ca II (8498.02) Ca II (8542.09) Ca II (8662.14) average
+(L⊙)
+(L⊙)
+(L⊙)
+(L⊙)
+(L⊙)
+(L⊙)
+(L⊙)
+(L⊙)
+(M⊙yr−1)
+TYC 7335-550-1
+–3.43
+–2.82
+–2.31
+–2.19
+–2.01
+–1.94
+–1.88
+–2.16
+–9.40
+Fig. 7: RV of our accretors (cyan dots), chromospherically-
+dominant targets (red dots), and the Lupus I core members
+(black dots).
+2MASS J15383733-3422022, our results are again in good
+agreement with those reported by Comer´on et al. (2013),
+but their difference emanates from the fact that Comer´on et
+al. (2013) measured AV = 1.2 mag for 2MASS J15383733-
+3422022, which results in a discrepancy in luminosity, mass,
+and radius.
+4.2. Equivalent widths
+The EWs of several lines are quoted in Table 5, and sepa-
+rately for TYC 7335-550-1, in Table 6, as for this star the
+flux and EW measurements were performed by subtracting
+the photospheric spectrum.
+We could not detect the Li i line in the spectra of some
+of our targets for various reasons, which can be i) solely
+due to the low SNR of their spectra; ii) based on the simu-
+lations conducted by Constantino et al. (2021), for initially
+lithium-rich stars we know that slow rotators could deplete
+their lithium (also considering their SpT) at early ages (<
+10 Myr), while fast rotators tend to retain their lithium; iii)
+a combination of the low SNR and fast rotation (which may
+be especially true for Gaia DR2 6014269268967059840),
+which would further complicate the issues associated with
+Li i detection; iv) a complex relationship between the ac-
+cretion processes, early angular momentum evolution, and
+possibly planet formation for young stars (∼ 5 Myr) that
+yet needs to be fully explored (Bouvier et al. 2016); v) no
+obvious relationship between the rotation of YSOs and the
+lithium depletion process (Binks et al. 2022).
+The non-detection of Li i in the spectra of some objects
+has been reported as a three-sigma upper limit on the flux
+of the lithium line which is a sensitive enough threshold for
+separating them from objects containing lithium.
+4.3. Evolutionary status of the targets
+The main properties and final status of all our targets are
+summarized in Table 10. Based on all the criteria discussed
+in Sect. 3.2, we confirm that all our objects are YSOs, with
+ages < 11 Myrs.
+The
+targets
+2MASS
+J15414827-3501458
+and
+UCAC4 273-083363 do not show the presence of the
+lithium line in the spectra, but their effective temperature
+is compatible with the possible presence of a large amount
+of Li depletion for fully convective pre-main sequence stars
+(Bildsten et al. 1997). Lithium depletion was investigated
+in several star forming regions, like some sub-groups of
+Orion (Palla et al. 2007; Sacco et al. 2007), but also
+in Lupus I and III (see, e.g., Biazzo et al. 2017, and
+references therein). Due to their very young age (< 4 Myr),
+10
+
+12
+10
+8
+6
+(s/w>)
+-2
+-4
+-6
+-8
+5.8
+6
+6.2
+6.4
+6.6
+6.8
+7
+7.2
+7.4
+7.6
+Parallax (mas)Majidi et al.: New members of the Lupus I cloud
+Table 10: Overall status checklist for our targets. The rotation column refers to fast (F) or slow (S) rotators.
+Name
+Membership
+Active
+Accreting
+Contains Li i
+Rotation
+Av
+Conclusion
+(UCL/Lup I)
+(yes/no)
+(yes/no)
+(yes/no)
+(F/S)
+(mag)
+2MASS J15383733-3422022
+Lup I
+yes
+no
+yes
+S
+0
+Genuine member of Lup I
+Sz 70
+Lup I
+yes
+yes
+yes
+S
+0.5
+Genuine Lup I member +
+wide companion candidate
+TYC 7335-550-1
+Lup I
+yes
+no
+yes
+S
+0.7
+Genuine member of Lup I +
+wide companion candidate
+2MASS J15361110-3444473
+?
+yes
+yes
+no
+S
+1.75
+Unresolved binary (?) +
+wide companion candidate
+2MASS J15523574-3344288
+Lup I
+yes
+no
+yes
+S
+0.5
+New member of Lup I
+2MASS J15551027-3455045
+Lup I
+yes
+yes
+?
+S
+0.75
+Genuine member of Lup I
+2MASS J16011870-3437332
+Lup I
+yes
+yes
+yes
+S
+0
+New member of Lup I
+UCAC4 269-083981
+Lup I
+yes
+no
+yes
+S
+0.5
+Genuine member of Lup I
+Gaia DR2 6010590577947703936
+Lup I
+yes
+no
+yes
+F
+0
+New member of Lup I
+2MASS J15414827-3501458
+Lup I
+yes
+no
+no
+F
+0
+Genuine member of Lup I
+UCAC4 273-083363
+Lup I
+yes
+no
+no
+S
+0
+Genuine member of Lup I
+Gaia DR2 6014269268967059840
+?
+yes
+no
+no
+F
+0
+?
+we
+therefore
+classify
+2MASS
+J15414827-3501458
+and
+UCAC4 273-083363 as Lupus I members. Newly discovered
+members of Lupus I in this work are 2MASS J15523574-
+3344288,
+2MASS
+J16011870-3437332,
+and
+Gaia
+DR2
+6010590577947703936.
+There are also two objects analyzed in this work that
+we could not identify either as a member of Lupus I or
+UCL. These are 2MASS J15361110-3444473, whose spec-
+trum indicates an unresolved binary star of spectral types
+M5.5 (VIS arm) and M8 (NIR arm), and we could not
+detect lithium in its spectrum (see Appendix C for more
+details on the analysis of this target). However, we would
+like to emphasize that 2MASS J15361110-3444473 is an ac-
+creting source that has consistent kinematic and physical
+properties with the genuine members of Lupus I, hence,
+there is a possibility that this target also qualifies as a
+new member of Lupus I. The other object is Gaia DR2
+6014269268967059840, for which we acquired a spectrum
+with poor SNR (see Sect. 2 for details on the observation
+conditions of this target). The poor SNR of its UVB spec-
+trum hindered us from carrying out any measurements on
+its Hβ and Hγ lines in emission (as reported in Table 5),
+which also leads to evaluating its accretion properties only
+according to its Hα emission line (as reported in Table 8).
+Therefore, the non-detection of lithium in its spectrum can
+be purely due the poor SNR in the VIS arm, and we do not
+approve nor rule out the possibility of this target being a
+member of Lupus I.
+We hence confirm that all our targets are YSOs, with
+Hydrogen lines in emission above the continuum. Therefore,
+this investigation suggests that although only four of our
+targets were retrieved as Hα emitters in the OmegaCAM
+survey (flagged in Table 2), it is likely that our entire sample
+of 43 candidate YSOs could include Hα emitters or objects
+with filled Hα profiles, which can only be confirmed by a
+high- or mid-resolution spectroscopic study or in deep X-
+ray surveys.
+As a further investigation to strengthen our argument,
+we cross-matched all of the Lupus I core members included
+in Table 1 with the OmegaCAM survey. Except for three
+objects, they were all retrieved in the survey as Hα emit-
+ters. These exceptional three core members are RXJ1529.7-
+3628 (which was out of the field of view of the survey), RX
+J1539.7-3450B and Sz 68/HT Lup C, for which only one
+object was resolved in the survey. Combining this result
+with the results of this paper, we emphasize the necessity
+of observing all our sample to characterize all the members
+of Lupus I that have escaped the Hα surveys.
+4.4. Accretion versus chromospheric–dominated objects
+We realized that four of our targets in the current sam-
+ple are accretors. We measured the Lacc of these tar-
+gets, in addition to our chromospherically-dominant objects
+(Table 8 and Table 9). The measured Lacc for all our tar-
+gets are displayed in Fig. 8. In the same figure, we have
+included the limits suggested by Manara et al. (2017b)
+for objects with Teff > 4000 K and Teff < 4000 K, be-
+low which the chromospheric activity of targets is domi-
+nant. All our four accretors exceed this limit for targets
+with Teff < 4000 K, confirming that they are accretion-
+dominated. The rest of our targets within the same ef-
+fective temperature range are below this threshold, which
+make them chromospheric-dominated objects, as expected.
+2MASS J15523574-3344288, however, lies exactly on the
+threshold between these two regimes, which is consistent
+with its significant Hα emission. We also emphasize that
+this target was retrieved in the OmegaCAM survey as an
+Hα emitter.
+Fig. 9 shows the
+˙Macc versus M∗ for the four accre-
+tors in our sample in comparison with the Lupus members.
+Among the four accretors, 2MASS J15551027-3455045 is
+the least massive target, and has a very high mass accretion
+rate in comparison with Lupus members of similar mass.
+This target also stands above the double power-law rela-
+tionship between
+˙Macc and M∗ established by Vorobyov &
+Basu (2009), based on modeling self-regulated accretion by
+gravitational torques in self-gravitating disks. As concluded
+by Alcal´a et al. (2017), only the strongest accretors stand
+above this model. Our three other accretors have values of
+mass accretion rates typical of Lupus accretors.
+Finally, it is worth noting that three of our accretors (Sz
+70, 2MASS J15361110-3444473, and 2MASS J16011870-
+3437332) have WHα(10%)>270 km/s (see Table 5), which
+is expected from accreting stars. Our chromospherically-
+dominant targets have much narrower Hα profiles.
+11
+
+Majidi et al.: New members of the Lupus I cloud
+Fig. 8: Log < Lacc/L∗ > vs Teff for all our targets. The
+cyan dots represent accretors, and the red dots represent
+chromospherically-dominant targets. The lines indicate the
+limit below which the chromospheric activity for a star is
+dominant (Manara et al. 2017b), for two regimes of stars
+with Teff ≤ 4000 K (the diagonal blue line) and those with
+Teff ≥ 4000 K (the horizontal orange line).
+Fig. 9: Log Macc(M⊙/yr) vs log M∗(M⊙) for the four accre-
+tors in our sample (cyan dots), together with the previously
+identified members of the Lupus (black dots). The blue
+crossed squares represent the substellar accreting compan-
+ions detected at wide orbits by Zhou et al. (2014) around
+GQ Tau, GSC 06214 00210 and DH Tau as labeled. 2MASS
+J15551027-3455045, GQ Lup c and 2MASS J16085953-
+3856275 are also labelled. 2MASS J15523574-3344288 is
+labelled as red dot. The continuous red line indicates the
+double power-law prediction of Vorobyov & Basu (2009),
+while the magenta dashed line shows the prediction of disk
+fragmentation model by Samatellos & Herczeg (2015).
+5. Discussion
+In this paper, we analyzed 12 objects observed by X-
+Shooter out of our original sample of 43 proposed new
+candidate members of Lupus I. We confirm that all these
+12 objects are YSOs, and ten out of 12 are members of
+Lupus I. We could not determine the membership of two of
+our targets, namely 2MASS J15361110-3444473 and Gaia
+DR2 6014269268967059840, as explained in the previous
+Section. We could not fully measure the accretion prop-
+erties of Gaia DR2 6014269268967059840 and hence our
+analysis in this regard for this specific target is not reliable.
+2MASS J15361110-3444473, on the other hand, is a rather
+(intrinsic) faint object to be followed up by any available
+spectrographs, but perhaps can be followed up with ALMA
+to understand whether it is surrounded by a disk. Although
+recognized to have an older age with respect to Lupus I
+members (9 Myr), it can be still strongly accreting matter,
+consistent with the members of γ Vel with age ∼10 Myr
+(Frasca et al. 2015). One of the interesting targets discussed
+in this work is TYC 7335-550-1, a lithium-rich K-type star
+with Hα in absorption and without IR excess. We would
+like to emphasize that YSOs with these particular charac-
+teristics would never appear in Hα imaging surveys such as
+OmegaCAM, although one of their main aims is to identify
+the members of young star forming regions. All the above
+points considered, we have fully characterized ten members
+of Lupus I in this work.
+In the following, we will discuss further qualities of our
+targets, which are mainly based on the data available in
+the literature in connection with the targets analyzed in
+this work.
+5.1. Spectral energy distributions / Circumstellar disks
+For all our objects, we also investigated whether there are
+hints of continuum flux excess suggestive of circumstellar
+disks. To this aim, we extracted their SEDs from literature
+which are collectively exhibited in Figs. 10 and 11. For this
+work, we only concentrate on the morphology and trends
+of the SEDs of our targets, as well as their near- to mid-
+infrared photometric data (published by 2MASS and WISE
+surveys). For generating the SEDs, we have used the follow-
+ing WISE filters: W1 (3.4 microns), W2 (4.6 microns), W3
+(12 microns), W4 (22 microns). In a parallel paper (Majidi
+et al. in prep), we will study the variability of these stars
+and model their disks.
+The photometric data for all four accretors significantly
+deviate from their BT-Settl spectral model (based on their
+Teff, log g, and zero metallicity) in W3 and W4 filters
+(with the average flux errors of 5e-17 W.m−2 and 1.7e-16
+W.m−2 respectively). This trend can be observed for our
+less massive, stronger accretors 2MASS J15551027-3455045
+and 2MASS J15361110-3444473 in all four WISE filters
+(W1, W2, W3, and W4). According to Sicilia-Aguilar et
+al. (2014), the morphology of the SEDs of all our four ac-
+cretors in addition to 2MASS J15523574-3344288 is com-
+patible with objects surrounded by full disks. This is further
+confirmed by the disk categorization of Bredall et al. (2020)
+based on Ks−W3 and Ks−W4 magnitudes for Lupus dip-
+pers, Lupus YSOs, Upper Scorpius and Taurus members.
+Hence, also according to Bredall et al. (2020), all our four
+accretors in addition to 2MASS J15523574-3344288 are sur-
+rounded by a full disk. Note, however, that the “valley”
+around W3 in the SED of 2MASS J15361110-3444473 is
+typical of those seen in transitional disks.
+For the rest of our targets, we have two categories
+of circumstellar disks based on the morphology of their
+SEDs further approved by their Ks − W3 and Ks − W4
+magnitudes: i) Evolved disks, which are characterized by
+only W4 excess with respect to the theoretical BT-Settl
+model, and are evident in the SEDs of 2MASS J15383733-
+3422022, Gaia DR2 6010590577947703936, and Gaia DR2
+6014269268967059840 (Fig. 11), ii) Debris disks, which are
+12
+
+-8
+(Mo yr-1)
+GQ Lup
+区
+GQ/Lup
+c
+区
+-10
+2MASS15551
+GSC 06214 b
+区
+2MASS16085
+-DH Tau
+b
+-12
+2
+0
+logM* (Mo)-1
+-1.5
+-2
+-2.5
+60
+-3
+-3.5
+-4
+5000
+4500
+4000
+3500
+3000
+2500
+Teff (K)Majidi et al.: New members of the Lupus I cloud
+Fig. 10: BT-Settl models (in grey) with the photometric data (red dots) for our accretors.
+characterized by little to no mid-infrared excess, and is ev-
+ident in the SEDs of TYC 7335-550-1, UCAC4 269-083981,
+2MASS J15414827-3501458, and UCAC4 273-083363 (Fig.
+11).
+5.2. High accretion in the low-mass regime
+Deriving
+˙Macc for the lowest mass accretors is relevant for
+the studies of disk evolution. There is growing evidence
+of a change in the slope of the M⋆– ˙Macc relationship for
+YSOs with ages of 2-3 Myr at M⋆<0.2 M⊙ (Manara et al.
+2017b and Alcal´a et al. 2017, and see Fig. 9). Such a break
+could be related to a faster disk evolution at the low-masses
+(e.g. Vorobyov & Basu (2009)). To verify this, the
+˙Macc–
+M⋆ relationship needs to be sampled at much lower M⋆ and
+˙Macc values than done so far.
+Our target 2MASS J15551027-3455045 is one of the
+lowest
+mass
+accretors
+in
+Lupus
+(see
+Fig.
+3).
+With
+M⋆=0.02 M⊙, 2MASS J16085953-3856275 is the accretor
+with comparable mass reported in the previous Lupus stud-
+ies (Alcal´a et al. 2017, 2019). Considering the very low mass
+of this YSO, its accretion rate
+˙Macc∼10−11 M⊙/yr (Alcal´a
+et al. 2019) is relatively high. Yet the ˙Macc value for 2MASS
+J15551027-3455045 is about an order of magnitude higher
+(see Fig. 9); hence, it is one of strongest accretors in Lupus
+in the mass range 0.02–0.03M⊙, i.e. close to the planetary
+mass regime. From modeling of a shock at the surface of
+a planetary-mass object, Aoyama et al. (2021) have pre-
+dicted much higher Lacc values than what the scaling Lacc–
+Lline relations for stars would predict. The relationships by
+these authors would yield an even higher ˙Macc value, almost
+an order of magnitude higher than our estimate. This ob-
+ject falls above the model prediction by Vorobyov & Basu
+(2009), in contrast with the idea of faster disk evolution at
+very low masses. However, statistics are still rather poor at
+this mass regime for a firm conclusion.
+Other very low-mass YSOs, companions to T Tauri
+stars, have been found to exhibit similar, or even higher
+rates of mass accretion (Betti et al. 2022; Zhou et al. 2014,
+see Fig. 9). To explain the very high levels of accretion
+observed in such sub-stellar and planetary-mass compan-
+ions, Samatellos & Herczeg (2015) modeled the accretion
+onto very low-mass objects that formed by the fragmenta-
+tion of the disk around the hosting star. During the early
+evolution the individual disks of sub-stellar companions,
+including those at the planetary-mass regime, accrete addi-
+tional material from the gas-rich parent disk, hence, their
+disks are more massive and their accretion rates are higher
+than if they were formed in isolation. Therefore, these very
+low-mass objects have disk masses and accretion rates that
+are independent of the mass of the central object and are
+higher than expected from the scaling relation
+˙Macc ∝ M 2
+⋆
+of more massive YSOs. These models predict that
+˙Macc is
+independent of M⋆.
+Using Gaia DR3, we have investigated whether 2MASS
+J15551027-3455045 might be a wide companion of another
+star, but it is an isolated object. Hence, the high mass ac-
+cretion rate cannot be explained in terms of the Samatellos
+& Herczeg (2015) scenario. Due to its intrinsic faintness,
+2MASS J15551027-3455045 would be an interesting target
+to be followed up by CUBES, which is a next-generation
+spectrograph suitable for investigating fainter, low-mass ac-
+creting YSOs (Alcal´a et al. 2022).
+13
+
+2MASSJ15551027-3455045
+-10
+Teff = 2700 K, log g = 3.5
+10.5
+PhotometricData
+ cm-2)
+-11
+(erg S-1
+11.5
+-12
+12.5
+-13
+13.5
+-14
+1000
+10000
+入 (nm)2MASSJ15361110-3444473
+Teff = 2900 K, log g = 4.5
+-11
+Photometric Data
+L cm-2)
+11.5
+12
+-12.5
+-13
+1000
+10000
+入 (nm)Sz 70
+6
+Teff = 3000 K, log g = 4.0
+PhotometricData
+9.5
+-10
+-10.5
+log 入 Flux
+11
+11.5
+-12
+1000
+10000
+入 (nm)2MASSJ16011870-3437332
+-10
+Teff = 3100 K, log g = 4.5
+Photometric Data
+10.5
+-11
+11.5
+log 入Flux
+12
+12.5
+13
+1000
+10000
+入 (nm)Majidi et al.: New members of the Lupus I cloud
+Fig. 11: BT-Settl models (in grey) with the photometric data (red dots) for our chromospherically-dominant targets.
+5.3. Possible wide companions
+While studying the kinematic properties of the targets, we
+also noticed that a few of our targets and core members
+of the Lupus I share similar kinematic properties, and can
+be considered as wide companion candidates. These wide
+companion candidates are presented in Table 12 and Table
+13, divided into two categories of candidates studied in this
+14
+
+TYC 7335-550-1
+8
+Teff = 4500 K, log g = 4.0
+Photometric Data
+(erg s-1 cm-2)
+-10
+log 入Flux 
+-11
+12
+13
+1000
+10000
+入 (nm)2MASSJ15523574-3344288
+-10
+Teff = 3000 K, log g = 4.5
+PhotometricData
+10.5
+-11
+-11.5
+log 入Flux
+12
+12.5
+-13
+1000
+10000
+入 (nm)UCAC4269-083981
+-9
+Teff = 3800 K, log g = 4.5
+9.5
+Photometric Data
+ cm-2)
+-10
+(erg s-1
+10.5
+-11
+log 入Flux
+11.5
+-12
+12.5
+13
+1000
+10000
+入 (nm)2MASSJ15383733-3422022
+-10
+Teff = 3100 K, log g = 4.5
+Photometric Data
+-10.5
+-11
+-11.5
+log 入Flux
+-12
+12.5
+13
+1000
+10000
+入 (nm)2MASSJ15414827-3501458
+-9
+Teff = 3200 K, log g = 4.5
+9.5
+PhotometricData
+ cm-2)
+-10
+(erg s-1
+10.5
+11
+log 入Flux
+11.5
+-12
+12.5
+13
+1000
+10000
+入 (nm)GaiaDR26010590577947703936
+-10
+Teff = 3100 K, log g = 4.5
+Photometric Data
+10.5
+-11
+-11.5
+log 入Flux
+-12.5
+13
+1000
+10000
+入 (nm)UCAC4273-083363
+-9
+Teff = 3000 K, log g = 4.5
+9.5
+Photometric Data
+ cm-2)
+-10
+10.5
+-11
+log 入Flux
+11.5
+-12
+12.5
+13
+1000
+10000
+入 (nm)GaiaDR26014269268967059840
+-10
+Teff = 3000 K, log g = 4.5
+10.5
+Photometric Data
+. cm-2)
+-11
+(erg s-1
+11.5
+-12
+12.5
+13
+13.5
+-14
+1000
+10000
+入 (nm)Majidi et al.: New members of the Lupus I cloud
+Table 11: Disk categorization of all our targets, in addition to their reddest colors available in the 2MASS and WISE
+catalogs.
+Name
+Ks − W3
+Ks − W4
+Bredall et al. (2020)
+Sicilia-Aguilar et al. (2014)
+mag
+mag
+Disk type
+SED/Disk type
+2MASS J15383733-3422022
+0.75
+3.93
+Evolved disk
+Sz 70
+2.28
+3.9
+Full disk
+Full disk
+TYC 7335-550-1
+0.20
+1.14
+Debris disk
+2MASS J15361110-3444473
+2.70
+5.04
+Full disk
+Full disk
+2MASS J15523574-3344288
+2.69
+4.31
+Full disk
+Full disk
+2MASS J15551027-3455045
+3.24
+5.7
+Full disk
+Full disk
+2MASS J16011870-3437332
+2.18
+4.09
+Full disk
+Full disk
+UCAC4 269-083981
+0.13
+1.06
+Debris disk
+Gaia DR2 6010590577947703936
+0.61
+3.79
+Evolved disk
+2MASS J15414827-3501458
+0.39
+1.16
+Debris disk
+UCAC4 273-083363
+0.4
+1.86
+Debris disk
+Gaia DR2 6014269268967059840
+0.89
+3.58
+Evolved disk
+Notes. The overall SED of 2MASS J15361110-3444473 may be affected by a possible unresolved M8-type companion.
+work and the Lupus I core members. In order to understand
+whether two objects with similar kinematic properties are
+gravitationally bound, we calculated their total velocity dif-
+ference (∆v) and compared it with the maximum total ve-
+locity difference (∆vmax) as a function of projected sepa-
+ration between the two binary components, suggested by
+Andrews et al. (2017). If ∆v exceeds ∆vmax, we do not ex-
+pect the two targets to be gravitationally bound. It should
+be noted, however, that the theoretical maximum velocity
+difference modeled by Andrews et al. (2017) is only for bina-
+ries of total mass 10 M⊙ in circular orbits. We summarize
+our results on identifying wide companions candidates in
+the Lupus I cloud as follows:
+Sz 70 and Sz 71 – Same as the GQ Lup triple system
+(Alcal´a et al. 2020), Sz 70 and Sz 71 (GW Lup) are located
+on the main filament of Lupus I. Sz 70 lies at a separation of
+32.32 arcseconds from GW Lup, and in between these ob-
+jects lies the X-ray source [KWS97] Lupus I 37 (Krautter
+et al. 1997) at a separation of 24.23 arcseconds from Sz 70.
+We conducted a chance projection study in Alcal´a et al.
+(2020, Appendix E), which was focused on understanding
+how probable it is to find a field object around a genuine
+member of Lupus I, lying on the same filament where GQ
+Lup stellar system and Sz 70/Sz 71 are located. The linear
+density of this filament is 0.0024 objects/arcsec, or an av-
+erage object separation of 418 arcsec, which is 13 times the
+observed separation between Sz 70 and Sz 71. As exhibited
+in Fig. 12, Sz 70 and Sz 71 do not qualify as gravitation-
+ally bound stars, but we would like to emphasize that the
+test proposed by Andrews et al. (2017) is only valid for
+gravitationally bound binaries, and not systems of higher
+multiplicities (if this is the case for this stellar system).
+Hence, we would consider this case as a wide companion
+candidate that cannot be confirmed or ruled out according
+to the available information.
+TYC
+7335-550-1
+and
+2MASS
+J15361110-
+3444473 – As discussed in Sect. 4, 2MASS J15361110-
+3444473 might be an unresolved binary, composed of an
+M6 (VIS spectrum) and an M8 (NIR spectrum) star. The
+RV calculated for this target based on the ROTFIT code
+is obtained by cross-correlations conducted on the VIS
+spectrum of this target, which is also used for calculating
+the maximum velocity difference between TYC 7335-550-1
+and 2MASS J15361110-3444473. As exhibited in Fig. 12,
+the two objects can be gravitationally bound. However,
+Fig. 12: Log-log plot of total velocity difference ∆v (km/s)
+versus projected separation s (au) for the wide companion
+candidates analyzed in this work, in addition to the genuine
+wide companions GQ Lup and GQ Lup C. ∆vmax (km/s)
+(orange line) indicates the maximum total velocity differ-
+ence that bound binaries with a total mass equal to 10 M⊙
+in circular orbits can possess (Andrews et al. 2017). Each
+point is marked as one of the wide companion candidates
+involved. For the detailed information, see Tables 12 and
+13.
+TYC 7335-550-1 has an age of ∼ 4 Myr and 2MASS
+J15361110-3444473 an age of ∼ 9 Myr, which states
+the two stellar systems are probably not coeval. Also,
+unlike TYC 7335-550-1, we could not determine whether
+2MASS J15361110-3444473 is a member of Lupus I due to
+many uncertainties explained earlier. Hence, any further
+comments on its physical association with TYC 7335-550-1
+would be misleading and inconclusive.
+Sz 65 and Sz 66 – At a separation of 6.45 arcseconds,
+with ∆V = 5.26±2.69 km/s, Sz 65 and Sz 66 (although
+coeval) according to the test suggested by Andrews et al.
+(2017) are not gravitationally bound. There are no other
+objects located in a close separation with respect to either
+Sz 65 or Sz 66. Hence, we rule out the possibility of Sz 65
+and Sz 66 as wide companion candidates.
+HT Lup A-B-C – This stellar system is located in
+an over-crowded region on the same filament of Lupus I
+as GQ Lup stellar system. In Gaia DR2 catalog, HT Lup
+15
+
+1.4
+1.2
+1
+(km/s)
+0.8
+(△ v)
+0.6
+0.4
+0.2 
+0
+-0.2
+2.6
+2.8
+3
+3.2
+3.4
+3.6
+3.8
+4
+log s (au)
+GQLupC
+SZ 66
+HT Lup
+Sz 70
+TYC 7335-550-1
+△ Vmax (km/s)Majidi et al.: New members of the Lupus I cloud
+Table 12: Kinematic properties of the Lupus I members from this work (measurement errors are displayed in parenthesis).
+Name
+α (J2000)
+δ (J2000)
+ϖ
+µα∗
+µδ
+RV
+Age
+∆V
+δ∆V
+S
+(h:m:s)
+(d:m:s)
+(mas)
+(mas/yr)
+(mas/yr)
+(km/s)
+(Myr)
+(km/s)
+(km/s)
+(′′)
+Sz 71/GW LUP∗
+15 46 44.73
+–34 30 35.68
+6.41(0.06)
+–14.03(0.10)
+–23.36(0.07)
+–3.30(1.90)
+2.0
+6.07
+3.24
+32.32
+Sz 70
+15 46 42.99
+–34 30 11.55
+6.09(0.21)
+–12.58(0.39)
+–22.16(0.25)
+1.1(2.6)
+0.5
+2MASS J15361110-3444473
+15 36 11.09
+–34 44 47.82
+5.83(0.29)
+–13.56(0.29)
+–20.21(0.23)
+6.9(2.6)
+9.77
+4.72
+3.47
+16.28
+TYC 7335-550-1
+15 36 11.55
+–34 45 20.54
+6.26(0.07)
+–13.93(2.43)
+–19.51(1.01)
+2.6(2.0)
+3.55
+∗ RV obtained by Frasca et al. (2017).
+Table 13: Core members of Lupus I sharing similar kinematic properties (measurement errors are displayed in parenthesis).
+Name
+α (J2000)
+δ (J2000)
+ϖ
+µα∗
+µδ
+RV
+Age
+∆V
+δ∆V
+S
+(h:m:s)
+(d:m:s)
+(mas)
+(mas/yr)
+(mas/yr)
+(km/s)
+(Myr)
+(km/s)
+(km/s)
+(′′)
+Sz 65/V∗ IK Lup∗
+15 39 27.77
+–34 46 17.21
+6.44(0.05)
+–13.27(0.12)
+–22.24(0.07)
+–2.70(2.00)
+1.9
+5.26
+2.69
+6.41
+Sz 66∗
+15 39 28.28
+–34 46 18.09
+6.36(0.09)
+–13.60(0.19)
+–21.56(0.12)
+2.40(1.80)
+3.9
+Sz 68/HT LUP A-B∗
+15 45 12.87
+–34 17 30.65
+6.49(0.06)
+–13.63(0.13)
+–21.60(0.08)
+–4.30(1.80)
+0.5
+6.30
+4.30
+2.82
+CD-33 10685C/HT Lup C∗∗
+15 45 12.67
+–34 17 29.37
+6.55(0.19)
+–15.43(0.22)
+–20.27(0.15)
+1.2(3.9)
+–
+∗ RV and age obtained by Frasca et al. (2017).
+∗∗ RV for this target is adopted from the optimal RV calculated by BANYAN Σ, considering HT Lup C is a member of UCL.
+A and B are not resolved separately, hence we assume the
+central star to be Sz 68 (or HT Lup A), composed of two
+unresolved stars, and adopt its stellar characteristics from
+Frasca et al. (2017). As genuine members of Lupus I, we
+assume all the components of this triple system to have an
+age consistent with the other bona fide members of Lupus I
+(≤ 2 Myr), and hence, to be coeval. However, the RVs used
+here should be taken with caution, both because HT Lup
+A-B are not resolved, and also because we have adopted
+the optimal RV calculated by BANYAN σ for HT Lup C
+considered as a member of UCL. With a separation of 2.82
+arc seconds, we have shown in Fig. 12 that as expected, this
+triple system is possibly gravitationally bound.
+We thus conclude that the possibility of Sz 70 & Sz 71
+being wide companions is rather low and for TYC 7335-
+550-1 & 2MASS J15361110-344447, follow-up studies on
+2MASS J15361110-344447 are required. As for the previ-
+ously identified members of Lupus I, we understood that
+Sz 65 and Sz 66 are not gravitationally bound, and HT
+Lup A-B-C are the components of a triple system.
+6. Conclusion
+The main conclusions of this paper can be summarized as
+follows:
+– Out of the 12 objects fully characterized in this work,
+ten are recognized as genuine members of Lupus I, and
+two remain ambiguous in terms of stellar properties.
+– Out of the ten members of Lupus I analyzed in this
+work, three were recognized to be accretors (Sz 70,
+2MASS J15551027-3455045, and 2MASS J16011870-
+3437332), and Sz 70 and 2MASS J15551027-3455045 are
+likely to be surrounded by full disks. 2MASS J15551027-
+3455045 is among the least massive accretors discovered
+so far in the Lupus complex, formed in full isolation and
+is an off-cloud member of Lupus I.
+– All of the three off-cloud targets included in our
+program
+turned
+out
+to
+be
+genuine
+members
+of
+Lupus I. These targets are 2MASS J15523574-3344288,
+2MASS J15551027-3455045, and 2MASS J16011870-
+3437332, with 2MASS J15551027-3455045 and 2MASS
+J16011870-3437332
+actively
+accreting
+matter,
+and
+2MASS J15523574-3344288 mildly accreting matter.
+Further investigation in this area may reveal a diffused
+population of M dwarfs close to the main filament of
+Lupus I. We thus would like to acknowledge that this
+work also contributes to revealing the diffused popula-
+tions of M-dwarfs around the Lupus cloud by Comer´on
+(2008).
+– Although the sample studied in this work is small, we
+proved that many interesting targets in young star form-
+ing regions can escape Hα surveys due to various rea-
+sons. Hence, using the kinematic properties of candi-
+date YSOs can play a key role in identifying the gen-
+uine members of the young stellar associations. This is
+specifically true for genuine members such as TYC 7335-
+550-1 that have Hα in absorption, and hence would not
+appear in Hα surveys.
+– We have identified a plausible binary system among
+the targets analyzed in this work, namely, TYC 7335-
+550-1 and 2MASS J15361110-3444473. It is noteworthy,
+however, that 2MASS J15361110-3444473 might be an
+unresolved binary, and its kinematic properties (espe-
+cially RV) should be revised with next-generation spec-
+trographs (due to its intrinsic faintness).
+– All the above points considered, we conclude that char-
+acterizing only a small portion of our sample has proved
+to have a high success rate for discovering the new mem-
+bers of Lupus I. This shows that the spectroscopy of our
+entire sample of 43 objects could have resulted in a far
+more solid investigation of the region in terms of de-
+termining the disk fraction, stellar properties, and the
+number of new members of Lupus I.
+Acknowledgements. FZM is grateful to Eugene Vasiliev for fruitful
+discussions on how to use Gaia catalogs. AFR is grateful to Giovanni
+Catanzaro for helping us with the analysis of TYC 7335-550-1. FZM is
+funded by ”Bando per il Finanziamento di Assegni di Ricerca Progetto
+Dipartimenti di Eccellenza Anno 2020” and is co-funded in agree-
+ment with ASI-INAF n.2019-29-HH.0 from 26 Nov/2019 for ”Italian
+participation in the operative phase of CHEOPS mission” (DOR -
+Prof. Piotto). A.B. acknowledges partial funding by the Deutsche
+Forschungsgemeinschaft Excellence Strategy - EXC 2094 - 390783311
+and the ANID BASAL project FB210003. JMA, AFR, CFM, KBI
+and ECO acknowledge ��nancial support from the project PRIN-
+INAF 2019 “Spectroscopically Tracing the Disk Dispersal Evolution”
+16
+
+Majidi et al.: New members of the Lupus I cloud
+(STRADE). CFM is funded by the European Union under the
+European Union’s Horizon Europe Research & Innovation Programme
+101039452 (WANDA). This work has also been supported by the
+PRIN-INAF 2019 ”Planetary systems at young ages (PLATEA)” and
+ASI-INAF agreement n.2018-16-HH.0. Views and opinions expressed
+are however those of the author(s) only and do not necessarily re-
+flect those of the European Union or the European Research Council.
+Neither the European Union nor the granting authority can be held
+responsible for them.
+This work has made use of data from the European Space
+Agency
+(ESA)
+mission
+Gaia
+(https://www.cosmos.esa.int/gaia),
+processed by the Gaia Data Processing and Analysis Consortium
+(DPAC,
+https://www.cosmos.esa.int/web/gaia/dpac/consortium).
+Funding for the DPAC has been provided by national institutions,
+in particular, the institutions participating in the Gaia Multilateral
+Agreement.
+This research has made use of the SIMBAD database and Vizier
+services, operated at CDS, Strasbourg, France. This research has
+made use of the services of the ESO Science Archive Facility.
+Finally, we would like to thank the anonymous referee who also
+contributed to this paper with his/her valuable comments.
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+
+Majidi et al.: New members of the Lupus I cloud
+Appendix A: Candidate members of Lupus I
+As we explained in Sect. 2, we proposed 43 objects to be ob-
+served with X-Shooter. Twelve out of these 43 objects were
+observed during a filler program, and in this work we fully
+characterized them. The rest of our targets in this sam-
+ple that were not observed are listed in Table A.1. Among
+these targets, only 2MASS J15464664-3210006 (Eisner et
+al. 2007) is partly characterized, and 20 objects are identi-
+fied as candidate YSOs using Gaia DR2 (Zari et al. 2018).
+Appendix B: Age estimation and isochrones
+For estimating the age of our targets we used multiple
+isochrones for the reasons explained in Sect. 3.2. In this
+Appendix, we present the ages of our targets using various
+isochrones. We repeat that the ages estimated for all our
+targets were overestimated by PARSEC models in compar-
+ison with all the other models with a considerable gap. We
+thus decided to remove the results achieved by the PARSEC
+models to avoid confusion. This is, however, a well-known
+problem of PARSEC isochrones that they overestimate the
+age of cool stars, and all our targets fall in this category.
+Appendix C: 2MASS J15361110-3444473
+Fig. C.1: Flux-calibrated, extinction-corrected NIR spec-
+trum of 2MASS J15361110-3444473 (in black) with its BT-
+Settl model (Teff = 2500 K and log g = 4.5, in grey).
+2MASS J15361110-3444473 is an M5.5 star according to
+its VIS spectrum (as we quantitatively indicated) and an
+M8 star based on its NIR spectrum (based on the fitting
+done with the BT-Settl model Teff = 2500 K and log g
+= 4.5, as exhibited in Fig. C.1), with a total extinction
+of AV = 1.75 mag. All the spectral typing and analysis
+that we have performed in this paper are based on the VIS
+spectrum of this target, especially the ROTFIT results are
+all based on the VIS spectrum. Hence, although we keep our
+analysis limited to the spectroscopy conducted on the VIS
+spectrum, we would like to emphasize that the possibility
+of this target being an unresolved binary (composed of two
+M dwarfs) with SpTs of M5.5 and M8 is viable. Considering
+the available data, we also cannot rule out the possibility
+that the star is heavily spotted instead of being a binary.
+Appendix D: Updates with Gaia DR3
+As stated in Sect. 2, we used the Gaia DR2 catalog to select
+our targets. Very recently, Gaia DR3 (Gaia Collaboration
+2021) became public and gave us the opportunity to check
+the catalog for any possible changes or updates on the
+kinematic or stellar properties of our objects analyzed in
+this work. We did not find any considerable difference be-
+tween the kinematic properties reported in both catalogs.
+However, we report the highlights of our search using these
+two catalogs in the following:
+TYC 7335-550-1 – as obtained in this work, for TYC
+7335-550-1 we obtained Teff = 4488 K, while in both Gaia
+DR2 and Gaia DR3 its reported temperature is 5000 K.
+The reported RV for TYC 7335-550-1 in Gaia DR2 is
+1.20±1.65 km/s, which is better constrained than the RV
+we report here (2.6±2.0 km/s). As the wide companion can-
+didate of 2MASS J15361110-3444473, we recalculated their
+∆v using the Gaia DR3 kinematic properties of TYC 7335-
+550-1, and it resulted in ∆v = 5.34±3.30 (km/s) which is
+consistent with the previous ∆v = 4.72±3.47 (km/s). For
+both of these calculations, we use the RVs calculated by
+ROTFIT.
+Sz 70 – has a high RUWE in both catalogs (4.86), but
+we saw no signs of binarity in the spectrum of Sz 70. Using
+the kinematic properties of Sz 70 reported in Gaia DR3
+and those of Sz 71 (which is also updated in Gaia DR3),
+we recalculated their maximum velocity difference, and it
+resulted in ∆v = 8.36±3.24 (km/s), which is consistent with
+the ∆v = 6.07±3.24 (km/s) calculated based on Gaia DR2.
+2MASS J15414827-3501458 – has a high RUWE
+(4.198) in both Gaia DR2 and Gaia DR3 catalogs, but we
+detected no signs of binarity in the spectrum of the object.
+We report that the kinematic properties of all our tar-
+gets (parallax and proper motions) are consistent within 3σ
+in the two catalogs. Also according to Manara et al. (2022),
+we do not expect the stellar physical parameters of our core
+sample to be changed with the astrometry reported in Gaia
+DR3.
+18
+
+-11
+Teff = 2500, log g = 4.5
+2MASS|15361110-3444473
+-11.2
+nm-1)
+ (erg s-1 cm-2 I
+11.4
+11.6
+log 入Flux 
+11.8
+-12
+-12.2
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+入 (nm)Majidi et al.: New members of the Lupus I cloud
+Table A.1: Astrometric properties of the candidate Lupus I members that were not observed by X-Shooter, with their
+errors in parentheses.
+Name
+α (J2000)
+δ (J2000)
+ϖ
+µα∗
+µδ
+J
+(h:m:s)
+(d:m:s)
+(mas)
+(mas/yr)
+(mas/yr)
+(mag)
+2MASS J15464664-3210006a
+15 46 46.64
+–32 10 00.62
+7.05(0.021)
+–19.47(0.023)
+–23.76(0.014)
+11.22
+Gaia DR2 6013000844869745664
+15 39 24.47
+–35 58 50.88
+6.62(0.039)
+–18.00(0.081)
+–22.23(0.057)
+10.11
+Gaia DR2 6013065853493820416b
+15 43 15.62
+–35 39 38.18
+6.88(0.015)
+–17.68(0.018)
+–24.51(0.012)
+10.20
+Gaia DR2 6011737574730221568c
+15 50 46.50
+–34 22 38.49
+6.69(0.019)
+–16.20(0.020)
+–22.52(0.015)
+10.74
+Gaia DR2 6012258330925877632d
+15 53 36.13
+–33 31 02.60
+6.92(0.016)
+–16.97(0.018)
+–24.57(0.016)
+10.75
+Gaia DR2 6039383622075982848e
+15 57 09.76
+–32 04 33.91
+6.72(0.02)
+–14.24(0.023)
+–23.58(0.015)
+10.56
+Gaia DR2 6011518462675791872f
+15 48 13.16
+–35 43 31.08
+6.62(0.023)
+–16.65(0.028)
+–24.31(0.023)
+11.48
+Gaia DR2 6011797738632729216g
+15 57 20.96
+–35 00 01.21
+6.71(0.027)
+–16.29(0.033)
+–24.21(0.024)
+11.65
+Gaia DR2 6014049985115937408
+15 34 59.21
+–34 58 16.16
+6.83(0.097)
+–17.76(0.16)
+–24.03(0.11)
+12.16
+Gaia DR2 6014830844535625344h
+15 47 58.08
+–33 46 59.53
+6.84(0.027)
+–17.73(0.031)
+–24.48(0.025)
+11.31
+Gaia DR2 6014224051546189568
+15 34 42.05
+–34 17 48.09
+6.66(0.098)
+–17.36(0.134)
+–23.67(0.094)
+11.94
+Gaia DR2 6009936093645659136
+15 43 49.43
+–36 48 38.64
+6.94(0.13)
+–20.45(0.28)
+–22.89(0.19)
+10.92
+Gaia DR2 6039633559115225344i
+15 52 59.02
+–31 38 33.57
+6.59(0.03)
+–18.34(0.036)
+–22.89(0.029)
+11.93
+Gaia DR2 6013187040287810944j
+15 37 53.31
+–35 55 12.42
+6.74(0.027)
+–17.9(0.03)
+–24.08(0.024)
+11.95
+Gaia DR2 6016139332082870272
+15 39 25.88
+–32 10 04.68
+6.42(0.40)
+–20.32(0.54)
+–23.65(0.37)
+10.81
+Gaia DR2 6013126738951338624k
+15 43 28.48
+–35 17 27.40
+6.77(0.032)
+–17.67(0.035)
+–24.48(0.022)
+11.91
+Gaia DR2 6013190201383772288
+15 37 53.00
+–35 52 28.70
+6.75(0.055)
+–19.08(0.13)
+–22.62(0.087)
+12.22
+Gaia DR2 6013077192207599232m
+15 43 11.42
+–35 26 34.43
+6.78(0.032)
+–17.32(0.034)
+–24.29(0.025)
+11.82
+Gaia DR2 6015181897983193728m
+15 51 57.84
+–33 29 33.17
+6.74(0.032)
+–16.22(0.039)
+–22.37(0.026)
+12.03
+Gaia DR2 6014590429442468096m
+15 45 06.91
+–35 06 21.73
+6.99(0.036)
+–16.97(0.042)
+–23.09(0.029)
+11.82
+Gaia DR2 6009995742152335232m
+15 44 26.97
+–36 25 42.75
+6.52(0.034)
+–18.30(0.043)
+–23.21(0.031)
+11.82
+Gaia DR2 6011607694917034112m
+15 50 00.76
+–35 29 19.71
+7.23(0.044)
+–20.18(0.052)
+–25.32(0.034)
+12.37
+Gaia DR2 6011695690208264320m
+15 47 59.03
+–34 56 38.36
+6.99(0.06)
+–17.93(0.069)
+–25.07(0.045)
+12.69
+Gaia DR2 6011261726715424128
+15 50 29.19
+–36 25 11.80
+6.92(0.11)
+–17.08(0.23)
+–23.52(0.16)
+13.32
+Gaia DR2 6015222957871475584
+15 48 46.12
+–33 18 35.48
+6.69(0.13)
+–19.21(0.26)
+–23.77(0.17)
+13.77
+Gaia DR2 6013030875279571328
+15 41 55.22
+–35 59 35.36
+6.97(0.12)
+–17.12(0.24)
+–25.52(0.14)
+13.17
+Gaia DR2 6014112107523072640m
+15 34 35.79
+–34 36 01.54
+6.88(0.084)
+–16.89(0.087)
+–24.841(0.066)
+13.14
+Gaia DR2 6012977136650130560m
+15 39 48.47
+–36 13 48.07
+6.94(0.10)
+–20.069(0.11)
+–23.61(0.069)
+12.81
+Gaia DR2 6015141830223216640
+15 50 19.17
+–33 50 07.12
+6.84(0.15)
+–17.29(0.29)
+–26.46(0.19)
+13.92
+Gaia DR2 6011581856393988352n
+15 48 06.26
+–35 15 48.15
+6.05(0.07)
+–12.22(0.084)
+–21.04(0.057)
+10.56
+Gaia DR2 6016191485871670400
+15 38 35.63
+–32 02 37.66
+6.53(0.26)
+–18.90(0.39)
+–23.38(0.28)
+14.35
+a 2MASS J15464664-3210006 is an M2, T Tauri star (Eisner et al. 2007).
+b aka UCAC4 272-080482, this target is a YSO candidate (Zari et al. 2018).
+c aka UCAC4 279-083370, this target is a YSO candidate (Zari et al. 2018).
+d aka UCAC4 283-086052, this target is a YSO candidate (Zari et al. 2018).
+e aka RX J1557.1-3204A, this target is a YSO candidate (Zari et al. 2018).
+f aka UCAC4 272-081081, this target is a YSO candidate (Zari et al. 2018).
+g aka UCAC4 275-083957, this target is a YSO candidate (Zari et al. 2018).
+h aka UCAC4 282-082547, this target is a YSO candidate (Zari et al. 2018).
+i aka UCAC4 292-084899, this target is a YSO candidate (Zari et al. 2018).
+j aka UCAC4 271-080669, this target is a YSO candidate (Zari et al. 2018).
+k aka UCAC4 274-080590, this target is a YSO candidate (Zari et al. 2018).
+l aka UCAC4 274-080590, this target is a YSO candidate (Zari et al. 2018).
+m This target is a YSO candidate (Zari et al. 2018).
+n aka UCAC4 274-081081, this target is a YSO candidate (Zari et al. 2018).
+19
+
+Majidi et al.: New members of the Lupus I cloud
+Table B.1: Ages of our targets estimated using various isochrones. The ages are all in Myr.
+Name
+Dartmouth
+Dartmouth
+MIST
+Baraffe
+std
+mag
+models
+2MASS J15383733-3422022
+11
+20
+12.6
+10.7
+Sz 70
+<1
+1
+<0.25
+0.5
+TYC 7335-550-1
+3
+5
+3.5
+3.55
+2MASS J15361110-3444473
+9
+20
+9
+9.77
+2MASS J15523574-3344288
+8
+13
+8
+6.3
+2MASS J15551027-3455045
+-
+-
+-a
+1.7
+2MASS J16011870-3437332
+9.5
+14
+9.5
+9.55
+UCAC4 269-083981
+4.5
+8
+3.5
+4.2
+Gaia DR2 6010590577947703936
+8
+14
+8
+8.8
+2MASS J15414827-3501458
+2.5
+3
+1.78
+1.82
+UCAC4 273-083363
+4.5
+8
+3.5
+3.63
+Gaia DR2 6014269268967059840
+8
+13
+8
+6.46
+a None of the three isochrones used here were able to reproduce the stellar parameters of this target due to its dimness.
+20
+

JNFRT4oBgHgl3EQfzTgG/content/tmp_files/2301.13649v1.pdf.txt

@@ -0,0 +1,636 @@
+Studies of New Physics in 𝑩0
+𝒒 − ¯𝑩0
+𝒒 Mixing and
+Implications for Leptonic Decays
+Kristof De Bruyn,𝑎,𝑏 Robert Fleischer,𝑎,𝑐 Eleftheria Malami𝑎,𝑑,∗ and Philine van Vliet𝑒
+𝑎Nikhef,
+Science Park 105, 1098 XG Amsterdam, Netherlands
+𝑏Van Swinderen Institute for Particle Physics and Gravity, University of Groningen,
+9747 Groningen, Netherlands
+𝑐Faculty of Science, Vrije Universiteit Amsterdam,
+1081 HV Amsterdam, Netherlands
+𝑑Center for Particle Physics Siegen (CPPS), Theoretische Physik 1, Universität Siegen,
+D-57068 Siegen, Germany
+𝑒Deutsches Elektronen-Synchrotron DESY,
+Notkestr. 85, 22607 Hamburg, Germany
+E-mail: Eleftheria.Malami@uni-siegen.de
+The phenomenon of 𝐵0
+𝑞- ¯𝐵0
+𝑞 mixing (𝑞 = 𝑑, 𝑠) provides a sensitive probe for physics beyond the
+Standard Model. We have a careful look at the determination of the Unitarity Triangle apex, which
+is needed for the Standard Model predictions of the 𝐵𝑞 mixing parameters, and explore how much
+space for New Physics is left through the current data. We study the impact of tensions between
+inclusive and exclusive determinations of the CKM matrix elements |𝑉𝑢𝑏| and |𝑉𝑐𝑏|, and focus on
+the 𝛾 angle extraction. We present various future scenarios and discuss the application of these
+results for leptonic rare 𝐵 decays, which allows us to minimise the CKM parameter impact in
+the New Physics searches. Performing future projections, we explore and illustrate the impact of
+increased precision on key input quantities. It will be exciting to see how more precise data in the
+future high-precision era of flavour physics can lead to a much sharper picture.
+8th Symposium on Prospects in the Physics of Discrete Symmetries (DISCRETE 2022)
+7-11 November, 2022
+Baden-Baden, Germany
+∗Speaker
+© Copyright owned by the author(s) under the terms of the Creative Commons
+Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).
+https://pos.sissa.it/
+arXiv:2301.13649v1  [hep-ph]  31 Jan 2023
+
+Studies of New Physics in 𝐵0
+𝑞 − ¯𝐵0
+𝑞 Mixing and Implications for Leptonic Decays
+Eleftheria Malami
+1.
+Introduction
+The phenomenon of 𝐵0
+𝑞- ¯𝐵0
+𝑞 mixing (where 𝑞 = 𝑑, 𝑠) arises only from loop processes in the
+Standard Model (SM) and is sensitive to possible New Physics (NP) contributions, which could
+enter the loop topologies or even at the tree level, for instance in 𝑍 ′ models. Associated to the mixing
+phenomenon are the mixing parameters and the CP-violating phases for which we have impressive
+experimental data. In this presentation, we follow Ref. [1] and explore the space allowed for NP
+by current measurements and the state-of-the-art parameters. In addition, we point out interesting
+connections to the studies of leptonic rare 𝐵 decays.
+In order to determine the parameter space of possible NP effects to 𝐵0
+𝑞– ¯𝐵0
+𝑞 mixing, we have to
+compare the SM predictions of the mixing parameters with the corresponding experimental values.
+For these SM predictions, a careful analysis of the Unitarity Triangle (UT) apex is required. We
+pay special attention to the different determinations of the Cabibbo-Kobayashi-Maskawa (CKM)
+parameters and the tensions that arise between the extractions of the |𝑉𝑢𝑏| and |𝑉𝑐𝑏| matrix elements
+through inclusive and exclusive semileptonic 𝐵 meson decays. These longstanding tensions have a
+profound impact on the whole analysis.
+2.
+Unitarity Triangle
+Using the parametrisation of the Particle Data Group (PDG), the UT apex is given as [2]:
+𝑅𝑏 𝑒𝑖𝛾 = ¯𝜌 + 𝑖 ¯𝜂 ,
+¯𝜌 ≡
+�
+1 − (𝜆2/2)
+�
+𝜌 ,
+¯𝜂 ≡
+�
+1 − (𝜆2/2)
+�
+𝜂 .
+(1)
+Here, 𝜌, 𝜂 and 𝜆 are the Wolfenstein parameters [3, 4], 𝑅𝑏 is the side from the origin to the apex of
+the UT, defined with the help of the CKM matrix elements 𝜆 ≡ |𝑉𝑢𝑠|, |𝑉𝑢𝑏| and |𝑉𝑐𝑏| as:
+𝑅𝑏 ≡
+�
+1 − 𝜆2
+2
+� 1
+𝜆
+����
+𝑉𝑢𝑏
+𝑉𝑐𝑏
+���� =
+√︃
+¯𝜌 2 + ¯𝜂 2 ,
+(2)
+and 𝛾 ≡ arg �−𝑉𝑢𝑑𝑉∗
+𝑢𝑏/𝑉𝑐𝑑𝑉∗
+𝑐𝑏
+� is the angle between the 𝑅𝑏 side and the UT basis.
+2.1 Determining the UT Apex Utilising 𝛾 and 𝑅𝑏
+In this subsection, we work in the SM and are interested in obtaining the UT apex in a way
+that is not affected by possible NP in 𝐵0
+𝑞- ¯𝐵0
+𝑞 mixing. One way of determining the apex is utilising
+the side 𝑅𝑏 and the angle 𝛾, which can both be determined from decays that proceed only via tree
+decays. The value of 𝛾 can be determined either from 𝐵 → 𝐷𝐾 decays or from a 𝐵 → 𝜋𝜋, 𝜌𝜋, 𝜌𝜌
+isospin analysis.
+More specifically, one option is to use the time-dependent 𝐵0
+𝑠 → 𝐷∓
+𝑠 𝐾± system, where mixing-
+induced CP violation plays a key role. Through interference effects caused by 𝐵0
+𝑞- ¯𝐵0
+𝑞 mixing, the
+CP asymmetry parameters allow the determination of 𝜙𝑠 + 𝛾, where 𝜙𝑠 is the 𝐵0
+𝑠- ¯𝐵0
+𝑠 mixing phase.
+Since 𝜙𝑠 is determined through the 𝐵0
+𝑠 → 𝐽/𝜓𝜙 channel, including penguin corrections [5, 6], 𝛾
+can be obtained in a theoretically clean way [7, 8]. However, the surprisingly large value arising in
+this case still needs to be further explored. An alternative way of getting the 𝛾 value is using the
+time-independent 𝐵 → 𝐷𝐾 transitions, where the sensitivity to 𝛾 comes from direct CP violation
+[9]. Last but not least, another interesting system is provided by 𝐵 → 𝜋𝜋, 𝜌𝜋, 𝜌𝜌 modes [10, 11],
+2
+
+Studies of New Physics in 𝐵0
+𝑞 − ¯𝐵0
+𝑞 Mixing and Implications for Leptonic Decays
+Eleftheria Malami
+which usually are used to determine 𝛼 from an isospin analysis. Actually this value corresponds to
+𝛾 when we use the 𝐵0
+𝑑- ¯𝐵0
+𝑑 mixing phase 𝜙𝑑, determined from 𝐵0
+𝑑 → 𝐽/𝜓𝐾0 [5, 6], taking penguin
+effects into account. Thus, we can convert the result 𝜙𝑑 + 2𝛾 into 𝛾. The value from the latter case
+is in good agreement with the one coming from 𝐵 → 𝐷𝐾 modes. Therefore, for our analysis, we
+average these two results [1]:
+𝛾avg = (68.4 ± 3.4)◦.
+(3)
+Regarding 𝑅𝑏 there are tensions between the various theoretical and experimental approaches.
+Even though there are different determinations of the |𝑉𝑢𝑠| element and the tensions between them
+are intriguing, they only have a negligible impact on NP studies in neutral 𝐵𝑞 mixing. Thus, we
+choose to work with the value |𝑉𝑢𝑠| = 0.22309 ± 0.00056 [12, 13]. Contrary to the |𝑉𝑢𝑠| case, the
+deviations between determinations of |𝑉𝑢𝑏| and |𝑉𝑐𝑏| from inclusive and exclusive semileptonic 𝐵
+decays, which are given as follows [14, 15]:
+|𝑉𝑢𝑏|incl = (4.19 ± 0.17) × 10−3 ,
+|𝑉𝑢𝑏|excl = (3.51 ± 0.12) × 10−3 ,
+differing by 3.9 𝜎,
+(4)
+|𝑉𝑐𝑏|incl = (42.16 ± 0.50) × 10−3 ,
+|𝑉𝑐𝑏|excl = (39.10 ± 0.50) × 10−3 ,
+differing by 4.3 𝜎,
+(5)
+have a significant impact on the allowed parameter space for NP in 𝐵0
+𝑞- ¯𝐵0
+𝑞 mixing. Trying to
+understand and resolve these tensions, another case is studied in the literature [15–18], which is a
+hybrid scenario combining the exclusive |𝑉𝑢𝑏| with the inclusive |𝑉𝑐𝑏| determination. Therefore,
+we consider for the rest of our analysis all these three cases. The corresponding 𝑅𝑏 results are:
+𝑅𝑏,incl = 0.434 ± 0.018 ,
+𝑅𝑏,excl = 0.392 ± 0.014 ,
+𝑅𝑏,hybrid = 0.364 ± 0.013 .
+(6)
+Making a fit to 𝑅𝑏 and 𝛾, the UT apex is determined [1]:
+Incl.
+¯𝜌 = 0.160 ± 0.025 ,
+¯𝜂 = 0.404 ± 0.022 ,
+(7)
+Excl.
+¯𝜌 = 0.144 ± 0.022 ,
+¯𝜂 = 0.365 ± 0.018 ,
+(8)
+Hybrid
+¯𝜌 = 0.134 ± 0.021 ,
+¯𝜂 = 0.338 ± 0.017 .
+(9)
+The results are illustrated in Fig. 1. The plot also shows the hyperbola coming from the |𝜀𝐾 |
+observable, which is related to indirect CP violation in the neutral kaon system and is highly
+sensitive to the |𝑉𝑐𝑏| numerical value. The hybrid case gives the most consistent picture of the
+UT apex within the SM, which illustrates the strong dependence on |𝑉𝑐𝑏|. In the future, this could
+help us to understand the inclusive-exclusive puzzle, if NP in the kaon system can be controlled or
+ignored.
+2.2 Determining the UT Apex Utilising 𝑅𝑏 and 𝑅𝑡
+An alternative way of determining the UT apex is utilising the 𝑅𝑡 side, which is defined as:
+𝑅𝑡 ≡ |𝑉𝑡𝑑𝑉𝑡𝑏/𝑉𝑐𝑑𝑉𝑐𝑏| =
+√︃
+(1 − ¯𝜌)2 + ¯𝜂 2.
+(10)
+3
+
+Studies of New Physics in 𝐵0
+𝑞 − ¯𝐵0
+𝑞 Mixing and Implications for Leptonic Decays
+Eleftheria Malami
+0
+0.2
+0.4
+0.6
+0.8
+1
+ρ
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+η
+avg
+γ
+b
+R
+Fit Solution
+|
+K
+ε|
+contours hold 39%, 87% CL
+| from Kl3
+us
+ & |V
+b
+Incl. R
+0
+0.2
+0.4
+0.6
+0.8
+1
+ρ
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+η
+avg
+γ
+b
+R
+Fit Solution
+|
+K
+ε|
+contours hold 39%, 87% CL
+| from Kl3
+us
+ & |V
+b
+Excl. R
+0
+0.2
+0.4
+0.6
+0.8
+1
+ρ
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+η
+avg
+γ
+b
+R
+Fit Solution
+|
+K
+ε|
+contours hold 39%, 87% CL
+| from Kl3
+us
+ & |V
+b
+Hybrid R
+Figure 1: Determination of the UT apex from the 𝑅𝑏 and 𝛾 measurements for the inclusive (left), exclusive
+(right) and hybrid (botttom) case [1].
+In this case, only information on the two UT sides 𝑅𝑏 and 𝑅𝑡 is required without needing any
+information from 𝛾. However, in order to get the 𝑅𝑡, we have to assume SM expressions for the
+mixing parameters Δ𝑚𝑑 and Δ𝑚𝑠. The numerical predictions are given in [1].
+The side 𝑅𝑡 can be written as
+𝑅𝑡 = 1
+𝜆
+����
+𝑉𝑡𝑑
+𝑉𝑡𝑠
+����
+�
+1 − 𝜆2
+2 (1 − 2 ¯𝜌)
+�
++ O
+�
+𝜆4�
+,
+(11)
+where
+����
+𝑉𝑡𝑑
+𝑉𝑡𝑠
+���� = 𝜉
+√︄
+𝑚𝐵𝑠Δ𝑚SM
+𝑑
+𝑚𝐵𝑑Δ𝑚SM
+𝑠
+.
+(12)
+Here the SU(3)-breaking parameter 𝜉 is the ratio of bag parameters and decay constants of the
+𝐵𝑑 and the 𝐵𝑠 systems that can be calculated on the lattice. The advantage of the ratio is that
+uncertainties cancel, making it cleaner than using individual results.
+Making a fit to the 𝑅𝑏 and 𝑅𝑡 sides, we obtain [1]:
+Incl.
+¯𝜌 = 0.180 ± 0.014 ,
+¯𝜂 = 0.395 ± 0.020 ,
+(13)
+Excl.
+¯𝜌 = 0.163 ± 0.013 ,
+¯𝜂 = 0.357 ± 0.017 ,
+(14)
+Hybrid
+¯𝜌 = 0.153 ± 0.013 ,
+¯𝜂 = 0.330 ± 0.016 .
+(15)
+We note that the UT apex determinations relying on 𝛾 are a factor 2 less precise than those without
+information from 𝛾. However, the determination through 𝑅𝑏 and 𝑅𝑡 requires the SM expressions
+of Δ𝑚𝑑 and Δ𝑚𝑠, thus ignores possible NP contributions in 𝐵0
+𝑞- ¯𝐵0
+𝑞 mixing.
+4
+
+Studies of New Physics in 𝐵0
+𝑞 − ¯𝐵0
+𝑞 Mixing and Implications for Leptonic Decays
+Eleftheria Malami
+0
+50
+100
+150
+200
+250
+300
+350
+]°
+ [
+σ
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+κ
+ System (Scenario I)
+d
+B
+ System (Scenario I)
+s
+B
+FUNP (Scenario II)
+contours hold 39%, 87% CL
+| from Kl3
+us
+ & |V
+b
+Incl. R
+0
+50
+100
+150
+200
+250
+300
+350
+]°
+ [
+σ
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+κ
+ System (Scenario I)
+d
+B
+ System (Scenario I)
+s
+B
+FUNP (Scenario II)
+contours hold 39%, 87% CL
+| from Kl3
+us
+ & |V
+b
+Excl. R
+0
+50
+100
+150
+200
+250
+300
+350
+]°
+ [
+σ
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+κ
+ System (Scenario I)
+d
+B
+ System (Scenario I)
+s
+B
+FUNP (Scenario II)
+contours hold 39%, 87% CL
+| from Kl3
+us
+ & |V
+b
+Hybrid R
+Figure 2: Comparing Scenario I and Scenario II fits for 𝜅𝑞 and 𝜎𝑞 for the inclusive (left), exclusive (right)
+and hybrid (bottom) case [1].
+3.
+NP in 𝐵0
+𝑞- ¯𝐵0
+𝑞 mixing
+The neutral 𝐵𝑞-meson mixing is a sensitive phenomenon for NP. In order to quantify its impact,
+we introduce NP parameters 𝜅𝑞, which describes the size of the NP effects, and 𝜎𝑞, which is a
+complex phase accounting for additional CP-violating effects. The generalised expressions of the
+mixing parameters take the following form [19]:
+Δ𝑚𝑞 = Δ𝑚SM
+𝑞
+��1 + 𝜅𝑞𝑒𝑖𝜎𝑞�� ,
+(16)
+𝜙𝑞 = 𝜙SM
+𝑞
++ 𝜙NP
+𝑞
+= 𝜙SM
+𝑞
++ arg �1 + 𝜅𝑞𝑒𝑖𝜎𝑞� .
+(17)
+This is a model independent parametrization. Utilising these relations, we explore two different NP
+scenarios; the first one is the most general case and the second one assumes Flavour Universal NP
+(FUNP) [1].
+Let us firstly discuss the general case, namely Scenario I. The only assumption here is that there
+is no NP in the angle 𝛾 and 𝑅𝑏. The determination from 𝑅𝑏 and 𝛾 does not rely on information from
+mixing. We make use of this determination to obtain the UT apex, which we then need for getting
+the SM predictions for the mixing parameters Δ𝑚𝑞 and 𝜙𝑞. Comparing them with their measured
+values, we can constrain the NP parameters. Here, the NP parameters (𝜅𝑑, 𝜎𝑑) and (𝜅𝑠, 𝜎𝑠) are
+determined independently from each other.
+In the second case, Scenario II, we have the FUNP assumption where we consider that the NP
+contributions are equal in the 𝐵𝑑 and 𝐵𝑠 systems, thus (𝜅𝑑, 𝜎𝑑) = (𝜅𝑠, 𝜎𝑠). This is not a Minimal
+Flavour Violation scenario but it can be realised in NP models with 𝑈(2) symmetry [20, 21]. The
+UT apex fit relies on 𝑅𝑏 and 𝑅𝑡, without using 𝛾 information, therefore possible NP in the angle 𝛾
+5
+
+Studies of New Physics in 𝐵0
+𝑞 − ¯𝐵0
+𝑞 Mixing and Implications for Leptonic Decays
+Eleftheria Malami
+will not affect the findings. Comparing the two scenarios, we have a test of the FUNP assumption
+and we see the impact of the assumptions on the constraints on the parameter space of NP in mixing.
+Fig. 2 illustrates this comparison of the two fits for 𝜅𝑞 and 𝜎𝑞 for the inclusive, the exclusive and
+the hybrid cases.
+4.
+Rare Leptonic Decays 𝐵0
+𝑞 → 𝜇+𝜇−
+The tensions between the CKM matrix elements have an impact not only on the UT apex
+determination and possible NP in 𝐵0
+𝑞- ¯𝐵0
+𝑞 mixing but also on the branching ratios of rare decays. A
+key example is the leptonic 𝐵0
+𝑞 → 𝜇+𝜇− transition. These modes are pure loop processes and helicity
+suppressed in the SM. This helicity suppression could be lifted by new scalar and pseudoscalar
+conttributions, therefore putting these decays in an outstanding position to probe NP in this sector.
+As these are decays of neutral 𝐵 mesons, 𝐵0
+𝑞- ¯𝐵0
+𝑞 mixing enters and leads to subtleties concerning the
+measurement of the experimental branching ratio and comparison with the theoretical prediction
+[22]. However, NP in 𝐵0
+𝑠- ¯𝐵0
+𝑠 mixing is included through the experimental values of the mixing
+parameters.
+The SM predictions require information on |𝑉𝑡𝑠| which we determine through |𝑉𝑐𝑏|, which
+again depends on inclusive and exclusive determinations. In order to minimise the dependence on
+|𝑉𝑐𝑏| and the UT apex, we create the following ratio with the 𝐵𝑠 mass difference Δ𝑚𝑠 [23–25]:
+R𝑠𝜇 ≡ ¯B(𝐵𝑠 → 𝜇+𝜇−)/Δ𝑚𝑠 .
+(18)
+Using this ratio, we can eliminate the leading dependence on the CKM elements but we have to
+correct for the possible NP contributions to 𝐵0
+𝑞- ¯𝐵0
+𝑞 mixing. This is now possible following our
+analysis in [1].
+So, we include NP effects in Δ𝑚𝑠 and then we can use the ratio R𝑠𝜇 to constrain NP in the
+scalar and pseudoscalar sector. We obtain the generalised expression:
+R𝑠𝜇 = RSM
+𝑠𝜇 ×
+1 + A𝜇𝜇
+ΔΓ𝑠 𝑦𝑠
+1 + 𝑦𝑠
+|𝑃𝑠
+𝜇𝜇|2 + |𝑆𝑠
+𝜇𝜇|2
+√︁
+1 + 2𝜅𝑠 cos 𝜎𝑠 + 𝜅2𝑠
+,
+(19)
+with 𝑃𝑠
+𝜇𝜇 ≡ |𝑃𝑠
+𝜇𝜇|𝑒𝑖𝜑𝑃, 𝑆𝑠
+𝜇𝜇 ≡ |𝑆𝑠
+𝜇𝜇|𝑒𝑖𝜑𝑆, where 𝜑𝑃, 𝜑𝑆 are CP-violating phases, and the observable
+A𝜇𝜇
+ΔΓ𝑠 in terms of the NP phase 𝜙NP
+𝑠 :
+A𝜇𝜇
+ΔΓ =
+|𝑃𝑠
+𝜇𝜇|2 cos(2𝜑𝑃 − 𝜙NP
+𝑠 ) − |𝑆𝑠
+𝜇𝜇|2 cos(2𝜑𝑆 − 𝜙NP
+𝑠 )
+|𝑃𝑠𝜇𝜇|2 + |𝑆𝑠𝜇𝜇|2
+.
+(20)
+The R𝑠𝜇 has only a dependence on the CKM matrix elements through the NP parameters 𝜅𝑞
+and 𝜎𝑞, determined as described above. Therefore, we have another constraint on the scalar and
+pseudoscalar contributions. The same strategy can be applied to the 𝐵0
+𝑑 → 𝜇+𝜇− channel once in
+the future accurate measurements of the branching ratio will become available.
+5.
+Future Prospects and Final Remarks
+It will be important in the future to achieve improved precision on the NP parameters 𝜅𝑞 and
+𝜎𝑞. In order to get a feeling of the prospects, we assume a hypothetical reduction of 50% on each
+6
+
+Studies of New Physics in 𝐵0
+𝑞 − ¯𝐵0
+𝑞 Mixing and Implications for Leptonic Decays
+Eleftheria Malami
+one of the three input parameters, which are the |𝑉𝑐𝑏|, the lattice calculations and the UT apex [1].
+We obtain interesting findings, which of course depend on these assumptions. In our studies, we
+demonstrate that in the 𝐵𝑑-system the apex plays a limiting factor and in order to fully explore the
+potentials of this system, progress on the UT apex has to be made. On the other hand, in the 𝐵𝑠-
+system we do not have this situation as the SM prediction of 𝜙𝑠 is more robust. Therefore, searches
+of NP in 𝐵0
+𝑠- ¯𝐵0
+𝑠 mixing are more promising than in the 𝐵𝑑-system but it is of key importance to
+constrain NP in both systems as much as possible.
+Another essential future prospect is related to the angle 𝛾. Improved precision on the input
+measurements might lead to significant discrepancies between the different 𝛾 determinations due
+to NP effects. In this case, averaging over the different results, as we did in this analysis, would
+no longer be justified. Therefore, the UT should then be revisited. Independent information from
+additional observables would be necessary to resolve such a situation. Exciting new opportunities
+might come up to search for NP, both in 𝛾 and in 𝐵0
+𝑞- ¯𝐵0
+𝑞 mixing, which is strongly correlated with
+the UT apex coordinates.
+Last but not least, the branching ratios of the 𝐵0
+𝑞 → 𝜇+𝜇− decays might offer interesting
+opportunities. The ratio of the branching fractions between 𝐵0
+𝑑 → 𝜇+𝜇− and 𝐵0
+𝑠 → 𝜇+𝜇− can
+provide an alternative way to determine the UT side 𝑅𝑡. Another useful application for the ratio of
+the branching fractions between these channels is the quantity [26]:
+𝑈𝑑𝑠
+𝜇𝜇 ∝
+�����
+𝑉𝑡𝑠
+𝑉𝑡𝑑
+����
+2 ¯B(𝐵𝑑 → 𝜇+𝜇−)
+¯B(𝐵𝑠 → 𝜇+𝜇−)
+�1/2
+(21)
+which requires knowledge of 𝑅𝑡 and offers a very powerful test of the SM, where 𝑈𝑑𝑠
+𝜇𝜇 = 1.
+In the future, 𝐵0
+𝑞- ¯𝐵0
+𝑞 mixing will remain a key element for constraining NP. It will be exciting
+to see how more precise data in the high-precision era of flavour physics ahead of us can lead to a
+much sharper picture.
+Acknowledgements
+We would like to thank the DISCRETE 2022 organisers for the invitation and for giving us the
+opportunity to present our studies. This research has been supported by the Netherlands Organisation
+for Scientific Research (NWO). PvV acknowledges support from the DFG through the Emmy
+Noether research project 400570283, and through the German-Israeli Project Cooperation (DIP).
+References
+[1] K. De Bruyn, R. Fleischer, E. Malami and P. van Vliet, 2022 J. Phys. G: Nucl. Part. Phys.
+https://doi.org/10.1088/1361-6471/acab1d
+[2] R. L. Workman et al. [Particle Data Group], PTEP 2022 (2022), 083C01
+[3] L. Wolfenstein, Phys. Rev. Lett. 51 (1983), 1945 doi:10.1103/PhysRevLett.51.1945
+[4] A. J. Buras, M. E. Lautenbacher and G. Ostermaier, Phys. Rev. D 50 (1994), 3433-3446
+7
+
+Studies of New Physics in 𝐵0
+𝑞 − ¯𝐵0
+𝑞 Mixing and Implications for Leptonic Decays
+Eleftheria Malami
+[5] M. Z. Barel, K. De Bruyn, R. Fleischer and E. Malami, [arXiv:2203.14652 [hep-ph]].
+[6] M. Z. Barel, K. De Bruyn, R. Fleischer and E. Malami, J. Phys. G 48 (2021) no.6, 065002
+[7] R. Fleischer and E. Malami, Phys. Rev. D 106 (2022) no.5, 056004
+[8] R. Fleischer and E. Malami, [arXiv:2110.04240 [hep-ph]].
+[9] R. Aaij et al. [LHCb], JHEP 12 (2021), 141
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+[11] J. Charles et al., Eur. Phys. J. C 77 (2017) no.8, 574
+[12] C. Y. Seng et al., Phys. Rev. D 105 (2022) no.1, 013005
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+[14] Y. Amhis et al. [HFLAV], [arXiv:2206.07501 [hep-ex]].
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+[19] P. Ball and R. Fleischer, Eur. Phys. J. C 48 (2006), 413-426
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+[22] K. De Bruyn et al., Phys. Rev. Lett. 109 (2012), 041801
+[23] A. J. Buras, Phys. Lett. B 566 (2003), 115-119
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+[26] R. Fleischer, R. Jaarsma and G. Tetlalmatzi-Xolocotzi, JHEP 05 (2017), 156
+8
+

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+page_content='Studies of New Physics in 𝑩0  𝒒 − ¯𝑩0  𝒒 Mixing and  Implications for Leptonic Decays  Kristof De Bruyn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='𝑏 Robert Fleischer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='𝑐 Eleftheria Malami𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='𝑑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='∗ and Philine van Vliet𝑒  𝑎Nikhef,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  Science Park 105,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' 1098 XG Amsterdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Netherlands  𝑏Van Swinderen Institute for Particle Physics and Gravity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' University of Groningen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  9747 Groningen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Netherlands  𝑐Faculty of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Vrije Universiteit Amsterdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  1081 HV Amsterdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Netherlands  𝑑Center for Particle Physics Siegen (CPPS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Theoretische Physik 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Universität Siegen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  D-57068 Siegen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Germany  𝑒Deutsches Elektronen-Synchrotron DESY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  Notkestr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' 85, 22607 Hamburg, Germany  E-mail: Eleftheria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='Malami@uni-siegen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='de  The phenomenon of 𝐵0  𝑞- ¯𝐵0  𝑞 mixing (𝑞 = 𝑑, 𝑠) provides a sensitive probe for physics beyond the  Standard Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' We have a careful look at the determination of the Unitarity Triangle apex, which  is needed for the Standard Model predictions of the 𝐵𝑞 mixing parameters, and explore how much  space for New Physics is left through the current data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' We study the impact of tensions between  inclusive and exclusive determinations of the CKM matrix elements |𝑉𝑢𝑏| and |𝑉𝑐𝑏|, and focus on  the 𝛾 angle extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' We present various future scenarios and discuss the application of these  results for leptonic rare 𝐵 decays, which allows us to minimise the CKM parameter impact in  the New Physics searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Performing future projections, we explore and illustrate the impact of  increased precision on key input quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' It will be exciting to see how more precise data in the  future high-precision era of flavour physics can lead to a much sharper picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  8th Symposium on Prospects in the Physics of Discrete Symmetries (DISCRETE 2022)  7-11 November, 2022  Baden-Baden, Germany  ∗Speaker  © Copyright owned by the author(s) under the terms of the Creative Commons  Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='it/  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='13649v1  [hep-ph]  31 Jan 2023  Studies of New Physics in 𝐵0  𝑞 − ¯𝐵0  𝑞 Mixing and Implications for Leptonic Decays  Eleftheria Malami  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  Introduction  The phenomenon of 𝐵0  𝑞- ¯𝐵0  𝑞 mixing (where 𝑞 = 𝑑, 𝑠) arises only from loop processes in the  Standard Model (SM) and is sensitive to possible New Physics (NP) contributions, which could  enter the loop topologies or even at the tree level, for instance in 𝑍 ′ models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Associated to the mixing  phenomenon are the mixing parameters and the CP-violating phases for which we have impressive  experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' In this presentation, we follow Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' [1] and explore the space allowed for NP  by current measurements and the state-of-the-art parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' In addition, we point out interesting  connections to the studies of leptonic rare 𝐵 decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  In order to determine the parameter space of possible NP effects to 𝐵0  𝑞– ¯𝐵0  𝑞 mixing, we have to  compare the SM predictions of the mixing parameters with the corresponding experimental values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  For these SM predictions, a careful analysis of the Unitarity Triangle (UT) apex is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' We  pay special attention to the different determinations of the Cabibbo-Kobayashi-Maskawa (CKM)  parameters and the tensions that arise between the extractions of the |𝑉𝑢𝑏| and |𝑉𝑐𝑏| matrix elements  through inclusive and exclusive semileptonic 𝐵 meson decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' These longstanding tensions have a  profound impact on the whole analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  Unitarity Triangle  Using the parametrisation of the Particle Data Group (PDG), the UT apex is given as [2]:  𝑅𝑏 𝑒𝑖𝛾 = ¯𝜌 + 𝑖 ¯𝜂 ,  ¯𝜌 ≡  �  1 − (𝜆2/2)  �  𝜌 ,  ¯𝜂 ≡  �  1 − (𝜆2/2)  �  𝜂 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  (1)  Here, 𝜌, 𝜂 and 𝜆 are the Wolfenstein parameters [3, 4], 𝑅𝑏 is the side from the origin to the apex of  the UT, defined with the help of the CKM matrix elements 𝜆 ≡ |𝑉𝑢𝑠|, |𝑉𝑢𝑏| and |𝑉𝑐𝑏| as:  𝑅𝑏 ≡  �  1 − 𝜆2  2  � 1  𝜆  ����  𝑉𝑢𝑏  𝑉𝑐𝑏  ���� =  √︃  ¯𝜌 2 + ¯𝜂 2 ,  (2)  and 𝛾 ≡ arg �−𝑉𝑢𝑑𝑉∗  𝑢𝑏/𝑉𝑐𝑑𝑉∗  𝑐𝑏  � is the angle between the 𝑅𝑏 side and the UT basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='1 Determining the UT Apex Utilising 𝛾 and 𝑅𝑏  In this subsection, we work in the SM and are interested in obtaining the UT apex in a way  that is not affected by possible NP in 𝐵0  𝑞- ¯𝐵0  𝑞 mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' One way of determining the apex is utilising  the side 𝑅𝑏 and the angle 𝛾, which can both be determined from decays that proceed only via tree  decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' The value of 𝛾 can be determined either from 𝐵 → 𝐷𝐾 decays or from a 𝐵 → 𝜋𝜋, 𝜌𝜋, 𝜌𝜌  isospin analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  More specifically, one option is to use the time-dependent 𝐵0  𝑠 → 𝐷∓  𝑠 𝐾± system, where mixing-  induced CP violation plays a key role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Through interference effects caused by 𝐵0  𝑞- ¯𝐵0  𝑞 mixing, the  CP asymmetry parameters allow the determination of 𝜙𝑠 + 𝛾, where 𝜙𝑠 is the 𝐵0  𝑠- ¯𝐵0  𝑠 mixing phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  Since 𝜙𝑠 is determined through the 𝐵0  𝑠 → 𝐽/𝜓𝜙 channel, including penguin corrections [5, 6], 𝛾  can be obtained in a theoretically clean way [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' However, the surprisingly large value arising in  this case still needs to be further explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' An alternative way of getting the 𝛾 value is using the  time-independent 𝐵 → 𝐷𝐾 transitions, where the sensitivity to 𝛾 comes from direct CP violation  [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Last but not least, another interesting system is provided by 𝐵 → 𝜋𝜋, 𝜌𝜋, 𝜌𝜌 modes [10, 11],  2  Studies of New Physics in 𝐵0  𝑞 − ¯𝐵0  𝑞 Mixing and Implications for Leptonic Decays  Eleftheria Malami  which usually are used to determine 𝛼 from an isospin analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Actually this value corresponds to  𝛾 when we use the 𝐵0  𝑑- ¯𝐵0  𝑑 mixing phase 𝜙𝑑, determined from 𝐵0  𝑑 → 𝐽/𝜓𝐾0 [5, 6], taking penguin  effects into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Thus, we can convert the result 𝜙𝑑 + 2𝛾 into 𝛾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' The value from the latter case  is in good agreement with the one coming from 𝐵 → 𝐷𝐾 modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Therefore, for our analysis, we  average these two results [1]:  𝛾avg = (68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='4 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='4)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  (3)  Regarding 𝑅𝑏 there are tensions between the various theoretical and experimental approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  Even though there are different determinations of the |𝑉𝑢𝑠| element and the tensions between them  are intriguing, they only have a negligible impact on NP studies in neutral 𝐵𝑞 mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Thus, we  choose to work with the value |𝑉𝑢𝑠| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='22309 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='00056 [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Contrary to the |𝑉𝑢𝑠| case, the  deviations between determinations of |𝑉𝑢𝑏| and |𝑉𝑐𝑏| from inclusive and exclusive semileptonic 𝐵  decays, which are given as follows [14, 15]:  |𝑉𝑢𝑏|incl = (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='17) × 10−3 ,  |𝑉𝑢𝑏|excl = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='12) × 10−3 ,  differing by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='9 𝜎,  (4)  |𝑉𝑐𝑏|incl = (42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='50) × 10−3 ,  |𝑉𝑐𝑏|excl = (39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='50) × 10−3 ,  differing by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='3 𝜎,  (5)  have a significant impact on the allowed parameter space for NP in 𝐵0  𝑞- ¯𝐵0  𝑞 mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Trying to  understand and resolve these tensions, another case is studied in the literature [15–18], which is a  hybrid scenario combining the exclusive |𝑉𝑢𝑏| with the inclusive |𝑉𝑐𝑏| determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Therefore,  we consider for the rest of our analysis all these three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' The corresponding 𝑅𝑏 results are:  𝑅𝑏,incl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='434 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='018 ,  𝑅𝑏,excl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='392 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='014 ,  𝑅𝑏,hybrid = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='364 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='013 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  (6)  Making a fit to 𝑅𝑏 and 𝛾, the UT apex is determined [1]:  Incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  ¯𝜌 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='160 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='025 ,  ¯𝜂 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='404 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='022 ,  (7)  Excl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  ¯𝜌 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='144 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='022 ,  ¯𝜂 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='365 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='018 ,  (8)  Hybrid  ¯𝜌 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='134 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='021 ,  ¯𝜂 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='338 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='017 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  (9)  The results are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' The plot also shows the hyperbola coming from the |𝜀𝐾 |  observable, which is related to indirect CP violation in the neutral kaon system and is highly  sensitive to the |𝑉𝑐𝑏| numerical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' The hybrid case gives the most consistent picture of the  UT apex within the SM, which illustrates the strong dependence on |𝑉𝑐𝑏|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' In the future, this could  help us to understand the inclusive-exclusive puzzle, if NP in the kaon system can be controlled or  ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='2 Determining the UT Apex Utilising 𝑅𝑏 and 𝑅𝑡  An alternative way of determining the UT apex is utilising the 𝑅𝑡 side, which is defined as:  𝑅𝑡 ≡ |𝑉𝑡𝑑𝑉𝑡𝑏/𝑉𝑐𝑑𝑉𝑐𝑏| =  √︃  (1 − ¯𝜌)2 + ¯𝜂 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  (10)  3  Studies of New Physics in 𝐵0  𝑞 − ¯𝐵0  𝑞 Mixing and Implications for Leptonic Decays  Eleftheria Malami  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='8  1  ρ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='7  η  avg  γ  b  R  Fit Solution  |  K  ε|  contours hold 39%, 87% CL  | from Kl3  us  & |V  b  Incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' R  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='8  1  ρ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='7  η  avg  γ  b  R  Fit Solution  |  K  ε|  contours hold 39%, 87% CL  | from Kl3  us  & |V  b  Excl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' R  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='8  1  ρ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='7  η  avg  γ  b  R  Fit Solution  |  K  ε|  contours hold 39%, 87% CL  | from Kl3  us  & |V  b  Hybrid R  Figure 1: Determination of the UT apex from the 𝑅𝑏 and 𝛾 measurements for the inclusive (left), exclusive  (right) and hybrid (botttom) case [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  In this case, only information on the two UT sides 𝑅𝑏 and 𝑅𝑡 is required without needing any  information from 𝛾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' However, in order to get the 𝑅𝑡, we have to assume SM expressions for the  mixing parameters Δ𝑚𝑑 and Δ𝑚𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' The numerical predictions are given in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  The side 𝑅𝑡 can be written as  𝑅𝑡 = 1  𝜆  ����  𝑉𝑡𝑑  𝑉𝑡𝑠  ����  �  1 − 𝜆2  2 (1 − 2 ¯𝜌)  �  + O  �  𝜆4�  ,  (11)  where  ����  𝑉𝑡𝑑  𝑉𝑡𝑠  ���� = 𝜉  √︄  𝑚𝐵𝑠Δ𝑚SM  𝑑  𝑚𝐵𝑑Δ𝑚SM  𝑠  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  (12)  Here the SU(3)-breaking parameter 𝜉 is the ratio of bag parameters and decay constants of the  𝐵𝑑 and the 𝐵𝑠 systems that can be calculated on the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' The advantage of the ratio is that  uncertainties cancel, making it cleaner than using individual results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  Making a fit to the 𝑅𝑏 and 𝑅𝑡 sides, we obtain [1]:  Incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  ¯𝜌 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='180 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='014 ,  ¯𝜂 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='395 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='020 ,  (13)  Excl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  ¯𝜌 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='163 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='013 ,  ¯𝜂 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='357 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='017 ,  (14)  Hybrid  ¯𝜌 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='153 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='013 ,  ¯𝜂 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='330 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='016 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  (15)  We note that the UT apex determinations relying on 𝛾 are a factor 2 less precise than those without  information from 𝛾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' However, the determination through 𝑅𝑏 and 𝑅𝑡 requires the SM expressions  of Δ𝑚𝑑 and Δ𝑚𝑠, thus ignores possible NP contributions in 𝐵0  𝑞- ¯𝐵0  𝑞 mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  4  Studies of New Physics in 𝐵0  𝑞 − ¯𝐵0  𝑞 Mixing and Implications for Leptonic Decays  Eleftheria Malami  0  50  100  150  200  250  300  350  ]°  [  σ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='5  κ  System (Scenario I)  d  B  System (Scenario I)  s  B  FUNP (Scenario II)  contours hold 39%, 87% CL  | from Kl3  us  & |V  b  Incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' R  0  50  100  150  200  250  300  350  ]°  [  σ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='5  κ  System (Scenario I)  d  B  System (Scenario I)  s  B  FUNP (Scenario II)  contours hold 39%, 87% CL  | from Kl3  us  & |V  b  Excl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' R  0  50  100  150  200  250  300  350  ]°  [  σ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='5  κ  System (Scenario I)  d  B  System (Scenario I)  s  B  FUNP (Scenario II)  contours hold 39%, 87% CL  | from Kl3  us  & |V  b  Hybrid R  Figure 2: Comparing Scenario I and Scenario II fits for 𝜅𝑞 and 𝜎𝑞 for the inclusive (left), exclusive (right)  and hybrid (bottom) case [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  NP in 𝐵0  𝑞- ¯𝐵0  𝑞 mixing  The neutral 𝐵𝑞-meson mixing is a sensitive phenomenon for NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' In order to quantify its impact,  we introduce NP parameters 𝜅𝑞, which describes the size of the NP effects, and 𝜎𝑞, which is a  complex phase accounting for additional CP-violating effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' The generalised expressions of the  mixing parameters take the following form [19]:  Δ𝑚𝑞 = Δ𝑚SM  𝑞  ��1 + 𝜅𝑞𝑒𝑖𝜎𝑞�� ,  (16)  𝜙𝑞 = 𝜙SM  𝑞  + 𝜙NP  𝑞  = 𝜙SM  𝑞  + arg �1 + 𝜅𝑞𝑒𝑖𝜎𝑞� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  (17)  This is a model independent parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Utilising these relations, we explore two different NP  scenarios;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' the first one is the most general case and the second one assumes Flavour Universal NP  (FUNP) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  Let us firstly discuss the general case, namely Scenario I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' The only assumption here is that there  is no NP in the angle 𝛾 and 𝑅𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' The determination from 𝑅𝑏 and 𝛾 does not rely on information from  mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' We make use of this determination to obtain the UT apex, which we then need for getting  the SM predictions for the mixing parameters Δ𝑚𝑞 and 𝜙𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Comparing them with their measured  values, we can constrain the NP parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Here, the NP parameters (𝜅𝑑, 𝜎𝑑) and (𝜅𝑠, 𝜎𝑠) are  determined independently from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  In the second case, Scenario II, we have the FUNP assumption where we consider that the NP  contributions are equal in the 𝐵𝑑 and 𝐵𝑠 systems, thus (𝜅𝑑, 𝜎𝑑) = (𝜅𝑠, 𝜎𝑠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' This is not a Minimal  Flavour Violation scenario but it can be realised in NP models with 𝑈(2) symmetry [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' The  UT apex fit relies on 𝑅𝑏 and 𝑅𝑡, without using 𝛾 information, therefore possible NP in the angle 𝛾  5  Studies of New Physics in 𝐵0  𝑞 − ¯𝐵0  𝑞 Mixing and Implications for Leptonic Decays  Eleftheria Malami  will not affect the findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Comparing the two scenarios, we have a test of the FUNP assumption  and we see the impact of the assumptions on the constraints on the parameter space of NP in mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' 2 illustrates this comparison of the two fits for 𝜅𝑞 and 𝜎𝑞 for the inclusive, the exclusive and  the hybrid cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  Rare Leptonic Decays 𝐵0  𝑞 → 𝜇+𝜇−  The tensions between the CKM matrix elements have an impact not only on the UT apex  determination and possible NP in 𝐵0  𝑞- ¯𝐵0  𝑞 mixing but also on the branching ratios of rare decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' A  key example is the leptonic 𝐵0  𝑞 → 𝜇+𝜇− transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' These modes are pure loop processes and helicity  suppressed in the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' This helicity suppression could be lifted by new scalar and pseudoscalar  conttributions, therefore putting these decays in an outstanding position to probe NP in this sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  As these are decays of neutral 𝐵 mesons, 𝐵0  𝑞- ¯𝐵0  𝑞 mixing enters and leads to subtleties concerning the  measurement of the experimental branching ratio and comparison with the theoretical prediction  [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' However, NP in 𝐵0  𝑠- ¯𝐵0  𝑠 mixing is included through the experimental values of the mixing  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  The SM predictions require information on |𝑉𝑡𝑠| which we determine through |𝑉𝑐𝑏|, which  again depends on inclusive and exclusive determinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' In order to minimise the dependence on  |𝑉𝑐𝑏| and the UT apex, we create the following ratio with the 𝐵𝑠 mass difference Δ𝑚𝑠 [23–25]:  R𝑠𝜇 ≡ ¯B(𝐵𝑠 → 𝜇+𝜇−)/Δ𝑚𝑠 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  (18)  Using this ratio, we can eliminate the leading dependence on the CKM elements but we have to  correct for the possible NP contributions to 𝐵0  𝑞- ¯𝐵0  𝑞 mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' This is now possible following our  analysis in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  So, we include NP effects in Δ𝑚𝑠 and then we can use the ratio R𝑠𝜇 to constrain NP in the  scalar and pseudoscalar sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' We obtain the generalised expression:  R𝑠𝜇 = RSM  𝑠𝜇 ×  1 + A𝜇𝜇  ΔΓ𝑠 𝑦𝑠  1 + 𝑦𝑠  |𝑃𝑠  𝜇𝜇|2 + |𝑆𝑠  𝜇𝜇|2  √︁  1 + 2𝜅𝑠 cos 𝜎𝑠 + 𝜅2𝑠  ,  (19)  with 𝑃𝑠  𝜇𝜇 ≡ |𝑃𝑠  𝜇𝜇|𝑒𝑖𝜑𝑃, 𝑆𝑠  𝜇𝜇 ≡ |𝑆𝑠  𝜇𝜇|𝑒𝑖𝜑𝑆, where 𝜑𝑃, 𝜑𝑆 are CP-violating phases, and the observable  A𝜇𝜇  ΔΓ𝑠 in terms of the NP phase 𝜙NP  𝑠 :  A𝜇𝜇  ΔΓ =  |𝑃𝑠  𝜇𝜇|2 cos(2𝜑𝑃 − 𝜙NP  𝑠 ) − |𝑆𝑠  𝜇𝜇|2 cos(2𝜑𝑆 − 𝜙NP  𝑠 )  |𝑃𝑠𝜇𝜇|2 + |𝑆𝑠𝜇𝜇|2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  (20)  The R𝑠𝜇 has only a dependence on the CKM matrix elements through the NP parameters 𝜅𝑞  and 𝜎𝑞, determined as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Therefore, we have another constraint on the scalar and  pseudoscalar contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' The same strategy can be applied to the 𝐵0  𝑑 → 𝜇+𝜇− channel once in  the future accurate measurements of the branching ratio will become available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  Future Prospects and Final Remarks  It will be important in the future to achieve improved precision on the NP parameters 𝜅𝑞 and  𝜎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' In order to get a feeling of the prospects, we assume a hypothetical reduction of 50% on each  6  Studies of New Physics in 𝐵0  𝑞 − ¯𝐵0  𝑞 Mixing and Implications for Leptonic Decays  Eleftheria Malami  one of the three input parameters, which are the |𝑉𝑐𝑏|, the lattice calculations and the UT apex [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  We obtain interesting findings, which of course depend on these assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' In our studies, we  demonstrate that in the 𝐵𝑑-system the apex plays a limiting factor and in order to fully explore the  potentials of this system, progress on the UT apex has to be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' On the other hand, in the 𝐵𝑠-  system we do not have this situation as the SM prediction of 𝜙𝑠 is more robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Therefore, searches  of NP in 𝐵0  𝑠- ¯𝐵0  𝑠 mixing are more promising than in the 𝐵𝑑-system but it is of key importance to  constrain NP in both systems as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  Another essential future prospect is related to the angle 𝛾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Improved precision on the input  measurements might lead to significant discrepancies between the different 𝛾 determinations due  to NP effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' In this case, averaging over the different results, as we did in this analysis, would  no longer be justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Therefore, the UT should then be revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Independent information from  additional observables would be necessary to resolve such a situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Exciting new opportunities  might come up to search for NP, both in 𝛾 and in 𝐵0  𝑞- ¯𝐵0  𝑞 mixing, which is strongly correlated with  the UT apex coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  Last but not least, the branching ratios of the 𝐵0  𝑞 → 𝜇+𝜇− decays might offer interesting  opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' The ratio of the branching fractions between 𝐵0  𝑑 → 𝜇+𝜇− and 𝐵0  𝑠 → 𝜇+𝜇− can  provide an alternative way to determine the UT side 𝑅𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' Another useful application for the ratio of  the branching fractions between these channels is the quantity [26]:  𝑈𝑑𝑠  𝜇𝜇 ∝  �����  𝑉𝑡𝑠  𝑉𝑡𝑑  ����  2 ¯B(𝐵𝑑 → 𝜇+𝜇−)  ¯B(𝐵𝑠 → 𝜇+𝜇−)  �1/2  (21)  which requires knowledge of 𝑅𝑡 and offers a very powerful test of the SM, where 𝑈𝑑𝑠  𝜇𝜇 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  In the future, 𝐵0  𝑞- ¯𝐵0  𝑞 mixing will remain a key element for constraining NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' It will be exciting  to see how more precise data in the high-precision era of flavour physics ahead of us can lead to a  much sharper picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content='  Acknowledgements  We would like to thank the DISCRETE 2022 organisers for the invitation and for giving us the  opportunity to present our studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' This research has been supported by the Netherlands Organisation  for Scientific Research (NWO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
+page_content=' PvV acknowledges support from the DFG through the Emmy  Noether research project 400570283, and through the German-Israeli Project Cooperation (DIP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'}
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+Spin-orbital order and excitons in magnetoresistive HoBi
+J. Gaudet,1, 2, 3, ∗ H.-Y. Yang,4 E. M. Smith,5 T. Halloran,1 J. P. Clancy,5 J. A. Rodriguez-Rivera,2, 3 Guangyong
+Xu,2 Y. Zhao,2, 3 W. C. Chen,2 G. Sala,6 A. A. Aczel,7 B. D. Gaulin,5, 8, 9 F. Tafti,4 and C. Broholm1, 2, 7
+1Institute for Quantum Matter and Department of Physics and Astronomy,
+Johns Hopkins University, Baltimore, MD 21218, USA
+2Center for Neutron Research, National Institute of Standards and Technology, MS 6100 Gaithersburg, Maryland 20899, USA
+3Department of Materials Science and Eng., University of Maryland, College Park, MD 20742-2115
+4Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA
+5Department of Physics and Astronomy, McMaster University, Hamilton, ON L8S 4M1, Canada
+6Spallation Neutron Source, Second Target Station, Oak Ridge National Laboratory, Oak Ridge, TN, 37831, USA
+7Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
+8Canadian Institute for Advanced Research, 661 University Avenue, Toronto, Ontario M5G 1M1, Canada.
+9Brockhouse Institute for Materials Research, Hamilton, ON L8S 4M1 Canada
+(Dated: January 13, 2023)
+The magnetism of the rock-salt fcc rare-earth monopnictide HoBi, a candidate topological material with
+extreme magnetoresistance, is investigated. From the Ho3+ non-Kramers J=8 spin-orbital multiplet, the cubic
+crystal electric field yields six nearly degenerate low-energy levels. These constitute an anisotropic magnetic
+moment with a Jahn-Teller-like coupling to the lattice. In the cubic phase for T > TN
+=
+5.72(1) K, the
+paramagnetic neutron scattering is centered at k = ( 1
+2
+1
+2
+1
+2) and was fit to dominant antiferromagnetic interactions
+between Ho spins separated by {100} and ferromagnetic interactions between spins displaced by { 1
+2
+1
+20}. For
+T < TN, a type-II AFM long-range order with k = ( 1
+2
+1
+2
+1
+2) develops along with a tetragonal lattice distortion.
+While neutron diffraction from a multi-domain sample cannot unambiguously determine the spin orientation
+within a domain, the bulk magnetization, structural distortion, and our measurements of the magnetic excitations
+all show the easy axis coincides with the tetragonal axis. The weakly dispersive excitons for T < TN can be
+accounted for by a spin Hamiltonian that includes the crystal electric field and exchange interactions within the
+Random Phase Approximation.
+I.
+INTRODUCTION
+In spite of their structural simplicity, the fcc rare-earth
+monopnictides (see Fig. 1), RX (R=Ce to Yb and X=N, As,
+P, Sb, and Bi1,2), display a wide variety of anisotropic mag-
+netism and electronic transport properties. The lattice param-
+eter varies by 30% across the pnictide series and this provides
+opportunities to tune the relative strength of crystal field and
+exchange interactions. In the 1960s to 1980s, the rare-earth
+monopnictides were studied to understand magnetic phases
+driven by oscillatory and highly anisotropic Ruderman-Kittel-
+Kasuya-Yosida (RKKY) exchange interactions 3–8. Work on
+CeSb for example gave rise to an extensive literature on the
+anisotropic nearest and next nearest neighbor Ising model
+(ANNNI)9.
+This work also resulted in progress towards
+a quantitative understanding of their anisotropic exchange
+interactions10.
+A recent resurgence of interest in these rare-earth monop-
+nictides is driven by their extreme magnetoresistance (XMR)
+and resistivity plateaus, and the possible connection to the
+3D topological state of the non-magnetic lanthanum monop-
+nictides LaX11,12. LaAs, LaSb, and LaBi have unsaturated
+XMR arising from near perfect electron-hole compensation
+and there is a topological transition from a trivial electronic
+band structure in LaAs to a topologically non-trivial band
+structure in LaBi13–17.
+Several studies have confirmed the
+presence of protected surface states in LaBi18–21. Since then,
+extensive works have been devoted to characterizing the XMR
+and topological states of various RX including for example
+CeX, HoX, and PrX. XMR has been found in each reported
+magnetic RX with characteristics that depend on the rare-earth
+ion22–30. The stabilization of topological non-trivial electronic
+bands generating protected surface states was proposed for
+several of the magnetic monopnictides29,31–33.
+FIG. 1. The rock-salt structure of the rare-earth monopnictide HoBi.
+Yellow and blue spheres respectively correspond to Ho and Bi. Spins
+interacting through the J1 and J2 exchange interaction are shown by
+the dashed black arrows. The k = ( 1
+2
+1
+2
+1
+2) magnetic order of the Ho3+
+spins is represented by the red arrows. The local spin orientations of
+the Ho3+ spins that are consistent with neutron diffraction are indi-
+cated for the Ho ion at (0,0,0). The magnetization, structural distor-
+tion, and inelastic neutron scattering however, provide clear evidence
+for easy [001] axis anisotropy.
+arXiv:2301.05141v1  [cond-mat.str-el]  12 Jan 2023
+
+2
+Here we study the magnetism of HoBi using modern neu-
+tron scattering techniques to gain insights into its unique
+magneto-transport properties29,34.
+Consistent with previ-
+ous works35–37, we confirm the antiferromagnetic (AFM)
+k
+=
+( 1
+2
+1
+2
+1
+2) structure and the associated tetragonal lattice
+distortion. Due to multi-domain averaging, our single-crystal
+neutron diffraction cannot unambiguously determine the local
+spin anisotropy of the k
+=
+( 1
+2
+1
+2
+1
+2) AFM structure. How-
+ever, we could resolve this ambiguity by measuring and mod-
+eling the magnetic excitations of HoBi, which take the form of
+weakly propagating spin-orbital excitons whose energies and
+intensities are sensitive to the local orientation of the Ho3+
+moments. Using this method, we found the k
+=
+( 1
+2
+1
+2
+1
+2)
+AFM structure has an Ising local spin anisotropy, which is
+consistent with the Ising easy-axis bulk magnetization and the
+tetragonal distortion. Through analysis of the paramagnetic
+diffuse scattering of HoBi and the crystal field excitons in the
+low T ordered state, we obtain a spin Hamiltonian with com-
+parable crystal field (CEF) and exchange energy scales.
+II.
+EXPERIMENTAL METHODS
+HoBi single crystals with mass of 10-50 mg were grown
+following a previously published procedure34. Single crys-
+tal low-temperature X-ray diffraction was performed using
+a Huber four-circle diffractometer with a Rigaku Rotaflex
+18 kW rotating copper anode X-ray generator and a Bicron
+point detector.
+We used a Ge (111) monochromator with
+d111 = 3.266 Å. The sample was aligned for diffraction in the
+(HHL) plane and mounted in a closed cycle cryogenic system
+with a base temperature of 2.17 K.
+We performed thermal neutron diffraction using the HB-1A
+triple-axis instrument at Oak Ridge National Laboratory. We
+used PG filtered 14.5 meV neutrons, and collected rocking
+scans at all accessible magnetic and nuclear Bragg positions
+in the (HHL) plane. Polarized neutron diffraction measure-
+ments were conducted with the triple-axis instrument BT-7
+at the Center for Neutron Research (NCNR), NIST. Nuclear
+spin-polarized 3He gas was used to polarize the incident neu-
+tron beam and to analyze the polarization of scattered neu-
+trons38,39. Horizontal guide fields were present throughout
+the beam path to allow measurements of the spin-flip (SF) and
+non-spin-flip (NSF) scattering cross-sections for incident neu-
+tron spins polarized parallel to momentum transfer Q. The
+flipping ratio measured at nuclear Bragg peaks was greater
+than 30.
+Cold neutron triple-axis experiments were performed using
+the SPINS and the MACS spectrometers at the NCNR. On
+both instruments we employed a fixed final neutron energy
+E f = 3.7 meV or 5 meV and measured the elastic and inelas-
+tic scattering for a single crystal of HoBi aligned for scattering
+within the (HHL) and the (HK0) plane in two different exper-
+iments. For the E f = 3.7 meV configuration, we used poly-
+crystalline cooled Be and BeO filters before and after the sam-
+ple, respectively. For the 5 meV configuration we only used
+a Be filter after the sample while the incident beam from the
+cold neutron source was unfiltered. For both experiments, we
+co-mounted 11 HoBi single crystals on an aluminum mount.
+We acquired background data using an identical mount with-
+out HoBi crystals. We used an ”orange” 4He flow cryostat to
+reach a base temperature of 1.6 K for these experiments.
+For the highest energy resolution and energy transfer, we
+performed time-of-flight neutron scattering experiments us-
+ing the CNCS spectrometer at Oak Ridge National Labora-
+tory. There we co-aligned two HoBi single crystals on an alu-
+minum mount and collected inelastic neutron scattering data
+with fixed incident energy Ei = 25 meV at T
+= 13 K with
+a total proton charge of 47 C. We used the high flux mode of
+operation of CNCS with a Fermi Chopper, Chopper 2, Chop-
+per 3, and a Double Disk frequency of 60, 60, 60, 300, and
+300 Hz respectively. The energy resolution (FWHM) at the
+elastic line for this configuration is 2.0(1) meV. Finally, we
+note that the error bars associated with the neutron scattering
+experiments represent one standard deviation.
+Both the magnetization and heat capacity measurements
+presented here were performed in a Quantum Design physical
+properties measurement system (PPMS). We used a PPMS di-
+lution refrigerator option for the low-temperature heat capac-
+ity.
+III.
+RESULTS AND ANALYSIS
+A.
+1st order phase transition
+FIG. 2. Low temperature heat capacity of HoBi collected using the
+long-pulse method. The red and blue curves respectively correspond
+to the warming and cooling protocol and shows a thermal hysteresis
+of 13(2) mK. The observation of a plateau at TN in the heating profile
+for both warming and cooling protocol (top inset panel) suggests a
+1st order phase transition in HoBi.
+The thermodynamic properties of HoBi were previously
+reported and a long-range k
+=
+( 1
+2
+1
+2
+1
+2) antiferromagnetic
+(AFM) order is known to occur concomitantly with a struc-
+tural distortion around TN
+=
+5.7 K14,35,36. The order of
+the transition, however, remains unknown. To determine the
+
+AT. = 13(2) mK
+HoBi
+5.6
+C
+K
+1000
+5.8
+Cp (J/mol K)
+6
+200
+400
+0
+Time (s)
+Warming
+500
+Cooling
+5.6
+5.7
+5.8
+5.9
+T(K)3
+order of the phase transition, we measured the temperature
+dependent specific heat capacity using the long-pulse heat
+method40.
+The resulting Cp data for HoBi is reported in
+Fig. 2 for both warming and cooling protocols.
+A sharp
+peak with a thermal hysteresis of 13(2) mK is observed in
+Cp. Correspondingly the inset shows a distinct plateau in the
+temperature versus time curves during heating and cooling.
+These observations indicate a 1st order phase transition at TN
+in HoBi.
+B.
+Paramagnetic phase
+To determine the magnetic interactions leading to this phase
+transition, we mapped the neutron elastic scattering for mo-
+mentum transfer Q covering the (HHL) plane and for temper-
+atures between 150 K and 1.6 K. Representative data sets are
+shown in Fig. 3.
+In
+the
+cubic
+paramagnetic
+phase
+for
+T
+=
+12 K
+>
+TN
+=
+5.72(1) K, the scattering is
+broad in Q and is centered at k = ( 1
+2
+1
+2
+1
+2) positions ((Fig. 3(a)).
+This indicates short-range AFM correlations preceding the
+long-range order.
+The ”butterfly” pattern of paramagnetic
+diffuse scattering is consistent with the equal time structure
+factor S(Q) of an fcc Heisenberg paramagnet with FM inter-
+actions between the first nearest-neighbor (n.n.) Ho3+ ions
+(J1), and AFM interactions between the 2nd n.n. (J2). Dashed
+lines in Fig. 1 indicate the lattice geometry associated with
+these interactions. The scattered intensity was modeled using
+I(Q) = 2
+3N| f(Q)|2 �
+ij⟨Si ·Sj⟩ cos(Q·rij) where N is the num-
+ber of spins, ri j is the displacement vector from Ho3+ site j to
+i, and f(Q) is the Ho3+ atomic form factor41. Including only
+self-correlations and correlations between spins separated by
+{100} and { 1
+2
+1
+20}, a ratio of ⟨Si ·Sj⟩{100}/⟨Si ·S j⟩{ 1
+2
+1
+2 0} = −2.2(2)
+was obtained at T = 12 K. The calculated magnetic diffuse
+scattering corresponding to the best fit shown in Fig. 3(d)
+accounts for all major features in the data (Fig. 3(a)) and the
+introduction of third n.n. correlations does not improve the
+fit significantly. A high temperature expansion allows us to
+associate the ratio of correlations to the ratio of the corre-
+sponding exchange interactions42,43 so that we may infer that
+J2/J1 ≈ −2.2(2). Even if some of the J1 bond interactions are
+frustrated, this resulting fitted ratio of exchange parameters
+stabilize a k = ( 1
+2
+1
+2
+1
+2) order, which is driven by the dominant
+AFM J2 interactions44–46.
+Upon cooling, the elastic magnetic scattering gets stronger
+(T = 5.5 K ≈ TN in Fig. 3(b)) and eventually forms mag-
+netic Bragg peaks (T = 1.6 K << TN in Fig. 3(c)) indi-
+cating long range magnetic order. To quantify the tempera-
+ture dependence of the diffuse and Bragg scattering, as shown
+in Fig. 3(e), we fitted the integrated intensity obtained from
+one-dimensional (HHH) scans acquired through the magnetic
+Bragg peak at Q = ( 1
+2
+1
+2
+1
+2). Each scan was fit to the sum of
+a Gaussian function and a Lorentzian function to describe the
+long and short range components of the spin correlations, and
+a linear background (needed to describe the temperature in-
+dependent nuclear and temperature-dependent magnetic inco-
+FIG. 3. The elastic diffuse neutron scattering from HoBi measured
+in the (HHL) reciprocal lattice plane at (a) 12 K, (b) 5.5 K, and (c)
+1.7 K with an incident neutrons energy of 3.7 meV . The scattering
+for panels (a,b,c) have been symmetrized to increase statistics. (d)
+Calculated paramagnetic diffuse scattering with J1/J2 = -2.17 on an
+fcc lattice where J1 is the first n.n. ferromagnetic interaction and J2
+is the 2nd n.n. antiferromagnetic interaction. Panel (e) is the elastic
+neutron scattering near the Q = ( 1
+2
+1
+2
+1
+2) Bragg peak acquired through
+scans along the the (HHH) direction. The data in panel (e) were fitted
+using a Lorentzian function for the diffuse scattering and a Gaussian
+function for the resolution limited Bragg component. The inferred
+integrated intensity for each component of the scattering are plotted
+in panel (f) as a function of temperature. The temperature depen-
+dence of the magnetic correlation length is plotted in the inset panel
+of (f).
+herent elastic scattering). The fits included as dashed curves
+in Fig. 3(e) provide a good account of the data.
+The temperature dependence of the integrated intensity of
+both the Bragg and the diffuse components of the scattering
+are reported in Fig. 3(f). The integrated intensity of the diffuse
+scattering (red markers) is peaked at TN where the appearance
+of Bragg scattering (blue markers) reveals the onset of long
+range order and translation symmetry breaking. The tempera-
+ture variation of the correlation length ξ, as inferred from the
+Lorentzian after correcting for resolution e���ects, is reported
+in the inset of Fig. 3(f). As expected, ξ increases dramatically
+at TN.
+
+do/dQ(b/sr/f.u.
+(a)
+(d)
+a.u.
+4
+0 1
+0
+4
+HoBi
+Calc.
+1.5
+1.5
+1
+1
+0.5
+0.5
+(T00)
+(T00)
+0
+0
+-0.5
+0.5
+-1
+-1.5
+-1.5
+12 K
+-0.5
+0
+0.5
+-1
+-0.5
+0
+0.5
+(b)
+(0HH)
+(HHO)
+(e
+●150K
+1.5
+·30K
+1
+10
+15 K
+10 K
+0.5
+● 6.7 K
+5.7 K
+[00
+0
+-0.5
+.6
+-1
+-1.5
+5.5 K
+-1
+-0.5
+0
+0.5
+1
+0.2
+0.4
+0.6
+0.8
+(c)
+(HHO)
+()
+(HHH)
+1.5
+1.5
+200
+6
+1
+wS 100
+(n'j/q)o
+0.5
+D
+4
+100)
+0
+0
+20
+ 40
+60.
+T(K)
+-0.5
+0.5
+2
+Elastic
+-1
+Diffuse
+-1.5
+1.7 K
+0
+-1
+-0.5
+0
+0.5
+1
+10
+100
+(HHO)
+T(K4
+C.
+Structural distortion
+A previous X-ray diffraction study revealed that a tetrago-
+nal distortion accompanies magnetic ordering in HoBi35. We
+confirmed the occurrence of this distortion in HoBi with a
+four-circle X-ray diffractometer experiment. The θ-2θ scans
+of various nuclear Bragg peaks were collected above and be-
+low TN with a base temperature of 5 K. Consistent with previ-
+ous work35, we observed a splitting of the (H00), (0K0), and
+(00L) nuclear Bragg peaks whereas the (HHH) Bragg peaks
+do not split. This indicates a tetragonal distortion and specifi-
+cally precludes a rhombohedral distortion.
+The temperature dependence of a longitudinal θ-2θ scan
+through the Q = (006) peak is plotted in Fig. 4(a). This is
+an unfiltered copper source with Kα1 and Kα2 radiation. Both
+components yield a split (006) peak below TN. The distortion
+was quantified by fitting the θ-2θ scans to Lorentzian func-
+tions while constraining the ratio of the Kα1 / Kα2t integrated
+intensity to be temperature independent and set by its fitted
+value obtained at high temperatures. Examples of these fits
+are included in Fig. 4(a). The temperature dependent lattice
+parameters inferred from this analysis are shown in Fig. 4(b).
+The order parameter-like temperature dependence is similar
+for both warming and cooling with no hysteresis detected
+down to the 100 mK temperature scale. For comparison the
+hysteresis detected through heat capacity measurements was
+13 mK (Fig. 2). A single (006) Bragg peak with a lattice pa-
+rameter of 6.2095(1) Å above TN, splits into two peaks with
+lattice parameters 6.2143(1) Å and 6.2075(1) Å below TN.
+Assuming an approximately volume conserving phase transi-
+tion implies that the lattice parameter that changes most is the
+c-axis. This indicates the structural unit cell elongates along
+the c-axis in the AFM state with c/a = 1.0011(1) at 5 K. We
+note that an orthorhombic distortion with the a and b axis dif-
+fering by less than 0.002 Å is not excluded by these data.
+A possible space group for HoBi below TN is the maximal
+tetragonal subgroup of the paramagnetic space group Fm3m,
+which is I4/mmm. The structural parameters in the tetragonal
+phase are aT = bT = 6.2075(1)/
+√
+2Å and cT = 6.2143(1) Å
+where the aT and bT axes are rotated by 45° relative to the
+a and b axes of the paramagnetic simple cubic cell. In this
+space group Ho3+ ions occupy a single 2a Wyckoff site and
+the magnetic ordering vector is k = ( 3
+20 3
+2). While we must
+use the tetragonal space group below TN, we continue to use
+the cubic unit cell to index wave vector transfer in the neu-
+tron scattering experiments, which do not resolve the multi-
+domain tetragonal distortion.
+D.
+Spin structure
+As described in the previous sections, the magnetic order
+has a characteristic wavevector k = ( 1
+2
+1
+2
+1
+2). In addition to the
+corresponding low T magnetic Bragg peaks, the intensities of
+all nuclear Bragg peaks are observed to increase below TN.
+The increase of intensity is approximately proportional to the
+intensity in the paramagnetic phase, which indicates it arises
+from secondary extinction release47. To check this hypothesis,
+FIG. 4. A series of θ-2θ X-ray diffraction scans through the Q = (006)
+Bragg peak. The inferred temperature dependence of the lattice pa-
+rameters is shown in panel (b). The neutron magnetic and nuclear
+refinement of HoBi are presented in (c) where the observed cross-
+sections for various Bragg peaks are plotted as a function of the cal-
+culated cross-sections. The inset in (c) reports the variation of the
+χ2 goodness of fit for the magnetic refinement of HoBi assuming a
+multi-domain k = ( 1
+2
+1
+2
+1
+2) spin structure with an easy axis defined
+by spherical coordinates θ and φ (φ = 0 corresponds to the [110] di-
+rection). Panel (d) shows the low-temperature magnetization versus
+field for fields applied parallel to the [001] and [110] directions. The
+data show that [001] is the easy axis.
+we performed polarized neutron diffraction on the (002) and
+(220) Bragg peaks below TN and found them to be exclusively
+nuclear in origin.
+We note that weak k = (001) Bragg peaks also onset at TN.
+Examples of these peaks include the (001) and (111) Bragg
+peaks (see Fig. 3(c)), which are forbidden within the Fm3m
+space group. These Bragg peaks are attributed to multiple
+magnetic scattering as their presence depends on both the em-
+ployed incident neutron wavelength and the scattering plane,
+and they are absent in powder neutron diffraction measure-
+ments36. The multiple scattering processes involve magnetic
+k = ( 1
+2
+1
+2
+1
+2) Bragg reflections so they occur only for T < TN.
+Referring to fcc close packing, the AFM k = ( 1
+2
+1
+2
+1
+2) spin
+structure can be described as an AFM stacking of FM trian-
+gular lattices. As the magnetic order and structural distortion
+in HoBi occur in a single 1st order phase transition, the direc-
+tion of the spins in each FM sheet is not constrained by the
+usual Landau argument for second order phase transitions. To
+determine the local spin orientation of the Ho3+ ions, we col-
+lected 18 rocking scans at different magnetic Bragg positions
+for a sample presumed to be in an unbiased multi-domain
+state. The data were compared to a cubic domain average
+of the calculated magnetic Bragg diffraction for a general spin
+orientation within one domain given by spherical angles θ, φ
+and k = ( 1
+2
+1
+2
+1
+2). Here θ = 0 corresponds to the tetragonal
+c-direction and θ = π/2 and φ = 0 corresponds to the [110]
+direction. Minimizing with respect to the moment size at each
+
+(a)
+(b)
+Kα
+Warming
+ 6K
+。 c (Tetragonal)
+HoBi
+6.214
+Cooling
+5.8 K
+600
+Q = (006)
+5.6 K
+(cts/s)
+. Par.
+6.212
+5 K
+400
+Ka2
+Latt.
+6.210
+a (Cubic)
+200
+6.208
+0
+a (Tetragonal)
+96
+96.4
+96.6
+5
+96.2
+5.5
+6
+6.5
+20
+T(K)
+(c)
+20
+(d)
+12
+●H[001]
+O H I [110]
+CCCCCCCCCCCCCCCCC
+15
+Magnetic
+Oobs(b/f.u.)
+Nuclear
+M(μB/Ho)
+8
+2
+X
+Xmin
+10
+180
+4
+90
+5
+45
+90
+0
+d
+0
+0
+5
+10
+15
+20
+0
+2
+4
+6
+Ocalc(b/f.u.)
+H(T)5
+point, the χ2 measure of fit quality is shown versus θ and φ
+in the inset panel of Fig. 4(c). The manifold of states rep-
+resented by the red arrows in Fig. 1 are indistinguishable by
+neutron diffraction. This degeneracy arises because the mag-
+netic diffraction intensity for a multi-domain sample only de-
+pends on the smallest angle between the spin and a ⟨111⟩ axis.
+From our refinement, we find this angle is 47(10)°. This is ex-
+perimentally indistinguishable from the angle between [001]
+and [111], which is 55°. This means the magnetic diffraction
+data are consistent with spins pointing along the [001] direc-
+tions, but also with many other directions including close to
+the [110] direction.
+Fortunately the spin anisotropy of the Ho3+ ions can be
+deduced from other pieces of information.
+First, the low-
+temperature magnetization of HoBi shown in Fig. 4(d) reveals
+the saturation magnetization is larger for fields along the [001]
+direction than along [110]. Second, the structural distortion
+also occurs along the [001] direction. Both of these measure-
+ments are consistent with spins oriented along the tetragonal
+cT-axis in the AFM ordered state. Additionally, in Sec. III F
+we show that a [001] easy axis anisotropy is needed to ac-
+curately model the inelastic neutron scattering spectrum be-
+low TN.
+We thus conclude the spins in the AFM type II
+order of HoBi are oriented along the cT direction, which is
+the direction of the structural elongation.
+The comparison
+between measured and calculated magnetic Bragg intensities
+is shown in Fig. 4(c). The corresponding spin structure is
+shown in Fig. 1. An ordered moment of 10.3(6) µB was de-
+termined, which is experimentally indistinguishable from the
+gJµB = 5
+4 · 8 µB = 10 µB saturation magnetization of Ho3+.
+E.
+Crystal electrical field interaction
+For Ho3+ ions, the J = 8 spin-orbit ground state manifold
+is (2J+1) = 17 fold degenerate under full rotation symmetry.
+This degeneracy is, however, lifted by the symmetry break-
+ing crystal electric fields (CEF). Using the Stevens operator
+formalism, the CEF Hamiltonian appropriate for Ho3+ in the
+high-temperature cubic phase of HoBi can be expressed as
+follows:
+ˆHcubic
+ce f
+= B4( ˆO0
+4 + 5 ˆO4
+4) + B6( ˆO0
+6 − 21 ˆO4
+6).
+(1)
+Here ˆOm
+n are Stevens operators48 that can be written in terms
+of the spin-orbital angular momentum operators ˆJ+, ˆJ− and
+ˆJz where ˆz ∥ c. The CEF parameters Bn are scalars of dimen-
+sion energy that dictate the strength of the different CEF terms
+and can be determined by fitting spectroscopic or thermo-
+magnetic data sensitive to the crystal field level scheme. Bn
+can also be estimated through the point-charge model49.
+Following Hutching’s formalism49 the point charge model
+yields
+B4 = 7|e||qBi|βJ⟨r4⟩
+64πϵ0d5
+Bi
+(2)
+and
+B6 = 3|e||qBi|γJ⟨r6⟩
+256πϵ0d7
+Bi
+.
+(3)
+Here e is the electron charge, qBi is the charge of the Bi ligand
+and ϵ0 is the vacuum permitivity. βJ and γJ are reduced matrix
+elements calculated in ref48 whereas the radial integrals for the
+4f state ⟨rn⟩ are tabulated in ref50. We used qBi =
+− 3e and
+the distance between a holmium ion and its first n.n. bismuth
+ion dBi = a/2 = 6.2093(1)/2 Å. Introducing these values in
+Eqs. 2 and 3 we obtain B4 = −2.2709(2) × 10−4 meV and
+B6 = −1.0468(1) × 10−7 meV.
+FIG. 5. Determination of the crystal electric field (CEF) level scheme
+for the J=8 Ho3+ ion in HoBi.
+(a) shows the results of a point
+charge (PC) calculation for the cubic and tetragonal phases. The cu-
+bic CEF scheme may be compared to the level scheme for the fitted
+CEF Hamiltonian of HoBi. Panel (b) and (c) respectively show the
+temperature dependence of the magnetic heat capacity (Cp) and the
+inverse magnetic susceptibility of HoBi compared to corresponding
+properties based on the fitted CEF Hamiltonian. The magnetic en-
+tropy obtained from integrating the Cp of HoBi is shown in the inset
+of (b). The measured (d) and calculated (e) inelastic neutron scatter-
+ing spectra of HoBi are shown for T = 12 K. The neutron inelastic
+scattering data were acquired using a 25 meV incident neutron beam.
+The corresponding CEF level scheme for Ho3+ in the cubic
+phase of HoBi is shown in Fig. 5(a). The Ho3+ J−multiplet
+is split into 4 triplets, 2 doublets, and 1 singlet that form three
+groups. Group I includes one doublet, one triplet, and one
+singlet between 0 and 0.2 meV. Group II is formed by two
+
+(a)HoBi
+P.C. cubic
+Fit cubic
+P.C. Tetragonal
+10
+888888888888888888 T
+D
+S
+8
+D
+D
+6
+4
+E
+2
+S
+D
+S
+0
+D
+D
+S
+(b)
+(c)
+60
+(J/mol/K)
+25
+R ln(17)
+20
+/emu)
+R ln(6)
+20
+15
+H=10 0e
+(J/mol/K)
+40
+10
+ (mol Oe/
+H II [001]
+mag
+S
+0
+10
+100
+10
+20
+T(K)
+%/ 1
+CEF fit
+CEF
+0
+0
+10
+100
+0
+100
+200
+300
+T(K)
+(e)
+T(K)
+(d)
+1
+12 K
+Data
+12 K
+Calc.
+12
+Ei=25 meV
+12
+hw (meV)
+I (a.u.)
+8
+8
+4
+0
+0
+0
+1
+3
+4
+2
+3
+4
+IQ(A)
+IQ(A)6
+triplets between 6 meV and 7 meV, and group III consists of a
+doublet and a triplet between 9 meV and 10 meV.
+The CEF Hamiltonian estimated from our point-charge cal-
+culation can reproduce the temperature dependence of the
+magnetic heat capacity Cp (Fig. 5(b)) and magnetic suscepti-
+bility χ (Fig. 5(c)). Obtained by integrating Cp/T, the temper-
+ature dependence of the entropy shown in the inset of Fig. 5(b)
+is informative. A first entropy plateau near 10 K is associ-
+ated with the sharp Cp anomaly at the phase transition to long
+range magnetic order. The corresponding change in entropy
+of ∆S = R ln 6 is that associated with the group I CEF states.
+The second plateau at S = R ln 17 is reached at room temper-
+ature and encompasses all of the entropy associated with the
+three groups of crystal field levels.
+For a more stringent test of the point charge model, we turn
+to inelastic neutron scattering. Fig. 5(d) shows the 12 K in-
+elastic neutron scattering spectrum with energy transfer rang-
+ing from 0 to 15 meV. At this temperature, the group II and
+III of CEF states are so scarcely populated that only CEF
+excitations originating from group I should be visible. No
+significant intrinsic broadening of the CEF excitations is ob-
+served and we note, also, that the experimental resolution is
+too coarse to resolve CEF levels within a group. The mag-
+netic neutron scattering cross section associated with CEF
+transition from group I to II and from group I to III can
+be computed based on the point charge CEF Hamiltonian
+(Imn ∝
+�
+i |⟨m|Ji|n⟩|2). This calculation predicts the cross
+section for transitions from group I to group II is 250 times
+stronger than for transitions from group I to group III. The in-
+tensity of the transition from I to III is thus predicted to be too
+weak to be detected. This explains why Fig. 5(d) shows just a
+single peak that we associate with transitions from group I to
+group II crystal field levels.
+While the measured 7.2 meV gap between group I and
+group II CEF levels is just 0.4 meV off from the point charge
+prediction of 6.8 meV, we can improve our estimate of the
+CEF Hamiltonian by simultaneously fitting B4 and B6 for
+the best possible account of the neutron scattering spectra
+(Fig. 5(d)), the specific heat data (Fig. 5(b)), and the mag-
+netic susceptibility data (Fig. 5(c)).
+The best fit parame-
+ters thus obtained are B4
+=
+− 2.24(1) × 10−4 meV and
+B6 = − 2.4(1) × 10−7 meV and with them the CEF Hamilto-
+nian provides an excellent account of all single ion properties
+that we’ve measured, as shown in Fig. 5.
+The CEF scheme obtained from our fit (Fig. 5(a)) is remark-
+ably similar to the point-charge calculation. Also a re-scaling
+of our CEF Hamiltonian for HoBi using Eq. 2 and Eq. 3 con-
+sidering only the different ligand spacing successfully predicts
+the level scheme for HoN ref51. This is in contrast with the
+praseodymium case where a pnictide ligand charge of q = −2e
+is needed to bring the point charge model into agreement with
+experimental data5. This indicates that holmium monopnic-
+tides are more ionic than praseodymium monopnictides.
+Finally, we estimated the effect of the tetragonal distortion
+on the CEF interaction in HoBi. We performed a point-charge
+calculation assuming that the first n.n. Ho-Bi bond is shorter
+along the a and b direction (da) as compared to the c direction
+(dc). The calculated CEF Hamiltonian can be written as:
+Htet
+ce f = |e||qBi|
+4πϵ0
+[αJ⟨r2⟩( 1
+d3c
+− 1
+d3a
+) ˆO0
+2+
+(4)
+βJ⟨r4⟩(( 1
+4d5c
++
+3
+16d5a
+) ˆO0
+4 +
+35
+16d5a
+ˆO4
+4)+
+γJ⟨r6⟩(( 1
+8d7c
+−
+5
+64d7a
+) ˆO0
+6 −
+63
+64d7a
+ˆO4
+6)].
+The corresponding level scheme is shown in Fig. 5(a). For
+this calculation, we used the lattice parameters determined
+in our high-resolution X-ray scattering experiment. The de-
+generacy of all the triplets and doublets associated with cubic
+symmetry is lifted. This results in four doublets and nine sin-
+glets and a significant broadening of each of the three groups
+of crystal field levels.
+F.
+Low energy spin dynamics
+We now turn our attention to the collective physics of HoBi,
+which we explore using inelastic magnetic neutron scatter-
+ing. Fig. 6(a) shows the temperature dependence of the in-
+elastic scattering for Q = ( 1
+2
+1
+2
+1
+2). Just above TN, the scat-
+tering is quasi-elastic with a physical (resolution corrected)
+FWHM of 0.30(5) meV. No inelastic intensity is observed up
+to 2 meV. This is consistent with the CEF energy scheme
+shown in Fig. 5(c). Below TN, the quasi-elastic scattering
+splits into an elastic and an inelastic component.
+To probe any dispersion of the low energy spin excitations,
+we acquired low energy spectra at momentum transfer Q cor-
+responding to high symmetry points in the Brillouin zone.
+Fig. 6(b) shows the spectrum consists of a peak that is broader
+than the experimental resolution (FWHM indicated by hor-
+izontal bar) and that shifts by less than the peak width be-
+tween the different values of Q. A gaussian fit finds the peak
+centered at 1.7(2) meV with a FWHM of 0.48(4) meV that
+exceeds the instrumental resolution (FWHM of 0.22 meV).
+The limited resolution and statistical accuracy of the data does
+not rule out the possibility of multiple dispersive components
+within the approximately Gaussian envelope of the peak.
+We also examined the higher energy excitations for T < TN
+by acquiring momentum resolved inelastic scattering data up
+to 11.5 meV. A representative slice through the data is dis-
+played as a color image versus Q along the (HH0) direction
+and energy transfer in Fig. 6(c). No dispersion is resolved.
+The data are similar to the high-temperature plot of intensity
+versus |Q| and ℏω in Fig. 5(d) though with additional inelastic
+features at 9.0(3) meV and 1.7(2) meV.
+Fig. 6(e) shows the momentum dependence of the inte-
+grated intensity of the 1.7 meV mode throughout the (HHL)
+zone.
+The Q dependence of the intensity is subtle albeit
+peaked at the magnetic ( 1
+2
+1
+2
+1
+2) zone center and smoothly de-
+creases with |Q| in accordance with the Ho3+ magnetic form
+factor41. We note that the 1.7 meV gap is about an order of
+magnitude greater than the predicted CEF gap arising from
+the tetragonal distortion. This indicates the phase transition is
+driven by the magnetic interactions, which we model below.
+
+7
+FIG. 6. The temperature dependence of the low energy inelastic neu-
+tron spectrum of HoBi at Q = ( 1
+2
+1
+2
+1
+2) is shown in panel (a). The spec-
+trum of neutron scattering at some high symmetry positions within
+the first Brillouin zone of HoBi are shown in (b).The horizontal
+black bar indicates the FWHM energy resolution of the spectrom-
+eter while the black dashed lines show the predicted spectrum based
+on the spin Hamiltonian presented in this work. The energies asso-
+ciated with each exciton are indicated by vertical black dashed lines.
+The observed and calculated inelastic neutron scattering spectrum
+up to 11.5 meV are respectively plotted in (c) and (d) for momentum
+transfer Q along the [HH0] direction. The observed and calculated
+momentum dependence of the 1.75 meV exciton scattering inten-
+sity is shown in (e) and (f). The energy integration for panel (e) is
+±0.25 meV.
+IV.
+MODELING SPIN DYNAMICS OF SPIN-ORBITAL
+EXCITONS
+The low-temperature excitations in HoBi are similar to
+other rare-earth metallic compounds where exchange interac-
+tions are strong enough to mix crystal field levels4,52,53. Be-
+cause components that are longitudinal with respect to the or-
+dered moment are involved, these are not conventional trans-
+verse spin wave excitations. They may be described as crystal
+field excitations that can propagate through the lattice due to
+inter-site interactions. We shall adopt the practice of calling
+these “crystal field exciton” or simply “exciton”54–56.
+A common theoretical approach to describing excitons in
+rare-earth magnets is to use a pseudo-boson theory where the
+exciton creation operator is a linear combination of single-ion
+operators53,57,58. In this theory, the Q = 0 single-ion opera-
+tors are obtained by diagonalizing the mean-field spin Hamil-
+tonian and the dispersion at finite Q is produced by the ex-
+change terms. We use this pseudo-boson theory to describe
+the magnetic excitation spectrum of HoBi below TN.
+The
+Hamiltonian Hs includes the single-ion tetragonal crystal field
+terms and isotropic exchange interactions. Hs is decomposed
+into a mean-field term (H0,k) and an interacting part (Hint) so
+Hs = �
+k H0,k + Hint where:
+H0,k = Htet
+ce f,k + (−1)kHzJk
+jz
+(5)
+and
+Hint =
+�
+j, j′,k,k′
+Jk,k′
+j, j′ Jk
+j · Jk′
+j′ −
+�
+j,k
+(−1)kHzJk
+jz.
+(6)
+Here j indexes the unit cell while k = 1, 2 specifies the
+anti-parallel sub-lattices of the AFM order (Fig. 1). We define
+Hz = 2 �
+r ZrJr⟨Jz⟩ where Jr and Zr are respectively the ex-
+change constant and coordination number associated with the
+rth neighbor. ⟨Jz⟩ is the thermal average of Jz on each site,
+which we found to be ⟨Jz⟩ = 8 in our diffraction and CEF
+analysis. By definition, Hint carries no mean value and so can
+be written in terms of creation (ˆa†
+n,k = |n, k⟩⟨0, k|) and annihila-
+tion (ˆan,k = |0, k⟩⟨n, k|) operators that connect the ground state
+|0, k⟩ and the excited eigenstates |n, k⟩ of ˆH0,k. In this case,
+ˆH0,k = �
+n Enˆa†
+n,kˆan,k where En,k is the eigenvalue of the |n, k⟩
+eigenstate of ˆHo,k. After writing ˆHs in terms of these operators
+and Fourier transforming it, we obtain:
+ˆHs = 1
+2
+�
+Q
+�
+ˆa†(Q)A(Q)ˆa(Q) + ˆa†(−Q)A(−Q)ˆa(−Q)
+(7)
++ˆa†(Q)B(Q)ˆa†(−Q) + ˆa(−Q)B(Q)ˆa(−Q)
+�
+with
+ˆA =
+ˆ∆ + 2ˆhzz + ˆh+− + ˆh−+ and
+ˆB = 2ˆhzz
++
+ˆh++
++
+ˆh−−
+where
+ˆ∆
+=
+En,kδk,k′δn,n′
+and
+ˆhαβ(k, k′, n, n′, Q) = J(Q)⟨k, n| ˆJα|0, k⟩⟨k′, 0| ˆJβ|n′, k′⟩.
+The procedure to compute the spin dynamics first consist of
+diagonalizing ˆH0,k to obtain the eigenvalues En,k and eigenvec-
+tors |n, k⟩ for Q = 0. At finite Q, the matrix ˆHs =
+� ˆA
+ˆB
+− ˆB − ˆA
+�
+is
+then computed and diagonalized to obtain the perturbed ener-
+gies (E˜n(Q)) and eigenstates |˜n(Q)⟩ for each exciton. We con-
+sider all the excited CEF states belonging to the (2J+1) spin-
+orbit manifold of HoBi so there are 32 creation and annihla-
+tion operators for each of the 2 Ho3+ spins within the magnetic
+unit cell. This give a Hilbert space of 64 states for ˆHs. The
+associated inelastic magnetic neutron scattering cross-section
+for a single magnetic domain is then53,57:
+d2σ
+dEdΩ = N(γr0)2 k f
+ki
+|g
+2 f(Q)|2
+(8)
+×
+�
+˜n,q,τm
+|⟨˜n(q)| ˆJQ|GS ⟩|2δ(E − E˜n(q))∆(Q − q − τm)
+Here N is the number of primitive magnetic unit cells, γ = -
+1.91 is the gyromagnetic ratio of the neutron, r0 = 2.818 ×
+
+(a)
+I (a.u.)
+0
+3
+HoBi
+AE
+(Aaw) m
+100
+7
+(a.u.)
+50
+0
+0
+2
+4
+6
+8
+10
+12
+1.5
+2
+2.5
+T(K)
+hw (meV)
+(c)
+(d)
+4
+Data
+Calc.
+10
+10
+(meV)
+8
+8
+(a.u.)
+6
+6
+hw
+4
+2
+2
+1.7 K
+0
+0
+0
+2
+3
+0
+2
+3
+0
+[HH0]
+[HHO]
+(f)
+(e)
+5
+1.75 meV
+1.75 meV
+Data
+Calc.
+2
+2
+1.7 K
+[00L]
+L
+1001
+(a.u.)
+I
+L
+UX
+U
+X
+0
+0
+0
+0.5
+1
+1.5
+0
+0.5
+1
+1.5
+2
+[HH0]
+[HH0]8
+10−15 m is the classical electron radius, τm is the magnetic
+zone center, q is the reduced momentum transfer within the
+first magnetic Brillouin zone, while k f and ki respectively are
+the scattered and incoming neutron wave vector. The mea-
+sured spectrum is subject to the finite resolution of the instru-
+ment which we account for by replacing the delta functions by
+a united normalized Gaussian functions with the Q-integrated
+energy resolution width. The final calculated spectrum was
+averaged over all possible magnetic domains.
+V.
+MICROSCOPIC SPIN HAMILTONIAN FOR HOLMIUM
+BISMUTH
+We determined the microscopic parameters of ˆHs for HoBi
+by fitting the Q = 0 spectrum consisting of three excitons
+at E1 = 1.7(2) meV, E2 = 7.4(2) meV and E3 = 9.0(3) meV
+with relative intensities I2/I1 = 5.5(3) and I2/I3 = 37(7). Em-
+ploying the ratio ∥J2/J1∥ = 2.17 obtained by analyzing the
+magnetic diffuse scattering (section III B) leaves just one free
+parameter. The tetragonal CEF Hamiltonian has six free pa-
+rameters that were initially estimated from the point-charge
+model. To reproduce the exact energies of the excitons at E2
+and E3, we allowed the CEF parameters to relax away from
+their point-charge values which results in many combinations
+of parameters consistent with the data. We estimated the ex-
+change constants by varying the CEF parameters away from
+their point-charge calculation values and keeping all solutions
+that have a χ2 within 20% (1/Nobs) of the global minimum.
+The exchange parameters refined to J1 = − 1.4(2) µeV and
+J2
+=
+3.0(5) µeV.
+A mean-field critical temperature of
+20(7) K is obtained from these parameters. For comparison,
+the actual ordering temperature is only TN
+=
+5.72(1) K.
+We hypothesize that fluctuations arising from competition be-
+tween the ferromagnetic J1 and the antiferromagnetic J2 in-
+teractions lead to the reduced critical temperature.
+The right column of Fig. 6 compares the optimized model
+for a multi-domain sample to the experimental data. Fig. 6(d)
+shows the full intensity versus ℏω and Q ∥ (HH0) for com-
+parison with Fig. 6(c). The position and relative intensity of
+the three modes are well reproduced. Looking more closely
+at the 1.75 meV mode, Fig. 6(b) compares the intensity ver-
+sus energy transfer at select high symmetry points in the Bril-
+louin zone. The vertical dashed lines show that multiple ex-
+citons contribute at each Q. This is generally consistent with
+the featured spectrum observed though there is more broad-
+ening/dispersion observed than reproduced by the model. In-
+clusion of anisotropic or longer range interactions might be
+needed to remedy this discrepancy though data with higher
+energy resolution is needed to justify the greater model com-
+plexity. Fig. 6(f) shows the calculated Q-dependent integrated
+intensity of the 1.75 meV mode. The dominant features of
+the experimental result in Fig. 6(e) are reproduced, includ-
+ing mainly the increase of scattered intensity at the magnetic
+zone centers. We note the presence of phonon scattering near
+Q
+=
+(002) that may account for the discrepancy between
+the calculation and the experimental data at that momentum
+point.
+VI.
+DISCUSSION AND CONCLUSION
+In this manuscript, we have characterized an antiferro-
+magnetic order and the associated crystal field excitons
+that develop below TN
+=
+5.72(1) K in the rare-earth
+monopnictide HoBi. This magnetic state is driven by strong
+2nd n.n. antiferromagnetic and weaker 1st n.n. ferromag-
+netic interactions, which we quantified via modeling of the
+diffuse paramagnetic and low temperature inelastic neutron
+scattering. The excitation spectrum is sensitive to the local
+orientation of the Ho3+ ordered spins, which allowed us
+to establish the Ising nature of the antiferromagnetic order
+in HoBi that cannot be deduced from neutron diffraction
+of a multi-domain sample.
+We used X-ray diffraction to
+provide evidence for a tetragonal structural distortion that
+accompanies magnetic ordering.
+Our CEF analysis and
+modelling of inelastic scattering data indicates the elongated
+c-axis coincides with the easy magnetic axis within a domain.
+The magnetic excitations that we have documented here
+surely have significant impacts on the magneto-transport
+properties of HoBi34. For example, we found strong quasi-
+elastic neutron scattering in the paramagnetic state.
+The
+associated short range correlated spin fluctuations, which
+may be accompanied by short range tetragonal lattice dis-
+tortions too given the non-Kramers nature of the Ho3+, are
+expected to enhance the electrical resistivity above TN. Below
+TN, these gapless fluctuations are replaced by a coherent
+exciton at 1.7(2) meV and correspondingly the electrical
+resistivity is reduced by an order of magnitude upon cooling
+below TN34. The field-dependence of spin-orbital excitons
+may be responsible for various features observed in the
+magnetoresistance of HoBi and more broadly in the rare-earth
+monopnictides23–29.
+VII.
+ACKNOWLEDGEMENTS
+This work was supported as part of the Institute for Quan-
+tum Matter, an Energy Frontier Research Center funded by the
+U.S. Department of Energy, Office of Science, Basic Energy
+Sciences Under Award No.DE-SC0019331. CB was further
+supported by the Gordon and Betty Moore foundation EPIQS
+program under GBMF9456. The work at Boston College was
+supported by the U.S. Department of Energy, Office of Basic
+Energy Sciences, Division of Physical Behavior of Materials
+under Award DE-SC0023124. This work was supported in
+part by the Natural Sciences and Engineering Research Coun-
+cil of Canada (NSERC). We acknowledge the support of the
+National Institute of Standards and Technology, U.S. Depart-
+ment of Commerce. Access to MACS was provided by the
+Center for High Resolution Neutron Scattering, a partnership
+between the National Institute of Standards and Technology
+and the National Science Foundation under Agreement No.
+DMR-1508249. The identification of any commercial prod-
+uct or trade name does not imply endorsement or recommen-
+dation by the National Institute of Standards and Technology.
+
+9
+A portion of this research used resources at the High Flux Iso-
+tope Reactor, a DOE Office of Science User Facility operated
+by the Oak Ridge National Laboratory.
+∗ Correspondence email address: Jonathan.Gaudet@nist.gov
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+

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@@ -0,0 +1,1307 @@
+arXiv:2301.12945v1  [math.CO]  30 Jan 2023
+CONTINUED FRACTIONS FOR PARTITION GENERATING
+FUNCTIONS
+GEOFFREY B CAMPBELL
+Dedicated to Professor Rodney J Baxter on his 83rd birthday.
+Abstract. We derive continued fractions for partition generating functions, uti-
+lizing both Euler’s techniques and Ramanujan’s techniques. Although our results
+are for integer partitions there is scope to extend this work to vector partitions,
+including for binary and n-ary partitions.
+1. Euler’s Continued Fraction
+Almost 290 years ago in 1737, Leonhard Euler wrote De fractionibus continuis dis-
+sertatio, which gave mathematics a first ever comprehensive account of the properties
+of continued fractions, and included the first proof that the number e is irrational.
+(See Sandifer [50]) Later, but still 275 years ago in 1748, Euler, in his Introductio in
+analysin infinitorum Vol. I, Chapter 18 [28], proved:
+(a) the equivalence of his continued fraction to a generalized infinite series,
+(b) every rational number can be written as a finite continued fraction, and
+(c) the continued fraction of an irrational number is infinite.
+2010 Mathematics Subject Classification. Primary: 11J70; Secondary: 05A15, 05E40, 11Y11,
+11P21.
+Key words and phrases. Continued fractions and generalizations. Exact enumeration problems,
+generating functions.
+Partitions of integers.
+Elementary theory of partitions.
+Combinatorial
+identities, bijective combinatorics. Lattice points in specified regions.
+Thanks are due to Professor Dr Henk Koppelaar, whose discussions and suggestions have been
+very helpful for the book for which this paper is essentially a chapter.
+1
+
+2
+GEOFFREY B CAMPBELL
+Euler’s continued fraction is the very nice identity, whose first few cases are:
+a0 + a0a1
+=
+a0/(1 − a1/(1 + a1))
+=
+a0
+1 −
+a1
+1 + a1
+;
+a0 + a0a1 + a0a1a2
+=
+a0/(1 − a1/(1 + a1 − a2/(1 + a2)))
+=
+a0
+1 −
+a1
+1 + a1 −
+a2
+1 + a2
+;
+a0 + a0a1 + a0a1a2 + a0a1a2a3
+=
+a0/(1 − a1/(1 + a1 − a2/(1 + a2 − a3/(1 + a3))))
+=
+a0
+1 −
+a1
+1 + a1 −
+a2
+1 + a2 −
+a3
+1 + a3
+.
+Hence, we can state Euler’s Continued Fraction in the following
+Theorem 1.1. If a0, a1, a3, ... an are defined functions such that no denominator
+is zero in the following equations then
+(1.1)
+n
+�
+k=0
+k
+�
+j=0
+aj = a0 + a0a1 + a0a1a2 + ... + a0a1...an
+= a0/(1 − a1/(1 + a1 − a2/(1 + a2 − a3/(1 + ... an−1/(1 + an−1 − an/(1 + an))))).
+=
+a0
+1 −
+a1
+1 + a1 −
+a2
+1 + a2 −
+a3
+1 + a3 −
+...
+...
+an−1
+1 + an−1 −
+an
+1 + an
+.
+Obviously, this lends itself to many of the elementary series that arise in school
+and university analysis.
+However, we shall put this to good use in applying it
+to partition generating functions. The fact of this theorem involving a finite sum
+allows us to incrementally extend the number of terms until we can infer the infinite
+versions of the theorem.
+Example 1: The exponential function is
+(1.2) exp(z) = 1 + z
+1! + z2
+2! + z3
+3! + ... = 1 +
+�z
+1
+�
++
+�z
+1
+� �z
+2
+�
++
+�z
+1
+� �z
+2
+� �z
+3
+�
++ ...
+= 1/
+�
+1 − z/
+�
+1 + z −
+�z
+2
+�
+/
+�
+1 +
+�z
+2
+�
+−
+�z
+3
+�
+/
+�
+1 +
+�z
+3
+�
+−
+�z
+4
+�
+/
+�
+1 +
+�z
+4
+�
+− ...
+�����
+.
+
+CONTINUED FRACTION PARTITION IDENTITIES
+3
+Applying an “equivalence transformation” that consists of clearing the fractions,
+this example is simplified to
+exp(z) = 1/(1 − z/(1 + z − z/(2 + z − 2z/(3 + z − 3z/(4 + z − . . .))))),
+or the equivalent statement
+exp(z) =
+1
+1 −
+z
+1 + z −
+z
+2 + z −
+2z
+3 + z −
+3z
+4 + z − . . .
+and we know this continued fraction converges uniformly on every bounded domain
+in the complex plane because it is equivalent to the power series for exp(z).
+Example 2: There is the well-known logarithmic function series
+(1.3)
+log
+�1 + z
+1 − z
+�
+= 2z(1
+1 + z2
+3 + z4
+5 + ...)
+= 2z(1 + (z2
+3 ) + (z2
+3 )(3z2
+5 ) + (z2
+3 )(3z2
+5 )(5z2
+7 ) + ...).
+Applying Euler’s continued fraction formula to this expression shows that:
+log
+�1 + z
+1 − z
+�
+= 2z/(1−(z2
+3 )/(1+(z2
+3 )−(3z2
+5 )/(1+(3z2
+5 )−(5z2
+7 )/(1+(5z2
+7 )−(7z2
+9 )/(1+(7z2
+9 )−...))))).
+Applying the “equivalence transformation” this example is simplified to
+log
+�1 + z
+1 − z
+�
+= 2z/(1−z2/(z2+3−(3z)2/(3z2+5−(5z)2/(5z2+7−(7z)2/(7z2+9−...)))))
+=
+2z
+1 −
+z2
+z2 + 3 −
+(3z)2
+3z2 + 5 −
+(5z)2
+5z2 + 7 −
+(7z)2
+7z2 + 9 − . . .
+Example 3: A continued fraction for π. We can use the previous example involving
+the principal branch of the natural logarithm function to construct a continued
+fraction representation of π. First we note that
+(i + 1)/(i − 1) = i,
+so
+then
+log((i + 1)/(i − 1)) = iπ/2.
+
+4
+GEOFFREY B CAMPBELL
+Setting z = i in the previous result, and remembering that i2 = −1, we obtain
+immediately
+π =
+4
+1 +
+12
+2 +
+32
+2 +
+52
+2 +
+72
+2 + . . .
+2. Euler’s continued fraction applied to partitions
+In this section we will technically do no more than apply the previous section.
+However, the theory of partitions is full of generating functions that are emenable to
+the Euler continued fraction. In a subsequent section we will examine Ramanujan
+type continued fractions, but firstly we will gather some ”low hanging fruit” from
+some elementary series-product identities.
+We begin with the well-known telescoping identities:
+If a1, a2, a3, ... , an, are functions chosen for nonzero denominators, then
+(2.1)
+1 +
+a1
+1 − a1
++
+a2
+(1 − a1)(1 − a2) + ... +
+an
+(1 − a1)(1 − a2)...(1 − an)
+=
+1
+(1 − a1)(1 − a2)(1 − a3)...(1 − an);
+and
+(2.2) 1 + a1 + a2(1 + a1) + a3(1 + a1)(1 + a2) + ... + an(1 + a1)(1 + a2)...(1 + an−1)
+= (1 + a1)(1 + a2)(1 + a3)...(1 + an).
+The series in (2.1) and (2.2) are already close to being in the required form to
+apply the Euler continued fraction since
+(2.3)
+1 +
+a1
+1 − a1
++
+a2
+(1 − a1)(1 − a2) + ... +
+an
+(1 − a1)(1 − a2)...(1 − an)
+= 1 +
+a1
+1 − a1
++
+a1
+1 − a1
+a2(1 − a1)
+a1(1 − a2) + ... +
+a1
+1 − a1
+a2(1 − a1)
+a1(1 − a2)...an(1 − an−1)
+an−1(1 − an);
+and
+(2.4) 1 + a1 + a2(1 + a1) + a3(1 + a1)(1 + a2) + ... + an(1 + a1)(1 + a2)...(1 + an−1)
+= 1+a1+a1
+a2(1 + a1)
+a1
++a1
+a2(1 + a1)
+a1
+a3(1 + a2)
+a2
++...+a1
+a2(1 + a1)
+a1
+a3(1 + a2)
+a2
+...an(1 + an−1)
+an−1
+.
+Hence combining (2.1) with (2.3) and then (2.2) with (2.4) respectively, we obtain
+(2.5)
+1
+(1 − a1)(1 − a2)(1 − a3)...(1 − an)
+
+CONTINUED FRACTION PARTITION IDENTITIES
+5
+=
+1
+1 −
+a1
+1−a1
+1 +
+a1
+1−a1 −
+a2(1−a1)
+a1(1−a2)
+1 + a2(1−a1)
+a1(1−a2) −
+a3(1−a2)
+a2(1−a3)
+1 + a3(1−a2)
+a2(1−a3) −
+...
+...
+an−1(1−an−2)
+an−2(1−an−1)
+1 + an−1(1−an−2)
+an−2(1−an−1) −
+an(1−an−1)
+an−1(1−an)
+1 + an(1−an−1)
+an−1(1−an)
+;
+and
+(2.6)
+(1 + a1)(1 + a2)(1 + a3)...(1 + an)
+=
+1
+1 −
+a1
+1 + a1 −
+a2(1+a1)
+a1
+1 + a2(1+a1)
+a1
+−
+a3(1+a2)
+a2
+1 + a3(1+a2)
+a2
+−
+...
+...
+an−1(1+an−2)
+an−2
+1 + an−1(1+an−2)
+an−2
+−
+an(1+an−1)
+an−1
+1 + an(1+an−1)
+an−1
+.
+After applying the “equivalence transformation” to both of (2.5) and then (2.6)
+to eliminate denominator terms, each continued fraction is simplified giving us the
+following two theorems.
+Theorem 2.1. If a1, a2, a3, ... , an, are functions chosen for nonzero denominators,
+then
+(2.7)
+1
+(1 − a1)(1 − a2)(1 − a3)...(1 − an)
+=
+1
+1 −
+a1
+1 −
+a2
+a1 + a2 − 2a1a2 −
+a1a3
+a2 + a3 − 2a2a3 −
+...
+...
+an−2an
+an−1 + an − 2an−1an
+.
+At first glance we can see this theorem as being applicable to generating functions
+for unrestricted partitions of various kinds. Similarly the next theorem applies for
+partitions of various sorts into distinct parts.
+
+6
+GEOFFREY B CAMPBELL
+Theorem 2.2. If a1, a2, a3, ... , an, are functions chosen for nonzero denominators,
+then
+(2.8)
+(1 + a1)(1 + a2)(1 + a3)...(1 + an)
+=
+1
+1 −
+a1
+1 + a1 −
+(1 + a1)a2
+a1 + a2 + a1a2 −
+(1 + a2)a3
+a2 + a3 + a2a3 −
+...
+...
+(1 + an−1)an
+an−1 + an + an−1an
+.
+There are many examples we could choose for substitution into theorems 2.2 and
+2.2. So, let’s start with the generating functions for unrestricted partitions, and for
+distinct partitions as follows.
+Corollary 2.1. If pn(k), is the number of unrestricted partitions of k into integers
+no greater than n, then
+(2.9)
+1
+(1 − q1)(1 − q2)(1 − q3)...(1 − qn) =
+∞
+�
+k=0
+pn(k)qk
+=
+1
+1 −
+q1
+1 −
+q2
+q1 + q2 − 2q1q2 −
+q1q3
+q2 + q3 − 2q2q3 −
+...
+...
+qn−2qn
+qn−1 + qn − 2qn−1qn
+.
+Corollary 2.2. If pn(D, k), is the number of distinct partitions of k into integers
+no greater than n, then
+(2.10)
+(1 + q1)(1 + q2)(1 + q3)...(1 + qn) =
+∞
+�
+k=0
+pn(D, k)qk
+=
+1
+1 −
+q1
+1 + q1 −
+(1 + q1)q2
+q1 + q2 + q1q2 −
+(1 + q2)q3
+q2 + q3 + q2q3 −
+...
+...
+(1 + qn−1)qn
+qn−1 + qn + qn−1qn
+.
+Next we choose the odd integer powers substituted into the two theorems.
+
+CONTINUED FRACTION PARTITION IDENTITIES
+7
+Corollary 2.3. If pn(O, k), is the number of unrestricted partitions of k into odd
+integers no greater than 2n − 1, then
+(2.11)
+1
+(1 − q1)(1 − q3)(1 − q5)...(1 − q2n−1) =
+∞
+�
+k=0
+pn(O, k)qk
+=
+1
+1 −
+q1
+1 −
+q3
+q1 + q3 − 2q1q3 −
+q1q5
+q3 + q5 − 2q3q5 −
+...
+...
+qn−2qn
+q2n−3 + q2n−1 − 2q2n−3q2n−1
+.
+Corollary 2.4. If pn(DO, k), is the number of distinct partitions of k into odd
+integers no greater than 2n − 1, then
+(2.12)
+(1 + q1)(1 + q3)(1 + q5)...(1 + q2n−1) =
+∞
+�
+k=0
+pn(DO, k)qk
+=
+1
+1 −
+q1
+1 + q1 −
+(1 + q1)q3
+q1 + q3 + q1q3 −
+(1 + q3)q5
+q3 + q5 + q3q5 −
+...
+...
+(1 + q2n−3)q2n−1
+q2n−3 + q2n−1 + q2n−3q2n−1
+.
+It is a well-known result due to Euler that p∞(DO, k) = p∞(O, k). Explicitly, as
+n → ∞ equations (2.12) and (2.11) are equal to each other.
+Next, let us give the cases covering binary partitions.
+Corollary 2.5. If bn(2, k), is the number of unrestricted binary partitions of k into
+non-negative powers of two no greater than 2n, then
+(2.13)
+1
+(1 − q1)(1 − q2)(1 − q4)...(1 − q2n) =
+∞
+�
+k=0
+bn(2, k)qk
+=
+1
+1 −
+q1
+1 −
+q2
+q1 + q2 − 2q1q2 −
+q1q4
+q2 + q4 − 2q2q4 −
+...
+...
+q2n−2q2n
+q2n−1 + q2n − 2q2n−1q2n
+.
+The following distinct binary partitions example is completely solvable.
+
+8
+GEOFFREY B CAMPBELL
+Corollary 2.6. If pn(2D, k), is the number of binary partitions of k into distinct
+non-negative powers of two no greater than 2n, then
+(2.14)
+(1 + q1)(1 + q2)(1 + q4)...(1 + q2n) = 1 − q2n+1
+1 − q
+=
+2n+1−1
+�
+k=0
+pn(2D, k)qk
+=
+1
+1 −
+q1
+1 + q1 −
+(1 + q1)q2
+q1 + q2 + q1q2 −
+(1 + q2)q4
+q2 + q4 + q2q4 −
+...
+...
+(1 + q2n−1)q2n
+q2n−1 + q2n + q2n−1q2n
+.
+Note that from (2.14) we have directly that
+pn(2D, k) =
+�
+1,
+when 0 ≤ k < 2n+1;
+0,
+when k ≥ 2n+1.
+The following distinct ternary partitions example is easily stated.
+Corollary 2.7. If pn(3D, k), is the number of ternary partitions of k into distinct
+non-negative powers of three no greater than 3n, then
+(2.15)
+(1 + q1)(1 + q3)(1 + q9)...(1 + q3n) =
+3n−1
+�
+k=0
+pn(3D, k)qk
+=
+1
+1 −
+q1
+1 + q1 −
+(1 + q1)q3
+q1 + q3 + q1q3 −
+(1 + q3)q9
+q3 + q9 + q3q9 −
+...
+...
+(1 + q3n−1)q3n
+q3n−1 + q3n + q3n−1q3n
+.
+Note that from (2.15) we have directly that
+pn(3D, k) =
+
+
+
+1,
+for 0 ≤ k < 3n+1; k is a sum of distinct powers of 3.
+0,
+for 0 ≤ k < 3n+1; k not a sum of distinct powers of 3.
+0,
+for k ≥ 3n+1.
+Clearly this topic of Euler Continued Fractions applied to partition generating
+functions is an interesting elementary study for students, and a possible tool for
+researchers. The above results are old, and have probably been well-worked over
+time.
+
+CONTINUED FRACTION PARTITION IDENTITIES
+9
+3. Rogers-Ramanujan Continued Fractions for partition functions
+The fraction given here was mentioned by Ramanujan in his second letter to
+Hardy (see Adiga et al. [2, p. xxviii]); namely
+(3.1)
+R(a, b) = 1 +
+bq
+1 + aq +
+bq2
+1 + aq2 +
+bq3
+1 + aq3 + bq4
+...
+.
+However, these now famous continued fractions, as with the Rogers-Ramanujan
+identities, were first discovered in 1894 by Rogers (see [49]). We define the functions
+G(q) and H(q) in the context of the Rogers–Ramanujan identities,
+(3.2)
+G(q) =
+∞
+�
+n=0
+qn2
+(1 − q)(1 − q2) · · · (1 − qn) =
+∞
+�
+n=0
+qn2
+(q : q)n
+=
+1
+(q; q5)(q4; q5) =
+∞
+�
+n=1
+1
+(1 − q5n−4)(1 − q5n−1),
+and
+(3.3)
+H(q) =
+∞
+�
+n=0
+qn2+n
+(1 − q)(1 − q2) · · ·(1 − qn) =
+∞
+�
+n=0
+qn2+n
+(q : q)n
+=
+1
+(q2; q5)(q3; q5) =
+∞
+�
+n=1
+1
+(1 − q5n−3)(1 − q5n−2).
+The Rogers–Ramanujan continued fraction is then,
+(3.4)
+R(q) = q
+11
+60 H(q)
+q
+−1
+60 G(q) = q
+1
+5
+∞
+�
+n=1
+(1 − q5n−4)(1 − q5n−1)
+(1 − q5n−3)(1 − q5n−2)
+= 1 +
+q
+1
+5
+1 +
+q
+1 +
+q2
+1 + q3
+...
+.
+So, we note that R(0, 1) leads us to the celebrated Rogers-Ramanujan contin-
+ued fraction, which has been researched by many (see Andrews [4, Chapter 7], for
+example). In the course of analyzing identities from Ramanujan’s Lost Notebook
+[7], Andrews and Berndt have discussed the fraction R(a, b), but mainly from the
+viewpoint of transformation formulas.
+Our emphasis here is on using (3.1) in a
+generalized approach to several partition identities, but there is a whole adjacent
+theory on particular values of these continued fractions determined from applying
+the theory of modular forms.
+Hence the examples, using ϕ as the golden ratio
+(
+√
+5 + 1)/2,
+
+10
+GEOFFREY B CAMPBELL
+(3.5)
+e− −π
+5
+1 +
+e−π
+1 +
+e−2π
+1 + e−3π
+...
+= 1
+2ϕ(
+√
+5 − ϕ3/2)(
+4√
+5 + ϕ3/2),
+(3.6)
+e− −2π
+5
+1 +
+e−2π
+1 +
+e−4π
+1 + e−6π
+...
+=
+4√
+5ϕ1/2 − ϕ,
+(3.7)
+e− −4π
+5
+1 +
+e−4π
+1 +
+e−8π
+1 + e−12π
+...
+= 1
+2ϕ(
+√
+5 − ϕ3/2)(−
+4√
+5 + ϕ3/2).
+So next we examine the continued fraction R(a, b) of Ramanujan and consider
+various restricted partition functions. For further reading, a good reference is Alladi
+and Gordon [3]. We use the continued fraction to give results for several partition
+identities, some of which generalize results of Bressoud [12] and G¨ollnitz [34]. We also
+give a combinatorial interpretation for the coefficients in the power series expansion
+of the reciprocal
+1
+R(−a,−b), extending a result of Odlyzko and Wilf [42]. The full
+description of this approach would add several more pages to our work, but [3]
+covers all of this very nicely.
+It turns out that Lebesgue’s identity plays a major role in our analysis with
+respect to the numerators and denominators of the finite continued fractions we
+consider.
+(3.8)
+�
+k≥0
+qk(k+1)/2 �k
+j=1(1 + bqj)
+(1 − q)(1 − q2)...(1 − qk) =
+�
+m≥1
+(1 + bq2m)(1 + qm).
+It is known that Lebesgue’s identity implies Ramanujan’s fraction R(a, b) has a
+product representation when a = 1. More precisely (3.14) and (3.15) (see below)
+yield
+(3.9)
+1 +
+bq
+1 + q +
+bq2
+1 + q2 +
+bq3
+1 + q3 + bq4
+...
+=
+∞
+�
+m=1
+(1 + bq2m−1)
+(1 + bq2m) .
+
+CONTINUED FRACTION PARTITION IDENTITIES
+11
+A neat case of (3.9) is obtained from q �→ q2 and b �→ bq−1 so then
+(3.10)
+1 +
+bq
+1 + q2 +
+bq3
+1 + q4 +
+bq5
+1 + q6 + bq7
+...
+=
+∞
+�
+m=1
+(1 + bq4m−3)
+(1 + bq4m−1).
+For a continued fraction F, let Pn/Qn denote its nth convergent, and suppose
+that limn→∞ Pn = P, limn→∞ Qn = Q in a suitable topology. We then say that F
+has numerator P and denominator Q, and write P = F N, Q = F D. Consider the
+fraction
+F(a, c) = 1 + a +
+acq
+1 + aq +
+acq2
+1 + aq2 +
+acq3
+1 + aq3 + acq4
+...
+.
+This can be written in the form
+F(a, c) = f(a, c)
+f(aq, c),
+where
+f(a, c) =
+�
+k≥0
+Akqk.
+We now compute the coefficients Ak = Ak(c, q), observing that f(a, c) satisfies
+the recurrence
+f(a, c) = (1 + a)f(aq, c) + acq f(aq2, c).
+Therefore the coefficients Ak satisfy
+Ak = qk Ak + qk−1Ak−1 q − cq2k−1 Ak−1,
+which is the same as
+Ak = qk−1(1 + cqk)
+(1 − qk)
+Ak−1.
+By iteration this yields
+F(a, c) =
+�
+k≥0
+akq
+k(k−1)
+2
+(−cq)k
+(q)k
+.
+Let c = a−1b. Then
+R(a, b) = f(a, a−1b)
+f(aq, a−1b) − a
+is Ramanujan’s fraction (3.1).
+Lemma 3.1. For the fraction R(a, b), the numerator is
+(3.11)
+RN(a, b) =
+�
+k≥0
+akqk(k+1)/2(−a−1b)k
+(q)k
+,
+and the denominator is
+(3.12)
+RD(a, b) =
+�
+k≥0
+akqk(k+1)/2(−a−1bq)k
+(q)k
+.
+
+12
+GEOFFREY B CAMPBELL
+Proof : The expansion (3.12) is an immediate consequence of
+(3.13)
+RD(a, b) = f(aq, a−1b).
+The expansion (3.11) is more complicated. To obtain it, observe that
+RN(a, b)
+=
+f(a, a−1b) − a f(aq, a−1b)
+=
+�
+k≥0
+akqk(k−1)/2(−a−1bq)k
+(q)k
+−
+�
+k≥0
+ak+1qk(k+1)/2(−a−1bq)k
+(q)k
+=
+1 +
+�
+k≥0
+ak+1qk(k+1)/2(−a−1bq)k
+(q)k
+�1 + a−1bqk+1
+1 − qk+1
+− 1
+�
+=
+1 +
+�
+k≥0
+ak+1q(k+1)(k+2)/2(−a−1bq)k(1 − a−1b)
+(q)k+1
+=
+�
+k≥0
+akqk(k+1)/2(−a−1b)k
+(q)k
+as required.
+■
+Andrews (see [5] and [6]) considered the expansions in lemma 3.1 while discussing
+a transformation formula of Ramanujan [47] for R(a, b). Our emphasis here is on
+the partition theorems that can be derived using R(a, b), and for this the following
+lemma is crucial.
+Lemma 3.2. For the fraction R(a, b), we also have the expansions
+(3.14)
+RN(a, b) =
+�
+i,j≥0
+aibjq(i2+i)/2+ij+j2
+(q)i(q)j
+,
+and the denominator is
+(3.15)
+RD(a, b) =
+�
+i,j≥0
+aibjq(i2+i)/2+ij+j2+j
+(q)i(q)j
+.
+Proof : To obtain (3.14) and (3.15) from (3.12) and (3.13) we use the q-binomial
+theorem,
+(−z)k =
+k
+�
+j=0
+zjqj(j−1)/2
+�k
+j
+�
+q
+with z = a−1b and z = a−1bq.
+(See Campbell [22] for the n-space q-binomial
+theorem.) Therefore
+RN(a, b)
+=
+�
+k≥0
+akqk(k+1)/2
+(q)k
+k
+�
+j=0
+a−jbjqj(j−1)/2(q)k
+(q)j(q)j−k
+=
+�
+i,j≥0
+aibjq(i+j)(i+j+1)/2
+(q)i(q)j
+,
+where i = k − j; this is equivalent to (3.12). To obtain (3.13), observe that
+(3.16)
+RD(a, b) = RN(a, bq)
+by comparing (3.14) and (3.15).
+
+CONTINUED FRACTION PARTITION IDENTITIES
+13
+The following two theorems relate successively to the numerator and the denom-
+inator of the fraction (3.1), so then to (3.14) and (3.15). For a proof of these see
+Alladi and Gordon [3].
+Theorem 3.1. (Numerator)
+Let AN(n; i, j) be the number of partitions of n into i + j distinct red parts and j
+distinct blue parts such that one of the blue parts may be zero and every blue part is
+≤ i + j − 1.
+Let BN(n; i, j) be the number of partitions of n into i distinct red parts and j
+distinct non-consecutive blue parts such that every red part is > j.
+Let CN(n; i, j) be the number of partitions of n into i red parts and j blue parts
+such that all parts are distinct and after each blue part there is a gap of at least 2.
+Then
+AN(n; i, j) = BN(n; i, j) = CN(n; i, j).
+Theorem 3.2. (Denominator)
+Let AD(n; i, j) be as in AN(n; i, j) except that every blue part is > 0 and ≤ i + j.
+Let BD(n; i, j) be as in BN(n; i, j) except that part 1 cannot be blue.
+Let CD(n; i, j) be as in CN(n; i, j) except that part 1 cannot be blue. Then
+AD(n; i, j) = BD(n; i, j) = CD(n; i, j).
+So reprising (3.10) namely
+1 +
+bq
+1 + q2 +
+bq3
+1 + q4 +
+bq5
+1 + q6 + bq7
+...
+=
+∞
+�
+m=1
+(1 + bq4m−3)
+(1 + bq4m−1)),
+we have interesting cancellations in numerator-denominator equations. That is,
+the numerator is given by
+�
+k≥0
+qk(k+1)(−bq−1; q2)k
+(q2; q2)k
+=
+∞
+�
+m=1
+(1 + bq4m−3)(1 + q2m)
+=
+∞
+�
+m=1
+(1 + bq4m−3)(1 + q4m−2)(1 + q4m)
+and the denominator is given by
+�
+k≥0
+qk(k+1)(−bq; q2)k
+(q2; q2)k
+=
+∞
+�
+m=1
+(1 + bq4m−1)(1 + q4m−2)(1 + q4m)
+with right sides having common factors that eliminate.
+This leads in particular to the continued fraction identity
+(3.17)
+1 +
+q
+1 + q2 +
+q3
+1 + q4 +
+q5
+1 + q6 + q7
+...
+=
+�
+j≡2,3,7 (mod8)(1 − qj)
+�
+j≡1,5,6 (mod8)(1 − qj).
+
+14
+GEOFFREY B CAMPBELL
+G¨o11nitz [34] states similar results, but (3.1) seems to have escaped attention. There
+is a continued fraction identity due to Gordon [33] and G¨o11nitz [34] which looks
+very similar to (3.17), namely
+(3.18)
+1 + q +
+q2
+1 + q3 +
+q4
+1 + q5 +
+q4
+1 + q7 + bq6
+...
+=
+�
+j≡3,4,5 (mod8)(1 − qj)
+�
+j≡1,4,7 (mod8)(1 − qj).
+However, this result first appears in Alladi and Gordon [3] almost 30 years after
+(3.1).
+4. Ramanujan’s three parameter continued fraction
+Ramanujan [45] obtained in addition to (3.1), the following continued fraction
+with three parameters a, b, q which has also a product representation
+(4.1)
+1 − ab +
+(a − bq)(b − aq)
+(1 − ab)(1 + q2) +
+(a − bq3)(b − aq3)
+(1 − ab)(1 + q4) +
+(a − bq5)(b − aq5)
+(1 − ab)(1 + q6) + (a − bq7)(b − aq7)
+...
+=
+∞
+�
+m=1
+(1 + a2q4m−3)(1 + b2q4m−3)
+(1 + a2q4m−1)(1 + b2q4m−1).
+This was proved only in 1985 by the reviewers of Chapter 16 of Ramanujan’s Second
+Notebook [2], 65 years after Ramanujan’s death. If we put a = 0 and replace b2 by
+−b in (4.1), we get (3.10). It seems there is still scope to study the combinatorial
+properties of the coefficients in the power series expansion of this fraction.
+References
+[1] ABRAMOWITZ, M., and STEGUN, I. Handbook of Mathematical Functions, Dover Publi-
+cations Inc., New York, 1972.
+[2] ADIGA,C. BERNDT,B. C.BHARGAVA,S. AND WATSON,G. N. ”Chapter 16 of Ramanu-
+jan’s Second Notebook: Theta Functions and q-Series”, Memoirs of the American Mathemat-
+ical Society, Vol. 315, Amer. Math. Soc., Providence, RI, 1985.
+[3] ALLADI, K. and GORDON H., Partition Identities and a Continued Fraction of Ramanujan,
+Journal of Combinatorial Theory, Series A 63, 275-300 (1993)
+[4] ANDREWS, G.E. The Theory of Partitions, Addison-Wesley Publishing Company, Advanced
+Book Program, Reading, Massachusetts, 1976.
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+[6] ANDREWS,G. E. Ramanujan’s ”Lost” Notebbook. III. The Rogers-Ramanujan continued frac-
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+[7] ANDREWS, G. E., and BERNDT, B. C. Ramanujan’s Lost Notebook: Part V Paperback
+(2018). Springer-Verlag, New York, ISBN-13: 978-3030085506.
+[8] ANDREWS, G.E. and ERIKSSON, K. Integer Partitions, Cambridge University Press, Cam-
+bridge, UK, New York, USA, Port Melbourne, Australia, Madrid, Spain, Cape Town, South
+Africa, 2004.
+
+CONTINUED FRACTION PARTITION IDENTITIES
+15
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+[10] BAXTER, R. J. Exactly Solved Models in Statistical Mechanics, Academic Press, New York,
+1982.
+[11] BIRKHOFF, G. and MACLAINE, S. A survey of modern algebra, fourth ed., N.Y., Macmillan,
+1977.
+[12] BRESSOUD, D.M. On a partition theorem of G¨ollnitz, J. Reine Angew. Math. 305 215-217,
+(1979).
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+La Trobe University, Melbourne, Australia, 1979.
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+(1994) 369-378.
+[17] CAMPBELL, G. B. A new class of infinite products, and Euler’s totient, International
+Journal of Mathematics and Mathematical Sciences, vol. 17, no. 3, pp. 417-422, 1994.
+https://doi.org/10.1155/S0161171294000591.
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+International Jour-
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+vol. 17,
+no. 4,
+pp. 637-654, 1994.
+https://doi.org/10.1155/S0161171294000918.
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+metical functions, Doctor of Philosophy Thesis, School of Mathematical Sciences, The Aus-
+tralian National University, October 1997.
+[20] CAMPBELL,
+G.
+B.
+A
+closer
+look
+at
+some
+new
+identities,
+International
+Journal
+of
+Mathematics
+and
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+Sciences,
+vol.
+21,
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+3,
+pp.
+581-586,
+1998.
+https://doi.org/10.1155/S0161171298000805.
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+International Jour-
+nal of Mathematics and Mathematical Sciences,
+vol. 23,
+no. 4,
+pp. 271-277, 2000.
+https://doi.org/10.1155/S0161171200000764.
+[22] CAMPBELL, G. B. Some n-space q-binomial theorem extensions and similar identities,
+arXiv:1906.07526v1 [math.NT], Jun 2019. (https://arxiv.org/abs/1906.07526)
+[23] CAMPBELL,
+G.
+B.
+An
+interview
+with
+Rodney
+James
+Baxter,
+Aust.
+Math.
+Soc.
+Gazette,
+Volume
+47,
+No1,
+pp24-32,
+March
+2020.
+(https://austms.org.au/wp-
+content/uploads/2020/07/471Web.pdf)
+[24] CAMPBELL,
+G.
+B.
+Fun
+with
+numbers:
+Rational
+solutions
+to
+xyyx
+=
+vwwv,
+Aust.
+Math.
+Soc.
+Gazette,
+Volume
+49,
+No5,
+pp210-211,
+November
+2022.
+(https://austms.org.au/publications/gazette/gazette495/)
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+p. 523, Oeuvres de Cauchy, 1re s´erie, T. VIII, Gauthier-Villars, Paris, 1893, 42- 50.
+[26] CHEEMA, M. S., Vector partitions and combinatorial identities, Math. Comp. 18, 1966 414-
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+Pure Math. 19, 1971, 37-39.
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+Lausannae (1748).
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+and its Applications, Vol 35, Cambridge University Press, (Cambridge - New York - Port
+Chester - Melbourne - Sydney), 1990.
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+rec., Vol II; reprinted in Werke 3 (1876), pp. 123–162.
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+doi:10.1080/10724117.1996.11974985. JSTOR 25678079.
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+464.
+[33] GORDON, B. Some continued fractions of the Rogers-Ramanujan type, Duke Math. J. 32
+(1965), 741-748.
+
+16
+GEOFFREY B CAMPBELL
+[34] G¨OLLNITZ, H. Partitionen mit Differenzenbedingungen, J. Reine Angew. Math. 225 (1967),
+154-190.
+[35] HARDY, G. H. An extension of a theorem on oscillating series, Collected Papers, Vol VI,
+Clarendon Press, Oxford, 1974, 500-506.
+[36] HARDY, G. H. On certain oscillating series, Collected Papers, Vol VI, Clarendon Press,
+Oxford, 1974, 146-167.
+[37] HARDY, G. H., and LITTLEWOOD, J. E. A further note on the converse of Abel’s theorem.
+Collected Papers of Hardy, Vol VI, Clarendon Press, Oxford, 1974, 699-716.
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+[39] HEINE, E. Handbuch der Kugelfunctionen, Theorie und Andwendungen, Vol. 1, Reimer,
+Berlin, 1878.
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+don Press ; New York : Oxford University Press, 1995.
+[41] MASSER, D. W. (1985). ”Open problems”. In Chen, W. W. L. (ed.). Proceedings of the
+Symposium on Analytic Number Theory. London: Imperial College.
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+Cambridge (1927); reprinted by Chelsea, New York, 1962.
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+numbers, Collected Papers of S. Ramanujan, Cambridge University Press, Cambridge (1927),
+179-199; reprinted by Chelsea, New York, 1962.
+[47] RAMANUJAN, S. ”The Lost Notebook, and Other Unpublished Papers,” Narosa, New Delhi,
+1988.
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+Monatsber. K¨onigl. Preuss. Akad. Wiss. Berlin, 671-680, Nov. 1859.
+[49] ROGERS, L. J. (1894). ”Second memoir on the expansion of certain infinite products”. Proc.
+London Math. Soc. 25: 318-343.
+[50] SANDIFER, C. E. (2006). ”Chapter 32: Who proved e is irrational?”. How Euler Did It
+(PDF). Mathematical Association of America. pp. 185–190. ISBN 978-0-88385-563-8. LCCN
+2007927658
+[51] SLOANE, N. J. A., The On-Line Encyclopedia of Integer Sequences (OEIS) Euler transform.
+https : //oeis.org/wiki/Euler transform.
+[52] SLOANE, N. J. A., The On-Line Encyclopedia of Integer Sequences (OEIS) sequence A061159
+Numerators in expansion of Euler transform of b(n)=1/2 https://oeis.org/A061159.
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+Mathematical Sciences Institute, The Australian National University, Canberra,
+ACT, 0200, Australia
+Email address: Geoffrey.Campbell@anu.edu.au
+

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@@ -0,0 +1,711 @@
+Direct electrical probing of anomalous Nernst conductivity
+Weinan Zhou,1, ∗ Asuka Miura,2, † Yuya Sakuraba,2 and Ken-ichi Uchida2, 3, ‡
+1International Center for Young Scientists, National Institute for Materials Science, Tsukuba 305-0047, Japan
+2Research Center for Magnetic and Spintronic Materials,
+National Institute for Materials Science, Tsukuba 305-0047, Japan
+3Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
+Despite the usefulness of the anomalous Nernst conductivity (αA
+xy) for studying electronic band
+structures and exploring magnetic materials with large transverse thermopower, there has not been a
+straightforward way to obtain αA
+xy in the experiment. Here, we propose a simple and versatile method
+enabling direct electrical probing of αA
+xy, which is realized by creating a closed circuit consisting
+of a target magnetic material and a non-magnetic conductor.
+This method was experimentally
+demonstrated on a thin film of magnetic Weyl semimetal Co2MnGa, where the closed circuit was
+formed simply by connecting both ends of the Co2MnGa film with a Au wire. A good approximation
+of αA
+xy was obtained, validating the proposed method and exhibiting its potential for aiding the
+further development of topological materials science and transverse thermoelectrics.
+The anomalous Nernst conductivity, i.e., the off-
+diagonal component of the thermoelectric conductivity
+tensor (αA
+xy) stemming from magnetic moments, de-
+scribes an intrinsic material property that directly con-
+verts a longitudinal temperature gradient into a trans-
+verse electric field in a magnetic material. It has been
+shown that αA
+xy is closely linked to the Berry curvature
+of the electronic bands; in comparison with the anoma-
+lous Hall conductivity, which is determined by all oc-
+cupied bands, αA
+xy can be more sensitive to the elec-
+tronic band structures close to the Fermi level, rendering
+it a valuable tool to study the topological features of
+magnetic materials through transport measurements [1–
+18]. In addition to this rapidly increasing interest from
+the viewpoint of fundamental physics, αA
+xy is regarded
+as a crucial parameter to explain unconventionally large
+transverse thermoelectric output in some magnetic mate-
+rials where intrinsic contribution plays a dominant role.
+Therefore, exploring magnetic materials with large val-
+ues of αA
+xy has become a major strategy for thermoelec-
+tric applications [19–23]. Due to the orthogonal relation-
+ship between the applied temperature gradient and gen-
+erated electric field, the transverse thermoelectric gen-
+eration module can be a simple slab or sheet, where no
+complicated three-dimensional structures are necessary
+unlike conventional Seebeck-effect-based modules. Thus,
+transverse thermoelectric modules could potentially cir-
+cumvent the problems of durability, flexibility, and cost
+that the Seebeck modules encounter [22–26], as well as
+be exploited for additional functionalities, such as heat
+flux sensing [23, 25, 27, 28]. Despite the significant role of
+αA
+xy in topological materials science and transverse ther-
+moelectrics, there has not been a straightforward way
+to experimentally obtain αA
+xy, and establishing such a
+method is of great importance.
+The conventional experimental method for estimat-
+ing αA
+xy consists of the measurements of the anomalous
+Nernst effect (ANE), anomalous Hall effect (AHE), See-
+beck effect (SE), and electrical resistivity of a magnetic
+material. The anomalous Nernst coefficient (SANE), i.e.,
+the transverse thermopower due to ANE, is expressed as
+SANE = ρxxαA
+xy − ρAHEαxx,
+(1)
+where ρxx, ρAHE, and αxx are the longitudinal resistivity,
+anomalous Hall resistivity, and diagonal component of
+the thermoelectric conductivity tensor, respectively. The
+first term on the right-hand side of Eq. (1) (SI = ρxxαA
+xy)
+is regarded as an intrinsic component of ANE, while the
+second term appears as a consequence of AHE acting on
+the longitudinal electric field induced by SE, which can
+be rewritten as SII = −SSEρAHE/ρxx [Fig. 1(a)] with SSE
+being the Seebeck coefficient. As a result, αA
+xy is obtained
+by experimentally measuring all four parameters of ρxx,
+ρAHE, SSE, and SANE, then calculating using Eq. (1).
+Many studies have exploited this conventional method
+to obtain αA
+xy of a variety of magnetic materials [2–5, 7–
+FIG. 1.
+(a) Schematic illustration of ANE in a magnetic
+material. The orange and green arrows represent the contri-
+bution from the SI and SII terms of SANE, while the black
+arrow represents the direction of magnetization (M). The +
+and − symbols indicate the accumulated electric charges due
+to SE and ANE. (b) Schematic illustration of the closed circuit
+in which a magnetic material (cyan) is electrically connected
+to a non-magnetic conductor (gray) at both ends along the
+direction of the applied temperature gradient (∇T).
+arXiv:2301.02465v1  [cond-mat.mtrl-sci]  6 Jan 2023
+
+(a)
+(b)
+S = -SsE PAHE IPxx
+e
+M
+e
+VT
+++++++
+S
+e
+e2
+19, 22, 23, 25, 28]. However, such a task could be cum-
+bersome, and sometimes challenging to complete, since
+it requires various experimental techniques and measure-
+ment systems.
+In this study, we propose a method to directly mea-
+sure the intrinsic component of ANE of a magnetic ma-
+terial and probe its αA
+xy with ease. This method is real-
+ized simply by creating a closed circuit consisting of the
+target magnetic material and a non-magnetic conductor,
+and then measuring transverse thermopower, as shown
+in Fig. 1(b). The formation of the closed circuit tunes
+the boundary conditions for electron transport, resulting
+in the direct emergence of αA
+xy reflecting the Berry curva-
+ture in the transverse thermopower. We experimentally
+demonstrated this method using a Co2MnGa thin film,
+and compared the result with the value of αA
+xy obtained
+using the conventional method. The proposed method
+grants easy access to αA
+xy, and could be a useful tool in
+studying topological features and transverse thermoelec-
+tric conversion properties of magnetic materials.
+When a magnetic material is electrically connected to
+a non-magnetic conductor at both ends along the direc-
+tion of the applied temperature gradient (∇T), a closed
+circuit is formed, and its total transverse thermopower
+measured at the magnetic material (Sy
+tot) is derived to
+be [29, 30]
+Sy
+tot = SANE −
+ρAHE
+ρC/r + ρM
+(SC − SM).
+(2)
+Here, ρC(M) and SC(M) are the longitudinal resistivity
+and Seebeck coefficient of the non-magnetic conductor
+(magnetic material), respectively. The size ratio r is de-
+termined by the geometry of the closed circuit, and in
+this case, can be expressed as r = (LM/LC) × (AC/AM),
+where LC(M) is the length of the non-magnetic conduc-
+tor (magnetic material) along the closed circuit [x axis
+in Fig. 1(b)] and AC(M) is the cross-section area of the
+non-magnetic conductor (magnetic material) perpendic-
+ular to the LC(M) direction [yz plane in Fig. 1(b)]. Pre-
+viously, thermoelectric materials have been connected to
+magnetic materials to create closed circuits in order to
+generate large transverse thermopower [29, 31], which is
+referred to as the Seebeck-driven transverse thermoelec-
+tric generation. However, Eq. (2) is still valid when a
+non-magnetic conductor having negligible SE is used in-
+stead of thermoelectric materials. If |SC| ≪ |SM| and we
+make ρC/r ≪ ρM through small ρC, large r, or both, the
+second term on the right-hand side of Eq. (2) is reduced
+to SMρAHE/ρM. By substituting Eq. (1) into Eq. (2), the
+SII term in SANE is canceled out, leaving only the SI term
+in Sy
+tot [Fig. 1(b)]. In other words, SE of the magnetic
+material is shunted by connecting to the non-magnetic
+conductor, leading to the disappearance of the SII term.
+Then, αA
+xy can be easily obtained as
+αA
+xy ≈ Sy
+tot
+ρM
+.
+(3)
+FIG. 2.
+(a) Schematic illustration of the sample structure
+and measurement setup for the experimental demonstration
+of the proposed method to directly probe αA
+xy. V1, V2, V3, and
+V4 represent four nanovoltmeters measuring the longitudinal
+thermoelectric signal, transverse thermoelectric signal, and
+resistance of two Pt wires, respectively. (b), (c) H dependence
+of the transverse electric field (Ey) divided by ∇T for the
+closed-circuit sample (b) and the reference sample (c). (d) H
+dependence of the transverse resistivity (ρyx) of the reference
+sample, showing AHE of Co2MnGa.
+(e) H dependence of
+the voltage from V1 of the closed-circuit (blue diamond) and
+reference (red square) samples. The magneto-Seebeck effect
+[32] in Co2MnGa was found to be negligibly small.
+In comparison with the conventional method based on
+Eq. (1), the method proposed here reduces the required
+parameters for obtaining αA
+xy from four to two. If ρM is
+known, a simple measurement of Sy
+tot in the closed circuit
+enables the direct probing of αA
+xy.
+We experimentally demonstrated the proposed method
+using a Co2MnGa thin film.
+We chose Co2MnGa be-
+cause it is known as a magnetic Weyl semimetal hav-
+ing substantial SI and SII terms contributing to its large
+SANE [7, 11, 14, 15]. The 26-nm-thick Co2MnGa thin
+film was epitaxially deposited on a single crystal MgO
+(100) substrate at room temperature by magnetron sput-
+
+(a)
+H
+V
+Au bonding wire
+Co2MnGa
+Au electrode
+MgO substrate
+Pt wire
+b
+E*/VT(μVK-1)
+K-1
+2
+2
+(μV
+0
+0
+2
+-2
+3
+2
+1
+0
+2
+3
+-3
+-2
+-1
+0
+1
+2
+μoH (T)
+HoH (T)
+20
+-135
+(d)
+(e)
+-130
+10
+Pyx (μQ cm)
+(μV)
+-125
++ Reference
+0
++ Closed circuit
+V
+-10
+-10
+-5
+-20
+-2
+-1
+0
+1
+2
+3
+-3
+-2
+-1
+0
+1
+2
+-3
+3
+μoH (T)
+μoH (T)3
+tering, followed by post annealing at 500◦C. After the
+sample was cooled down to room temperature, a 2-nm-
+thick Al capping layer was deposited to prevent oxidiza-
+tion. The composition of Co2MnGa was determined to be
+Co45.7Mn25.4Ga28.9 by X-ray fluorescence spectroscopy.
+The 111 superlattice peak of Co2MnGa was con���rmed
+in the X-ray diffraction pattern, indicating the forma-
+tion of L21 atomic ordering.
+Then, we patterned the
+Co2MnGa film into a 2-mm-wide and 8-mm-long Hall
+bar structure using photolithography and Ar ion milling,
+followed by the formation of Au electrodes through a lift-
+off process. On-chip thermometers made of Pt wires were
+subsequently formed through a lift-off process at the po-
+sitions corresponding to the electrodes of the Hall bar
+along the x axis, as shown in Fig. 2(a). In order to cre-
+ate the closed circuit, we simply connected both ends of
+the the Co2MnGa film along the x axis with a 30-µm-
+diameter Au bonding wire. Here, the Co2MnGa is the
+magnetic material under study, while the Au wire serves
+as the non-magnetic conductor. The electrical resistivity
+of Au wire is 2.3 µΩ cm at room temperature, two orders
+of magnitude smaller than that of the Co2MnGa film,
+which was measured to be ρM = 222.589±0.001 µΩ cm.
+Meanwhile, we assumed a 30-µm-diameter circle as AC,
+and estimated LC = 12 mm for the Au wire, leading to
+estimation of r = 7. Together with SC = 2.0 µV K−1 of
+Au [33] and experimentally measured SM = −32.7 ± 0.2
+µV K−1 for Co2MnGa, the close circuit satisfies the as-
+sumptions of |SC| ≪ |SM| and ρC/r ≪ ρM for Eq. (3).
+To measure the transverse thermopower, we set the sam-
+ple on a home-made holder, where one side of the sample
+was thermally connected to a Cu block then to a heat
+sink while the other side was thermally connected to a
+heater and insulated from the heat sink by a bakelite
+plate, similar to the one used in Ref. 34. When a charge
+current is applied to the heater, ∇T along the x axis
+is generated in the sample. To evaluate ∇T, we placed
+the holder in a physical property measurement system
+(PPMS; Quantum Design), and first calibrated the on-
+chip thermometers by measuring the resistance of the Pt
+wires as a function of temperature using the four-terminal
+method under zero magnetic field (H). Then, we set the
+temperature of PPMS at 295 K, applied the current to
+the heater, and swept H along the z axis while monitor-
+ing the longitudinal and transverse thermoelectric signals
+from the closed circuit with two nanovoltmeters, V1 and
+V2, respectively. The measured resistance of the Pt wires
+during the sweep of H was used to obtain ∇T. As a ref-
+erence, the same measuring process was carried out with-
+out the Au wire connecting both ends of the Co2MnGa
+film; this is the conventional ANE measurement.
+The
+average temperature and ∇T of the closed-circuit (ref-
+erence) sample were 302.56±0.02 (302.01±0.02) K and
+0.977±0.005 (0.937±0.004) K mm−1, respectively. For
+the reference sample, the ρM and ρAHE were separately
+measured at room temperature.
+FIG. 3.
+(a) SANE and SI of the reference sample in compar-
+ison with Sy
+tot of the closed-circuit sample. (b) αA
+xy obtained
+using the conventional method and Sy
+tot/ρM, which approxi-
+mately corresponds to αA
+xy through Eq. (3).
+Figures 2(b) and 2(c) show the H dependence of the
+transverse electric field (Ey) divided by ∇T for the
+closed-circuit and reference samples, respectively.
+The
+observed signal of the reference sample showed the H-
+odd dependence and saturation at |µ0H| ∼ 1 T, which
+is attributed to ANE of Co2MnGa in the open circuit
+condition. By contrast, the signal of the closed-circuit
+sample is smaller than that of the reference sample, al-
+though the shapes of the H dependence of the signals
+are similar to each other. The curve in Fig. 2(b) also
+saturates at |µ0H| ∼ 1 T along the z axis, suggesting
+the transverse thermopower of the closed-circuit sample
+is determined by the magnetization (M) of Co2MnGa
+as well. Figure 2(d) shows the H dependence of ρyx of
+Co2MnGa measured using the reference sample, where
+the signal is mostly due to AHE of Co2MnGa. The Sy
+tot,
+SANE, and ρAHE values were evaluated by extrapolating
+the curves in Figs. 2(b)-2(d) at high H after the sat-
+uration of M down to zero H. Figure 2(e) shows the
+longitudinal thermopower from V1 measured at the same
+time when the results in Figs. 2(b) and 2(c) were ob-
+tained. In case of the reference sample, this voltage was
+due to SE of the Co2MnGa-Au thermocouple (note that
+similar Au bonding wires were used to connect the elec-
+trodes of the sample to the home-made holder), and SM
+can be calculated by dividing the voltage at zero H with
+the corresponding temperature difference then adding SC
+of Au. On the other hand, the magnitude of the longitu-
+dinal thermopower of the closed-circuit sample was dra-
+matically reduced, indicating that SE of Co2MnGa was
+indeed shunted by the connection to the Au wire at both
+ends.
+By applying Eq. (3) to the experimental results of
+the closed-circuit sample, we were able to probe αA
+xy
+of Co2MnGa with ease. The values obtained using the
+proposed method and the conventional method are com-
+pared in Fig. 3.
+SANE of Co2MnGa was estimated to
+be 4.09±0.02 µV K−1, consistent with the previously re-
+ported result of the sample having similar composition
+
+5
+1.4
+a
+(b)
+1.2
+1.0
+3
+0.8
+0.6
+2
+0.4
+0.2
+0
+0
+S,
+PANE
+xy4
+FIG. 4.
+Size ratio r dependence of Sy
+tot calculated using
+Eq. (2) (cyan line) in comparison with SI of Co2MnGa ob-
+tained in the experiment (black dashed line).
+Sy
+tot of the
+closed-circuit sample (blue circle) is also plotted at the corre-
+sponding r.
+[15]. Meanwhile, Sy
+tot = 1.89±0.01 µV K−1 of the closed
+circuit is smaller than SANE, but comparable to its SI =
+2.01±0.02 µV K−1 [Fig. 3(a)]. For αA
+xy, the value based
+on Eq. (3) was calculated to be 0.848±0.005 A m−1 K−1,
+while 0.905±0.010 A m−1 K−1 was obtained using Eq. (1)
+of the conventional method [Fig. 3(b)]. As one can see,
+the proposed method exhibits a close approximation of
+αA
+xy, although the value is slightly smaller than that ob-
+tained from the conventional method: the difference is
+∼6%. To understand this difference, we calculated Sy
+tot of
+the closed circuit as a function of r using Eq. (2) and ma-
+terial parameters of Co2MnGa and Au, then compared
+it with the SI term from the conventional method, as
+shown in Fig. 4. The experimentally measured Sy
+tot is
+also plotted at its corresponding r = 7. One can see a
+quantitative agreement in Sy
+tot between the experiment
+and calculation. As r increases, the calculated Sy
+tot de-
+creases from the initial value ∼SANE of Co2MnGa down
+to ∼Sy
+tot measured in the experiment. The tendency of
+the curve suggests that the r of the closed circuit used
+for the demonstration is large enough to neglect the in-
+fluence of ρC. On the other hand, the difference between
+the calculated Sy
+tot and SI at large r is attributed to finite
+SC of Au. The Sy
+tot value being slightly smaller than SI is
+consistent with the fact that SC of Au is positive and op-
+posite to SM of Co2MnGa in sign. These results indicate
+that we should be mindful to the Seebeck coefficient of
+the magnetic material and non-magnetic conductor while
+using the proposed method, as SC being much smaller
+in magnitude than SM is important to achieve a better
+approximation. A non-magnetic conductor having zero
+SC would be an ideal material for the proposed method,
+which could further reduce the difference in αA
+xy.
+As shown above, the proposed method can be eas-
+ily implemented in the experiment to directly measure
+the SI term of a magnetic thin film and probe its αA
+xy.
+While multiple measurement setups are required to use
+the conventional method and evaluate the material pa-
+rameters in Eq. (1), the proposed method can be carried
+out mostly on one setup. This would lead to better relia-
+bility and reproductivity of the results as well as consid-
+erable time and effort saving for the experiment, which
+is especially beneficial for high-throughput materials re-
+search. In addition, using the first-principles calculations
+to obtain the Berry curvature and derive αA
+xy has been
+popularized in recent years and plays an important role in
+exploiting and predicting materials with valuable prop-
+erties. The proposed method could make αA
+xy a direct
+observable in the experiment, thereby enabling fast and
+straightforward comparison with the theory and promot-
+ing further understanding of the matter. It is worth men-
+tioning that although the experimental demonstration
+was done on a magnetic thin film, the proposed method
+should also be applicable to study bulk materials, as long
+as the assumptions of |SC| ≪ |SM| and ρC/r ≪ ρM for
+Eq. (3) are satisfied.
+In summary, we have proposed a method to directly
+probe αA
+xy of a magnetic material, which is realized sim-
+ply by connecting both ends of the magnetic material
+along the direction of ∇T with a non-magnetic conduc-
+tor to create a closed circuit. Sy
+tot of the closed circuit
+approximates the SI term of the magnetic material, and
+αA
+xy can be easily obtained from Sy
+tot and ρM, in con-
+trast to four different parameters required in the conven-
+tional method. The proposed method was experimentally
+demonstrated to probe αA
+xy of a Co2MnGa thin film. The
+closed circuit was easily realized using a Au wire, and a
+good approximation was obtained for both SI and αA
+xy,
+validating this method. Further analysis of the results
+revealed that the small difference was due to finite SC,
+and provided guides for the utilization of the proposed
+method. As the popularity of using αA
+xy is growing, our
+finding could become a powerful tool propelling studies
+of topological materials science and application of trans-
+verse thermoelectric phenomena.
+The authors thank R. Toyama and T. Hirai for their
+support in sample preparation and measurement. This
+work was supported by JST CREST “Creation of In-
+novative Core Technologies for Nano-enabled Thermal
+Management” (Grant No. JPMJCR17I1), JST ERATO
+“Magnetic Thermal Management Materials” (Grant No.
+JPMJER2201), JSPS KAKENHI Grant-in-Aid for Sci-
+entific Research (B) (Grant No.
+JP21H01608) and
+Grant-in-Aid for Research Activity Start-up (Grant No.
+JP22K20494), and NEC Corporation.
+∗ ZHOU.Weinan@nims.go.jp
+† Present address:
+Integrated Research for Energy and
+Environment Advanced Technology, Kyushu Institute of
+Technology, Fukuoka 804-8550, Japan
+‡ UCHIDA.Kenichi@nims.go.jp
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+

R9FRT4oBgHgl3EQfLjfi/content/tmp_files/2301.13503v1.pdf.txt

@@ -0,0 +1,2011 @@
+Identical Bands Around the Isobaric Rare Earth Even-Even Nuclei
+with the Mass Number A = 164
+M. A. Abdelsalam⋆, H. A. Ghanim⋆, M. Kotb⋆, and A. M. Khalaf⋆
+⋆Physics Department, Faculty of Science, Al-Azhar University, Cairo, Egypt
+Corresponding author: mahmoudkotb@azhar.edu.eg
+Abstract
+Eight pairs of rare-earth normally - deformed (ND) nuclei around the isobaric nuclei with A = 164
+and have identical values of F-spin, ± F0 and Np Nn (Np and Nn are the number of valence protons and
+valence neutrons respectively ) have been studied. These pairs of identical bands (IB’s) cover 16 mass
+units and are classified as (i) 3 pairs of nuclei separated by (2p,2n) :(162Y b −166 Hf), (162Er −166 Y b),
+(162Dy −166 Er) (ii) 2 pairs of nuclei separated by (4p,4n): (160Dy −168 Y b), (160Er −168 Hf) (iii) 2 pairs
+of nuclei separated by (6p,6n): (158Er −170 W) (158Dy −170 Hf) and (iv) one pair of nuclei separated
+by (8p,8n): (156Dy −172 W).
+We suggested a theoretical collective rotational formula containing three parameters (CRF3) as an
+extended version of Bohr-Mottelson model to calculate the ground state positive parity excitation en-
+ergies. Also, the sd-version of the interacting boson model (IBM) has been used to describe the nuclear
+shapes by using the intrinsic coherent-state. The optimized models parameters for each nucleus are
+adjusted by using a simulation search program to minimize the root mean square deviation between
+the theoretical calculation and experimental excitation energies. The best adopted model parameters
+of the CRF3 are used to calculate the rotational frequencies ¯hω, the kinematic J(1) and dynamic J(2)
+moments of inertia and the evolution of J(1) and J(2) with increasing ¯hω are systematically analyzed.
+A smooth gradual increase in both moments of inertia was seen.
+The calculated results agree excellently with the experimental ones which give strong support to
+the suggested CRF3.
+The adopted IBM parameters are used to calculate the potential energy surfaces (PES’s) which
+describe the nuclear deformation. The PES’s for our nuclei shows two wells corresponding to prolate
+and oblate sides which indicate that these nuclei are deformed and have rotational behaviors.
+The correlation quantities which identify the IB’s are extracted. It is found that the nuclei having
+NpNn/△ where △ is the average pairing gap, exhibit identical excitation energies and energy ratios in
+their ground state rotational bands.
+Keywords : Interacting Boson model (IBM) - Identical Bands - Potential Energy Surface
+1
+Introduction
+The discovery of rotational bands in adjacent even-even and odd-mass superdeformed (SD) nuclei in
+which the γ-ray transition energies are nearly identical to within a few KeV was an exotic and unex-
+pected phenomenon in nuclear structure physics [1–5]. Since the identical bands (IB’s) have essentially
+identical transition energies, then the associated dynamical moment of inertia are thus identical. Sev-
+eral explanations were put forward [4–12] to understand the origin of IB’s phenomenon assuming the
+occurrence of such IB’s to be a specific property of the SD states in nuclei. The explanations of these IB’s
+includes: the Coriolis force, the particle alignment and pairing [13], the roles of special high-N orbitals of
+intruder configuration and band crossing [14–17], the pseudo-spin in supersymmetry [7, 18, 19] and the
+supersymmetry with many-body interactions [20].
+Soon the phenomenon of low-spin identical bands was found in pairs of even-even normal deformed
+(ND) nuclei [21], and in neighboring even-even and odd-mass nuclei in rare-earth region where they have
+similar moments of inertia [22,23]. If was noted that low spin IB’s are not limited to nearby nuclei but are
+widespread and found in pairs of even-even nucleoside as separated by 24 mass unit (like 156Dy,180 Os)
+1
+arXiv:2301.13503v1  [nucl-th]  31 Jan 2023
+
+[24]. Attempts were made to understand the low-spin IB’s in terms of some simple systematics of the
+moments of inertia in the rare-earth region [25–30] or from several types of consideration [31].
+For the description of normally deformed (ND) bands, some useful models were proposed. Bohr and
+Mottelson [32] pointed out that, under the adiabatic approximation, the rotational energy of an axially
+symmetric nucleus may be expanded for K = 0 band as a power series in the I(I+1) term. The expansion
+for the K ̸= 0 band takes the same form, but includes a band head energy and the I(I+1) is replaced by
+�
+I(I + 1) − K2�
+. Another useful models for nuclear rotational spectra are the particle-rotor model (PRM)
+[33], the variable moment of inertia (VMI) model [34, 35], the soft rotor model [36] and the interacting
+boson model [37].
+In the concept of F-spin and its projection [38] any pairs of conjugate nuclei with the same F-spin and
+F0 values in any F-multiplet will have the same NpNn [24, 39, 40] where Np and Nn are respectively the
+number of valence protons and valence neutrons. The product NpNn was used in the classification of the
+changes that occur in nuclear structure [41,42]. It was assumed that [25,43] the moment and the P-factor
+depends also on the product NpNn.
+The purpose of the present paper is (i) to analyse the excitation energies for even-even normally de-
+formed nuclei in rare earth region in framework of suggested new collective rotational formula (CRF3).
+(ii) to exhibit the occurrence of IB’s in eight pairs of nuclei in rare earth region. (iii) to present the parame-
+ters which characterize the appearance of IB’s. (iv) use the sd version of interacting boson model (sdIBM)
+to calculate the potential energy surfaces (PES’s).
+2
+Outline of the Suggested Collective Rotational Formula with Three Pa-
+rameters (CRF3)
+Rotational states in normal deformed (ND) nuclei can be characterized by their excitation energies E(I)
+as a function of spin I, which generally lie low as compared to the single-particle excitation. In the strong
+coupling limit, the rotational ground state energy for an axially symmetric even-even nucleus obeys the
+I(I+1) rule, i.e form bands of levels that fulfill the relation
+E(I) = ¯h2
+2J I(I + 1) = α Î
+2
+(1)
+where α = ¯h2/2J and Î = I(I+1)
+The relation (1) defines in addition the nuclear moment of inertia J as a constant for an ideal rotor.
+This simple rotational formula gives deviations from experimental data, So Bohr and Mottelson pointed
+out that agreement was improved by adding to it a second team to yield
+E(I) = αI(I + 1) + β[I(I + 1)]2
+= α Î
+2 + β Î
+4
+E(I) = α Î
+2(1 + γ Î
+2)
+(2)
+where γ = β/α
+Since the moment of inertia J increases on rotation of the nucleus, the observed deviations from the
+experiment were still more evident.
+According to the variable moment of inertia(VMI) model [34, 35], there is a gradual increase in mo-
+ment of inertia J with increasing the spin I, so we suggest that the moment inertia J can be written as
+J = J(I) = J (1 + σ Î
+2)
+(3)
+Substituting in equation (2), yield
+E(I) = α Î
+2
+�
+1 + γ Î
+2
+1 + σ Î
+2
+�
+(4)
+Therefore, the two-term Bohr-Mottelson formula becomes an extended new formula with three pa-
+rameters. We denote formula (4) as the collective rotational formula with three parameters (CRF3). The
+parameters are α, β, γ.
+2
+
+The suggested CRF3 is more general because it leads to the following three predictions:
+a) when σ = γ it gives pure rigid rotor equation(1)
+b) when σ = 0 it gives the two parameters Bohr-Mottelson equation (2)
+c) when γ = 0 it gives soft rotor model [36]
+E(I) = ¯h2
+2J
+I(I + 1)
+1 + σ(I + I2)
+(5)
+Two types of moments of inertia were suggested by Bohr-Mottelson which reflect two different as-
+pects of nuclear dynamics. The first moment of inertia is the kinematic J(1), it is equal to the inverse of
+the slope of the curve of energy E versus Î
+2 (or I(I+1)) times ¯h2/2, while the second moment of inertia is
+the dynamic J(2), it is related to the curvature in the curve of E versus Î (or
+�
+I(I + 1) ).
+The kinematic J(1)) and dynamic J(2) moments of inertia are defined as:
+J(1) = ¯h2
+2
+�
+dE
+dI(I + 1)
+�−1
+= ¯h
+�
+I(I + 1)
+ω
+= ¯h2
+2
+�dE
+dÎ
+2
+�−1
+= ¯h Î
+ω
+(6)
+J(2) = ¯h2
+�
+d2E
+d(
+�
+I(I + 1))2
+�−1
+= ¯hd
+�
+I(I + 1)
+dω
+= ¯h2
+�d2E
+dÎ
+2
+�−1
+= ¯h dÎ
+dω
+(7)
+In the case of our CRF3, the two moments of inertia becomes
+J(1)(I) = ¯h2
+2α
+(1 + σÎ
+2)2
+[1 + γÎ
+2(2 + σÎ
+2)]
+(8)
+J(2)(I) = ¯h2
+2α
+(1 + σÎ
+2)3
+[(1 + 6γÎ
+2) + σÎ
+2(3γÎ
+2 + αγÎ
+4 − 3)]
+(9)
+Experimentally ¯hω, J(1)and J(2) are extracted in terms of the transition energy Eγ(I) = E(I)−E(I−2)
+as:
+¯hω(I) = 1
+4[Eγ(I + 2) + Eγ(I)]
+(MeV )
+(10)
+J(1)(I) = 2I − 1
+Eγ(I)
+(¯h2MeV −1)
+(11)
+J(2)(I) =
+4
+Eγ(I + 2) − Eγ(I)
+(¯h2MeV −1)
+(12)
+As a special case, the lowest dynamical moment of inertia reads
+J(2)
+lowest =
+4
+Eγ(4+
+1 → 2+
+1 ) − Eγ(2+
+1 → 0+
+1 )
+(13)
+3
+Determination of Ground State Band Properties of Even-Even Nuclei and
+the Physical Identical Parameters
+In order to understand the behavior of low lying states of an axially symmetric normally deformed nuclei,
+it is insightful to examine some physical observables which exist in a pair of IB’s, the observables include:
+1. The P- Factor, Structure Factor (SF), and Saturation Parameter (SP)
+Casten [43] introduced the P-Factor
+P =
+NpNn
+Np + Nn
+(14)
+3
+
+where Np and Nn are the numbers of valence protons and valence neutrons respectively which are
+counted as particles or holes from the nearest closed shell
+Np = min[(Z − 50), (82 − Z)]
+(15)
+Nn = min[(N − 82), (126 − N)]
+(16)
+The P- Factor represents the average number of interactions of each valence nucleon with those of the
+other type. It can be viewed as the ratio of the number of valences p-n residual interactions to the number
+of valence like-nucleon pairing interactions, or if the p-n and pairing interactions are orbit independent,
+then P is proportional to the ratio of the integrated p-n interaction strength to the integrated pairing
+interaction strength. The nuclear collectivity and deformation depend sensitively on the P- Factor.
+The structure factor (SF) and the saturation parameter (SP) are given by
+SF = NpNn(Np + Nn)
+(17)
+SP =
+�
+1 +
+SF
+SFmax
+�−1
+(18)
+It is found that the lowest dynamical moment of inertia J(2)
+lowest is proportional to
+√
+SF.
+2. The Concept of F-Spin
+A nucleus with Np valence protons and Nn valence neutrons has a total boson number
+NB = Np + Nn
+2
+= Nπ + Nν
+(19)
+The Nπ proton bosons and neutron bosons are assigned F-Spin, F =
+1
+2 with projection F0 = + 1
+2
+for proton bosons and F0 = − 1
+2 for neutron bosons. A given nucleus is characterized by two quantum
+numbers [38]:
+F = Nπ + Nν
+2
+and its projection F0 = Nπ − Nν
+2
+Squaring and subtracting, yield
+4(F 2 − F 2
+0 ) = 4NπNν = NpNn
+(20)
+That is any pair of conjugate nuclei with the same F-spin and F0 values in any F-spin multiplet have
+identical NpNn values.
+In our chosen nuclei, the F-spin multiplet is given by: (A+4, Z+2), (A+8, Z+4), (A+12, Z+6) and (A+16,
+Z+8) for Dy, Er, Yb, Hf, and W isotopes.
+Any pair of nuclei which show identical excitation energies have nearly equal value of the product of
+their valence nucleon numbers Np and Nn [41]. However, the analysis of experimental data shows that
+the converse is not true. The simple quantity NpNn helps also in the evolution of nuclear deformation
+and collectivity in nuclei [40]. On the other hand, the product NpNn or the P- Factor plays an important
+role in studying the orbit dependence, shell gaps, and intruder orbitals.
+3. Pairing Interaction Energy
+The pairing interaction energy △ in an even-even nucleus is the average pairing gap ((△p + △n)/2
+where △p and △n are respectively the proton and neutron pairing gaps which are determined from the
+difference in binding energies of the neighboring odd and even nuclei
+△p = 1
+4[B(N, Z − 2) − 3B(N, Z − 1) + 3B(N, Z) − B(N, Z + 1)]
+(21)
+△n = 1
+4[B(N − 2, Z) − 3B(N − 1, Z) + 3B(N, Z) − B(N + 1, Z)]
+(22)
+The pairing gaps △p and △n are determined empirically from the relation
+△p ≃ △n = 12
+√
+A
+(MeV )
+(23)
+The average pairing gap of the nucleus is then
+4
+
+△ = △p + △n
+2
+= 12
+√
+A
+MeV
+(24)
+It is observed that [39, 43] the even-even nuclei belong to different mass number having identical
+(NpNn/△) values exhibit identical excitation energies and identical energy ratios.
+4. Quadrupole Transition Probabilities and Deformation Parameters
+The quadrupole transition probability per unit time for the transition Ii → If is given by
+T(E2) = 4π
+75
+�5
+¯h
+� �E2+
+1
+¯hc
+�5
+B(E2; Ii → If)
+(25)
+where B(E2) is the reduced transition probability and E2+
+1 is the energy of the 2+
+1 state.
+Experimentally T(E2) for transition 2+
+1 → 0+
+1 is obtained by
+T(E2, 2+
+1 → 0+
+1 ) =
+ln2
+(1 + α)T1/2
+=
+0.693
+(1 + α)T1/2
+(26)
+where α is the total conversion coefficient taken from the tabulated values given by Rose [44] and T1/2
+is the lifetime of the rotational level.
+The B(E2, 2+
+1 → 0+
+1 ) values carry important information about the collectivity of nuclear rotation and
+can be extracted from the equations (25,26).
+The relation between the intrinsic nuclear quadrupole moment Q0 and B(E2) is given by
+Q2
+0 = 16π
+e B(E2, 2+
+1 → 0+
+1 )
+(27)
+Practically the most reliable method of determining the quadrupole deformation parameter β2 in
+framework of geometric collective model (GCM) is to extract β2 from Q0 according to the formula
+β2(exp) =
+√
+5π
+3ZR2
+0
+Q0
+(28)
+assuming a uniformly charged nucleus of spheroidal shape, where the nuclear radius has the value
+R0 = 1.2A1/3(fm) and Z is the nuclear charge number.
+The expression (28) for β2 is widely used to compare the quadrupole deformation of different nuclei.
+It is noticed that the B(E2, 2+
+1 → 0+
+1 ) values increase when going from the closed shell at N=82 toward
+midshell where maximum values are occur, while from midshell toward the shell closure at N= 126 its
+values are decreases.
+In a second way , specially where the B(E2, 2+
+1 → 0+
+1 ) value is not known, we estimate β by using the
+approximate empirical Grodzins relation [45]:
+E2+
+1 B(E2, 2+
+1 → 0+
+1 ) = 2.5 × 10−3 Z2
+A
+(29)
+where
+B(E2, 2+
+1 → 0+
+1 ) =
+1
+16πe2Q2
+0 =
+9
+80π2 e2Z2R4
+0β2
+(in units of e2b2)
+(30)
+We can relate β and E2+
+1 as:
+β2
+G =
+1224
+E2+
+1 A7/3
+(31)
+where E2+
+1 is in MeV.
+Also β2 can be determined by using the SU(3) rotational limit of interacting boson model(IBM) [37],
+the square of the deformation parameter β2 in a state of angular momentum I is given by [46]:
+⟨β2⟩I =
+α2
+6(2N − 1)[I(I + 1) + 8N2
+B + 22NB − 15]
+(32)
+5
+
+where NB is the total number of valence bosons and α is a normalization constant (α = 0.101 for rare-
+earth nuclei). The expectation value of β2 in the ground state becomes
+⟨β2⟩0 = α2 8N2
+B + 22NB − 15
+6(2N − 1)
+(33)
+which is an almost linearly increasing function of the boson number NB and has the same value for
+nuclei having the same number of valence nucleons
+N = [Np + Nn], N = [(Np − 1) + (Nn − 1)]
+(34)
+It is evident that βIBM extracted from IBM is much larger than βGCM extracted from GCM because
+βGCM refer to the deformation of all A nucleons while βIBM describe only 2N valence bosons, the ap-
+proximate relation between them is given by:
+βGCM = 1.18
+�2N
+A
+�
+βIBM
+(35)
+The deformation parameter β reflects the equilibrium shape and structure of the nucleus such as the
+energy ratio R4/2 = E(4+
+1 )/E(2+
+1 ) and the reduced transition probability B(E2, 2+
+1 → 0+
+1 ) which are the
+best indicators to exhibit the collective properties of the even-even nuclei.
+5. Energy Ratios and Percentage Difference in Transition Energies
+The energy ratios and the percentage difference in transition energies give the characteristic of the
+evolution of the collectivity in the even-even nuclei. Only deformed nuclei show rotational levels and
+particularly the even-even nuclei display a simple structure energies proportional to I(I+1) with only
+even values of the spin I considering that the moment of inertia is constant (rigid rotator), therefore
+the energy ratio R4/2 = 3.333. The observed moment of inertia extracted from the experiment is only
+one-quarter to one-half of what one would expect from a rigid rotator which means that not the whole
+nucleons are participating in the collective motion.
+On the other hand for an ideal harmonic quadrupole spectrum for spherical nuclei a system of
+equidistant states is formed by the composition of vibrational quanta. The first excited state is 2+
+1 fol-
+lowed by the degenerate 0+
+2 , 2+
+2 , 4+
+1 , and so forth. Therefore energy ratioR4/2 = 2.
+To compare level spacing in two nuclei with masses A1, and A2 where A2 > A1, we define the per-
+centage differences ratios in transition energies as :
+δ = △Eγ(I)
+Eγ2(I)
+(36)
+where
+Eγ = E(I) − E(I − 2)
+(37)
+△Eγ(I) = Eγ1(I) − Eγ2(I)
+(38)
+So that
+Eγ1 = (1 + δ)Eγ2
+(39)
+For rigid rotor the ratio
+δR =
+�A2
+A1
+�5/3
+− 1
+(40)
+define the fractional change in A5/3.
+The fractional change in transition energies δ divided by the rigid rotor ratio δR is denoted by δγ. If
+the spacings are identical, then δ = 0, δγ = 0 and if they scale as A5/3 then δγ=1.
+Similarly, the percentage difference in kinematic moment of inertia J(1) is given by
+K = −△J(1)(I)
+J(1)
+2 (I)
+(41)
+6
+
+where
+J(1)(I) = 2I − 1
+Eγ(I)
+(42)
+△J(1)(I) = J(1)
+1 (I) − J(1)
+2 (I)
+(43)
+So that
+J(2)
+2
+= (1 + K)J(1)
+1
+(44)
+Substituting for J(1), yield K = δ.
+4
+The Interacting Boson Model to Calculate the Potential Energy Surfaces
+and Electric Quadrupole Transition Probability
+We consider the Hamiltonian of the first order U(5)- SU(3) quantum shape phase transition in the form
+H = ϵdˆnd + a2 ˆQ(x) ˆQ(x)
+(45)
+where ˆnd and ˆQ(x) are respectively the d-boson number operator and quadrupole operator defined as
+ˆnd =
+�
+µ
+d†
+µ
+∼
+dµ
+(46)
+ˆQ(x) =
+�
+d†s + s† ∼
+d
+�(2)
++ x
+�
+d†×
+∼
+d
+�(2)
+(47)
+where
+�
+s†, d†�
+and
+�
+s,
+∼
+d
+�
+are the boson creation and annihilation operators respectively, and x is
+the structure parameter of the quadrupole operator of IBM (x for pure rotational SU(3) limit is equal to
+−
+√
+7/2). Here dµ = (−1)µd−µ and standard notation of angular momentum coupling is used.
+To get the potential energy surface (PES) of the Hamiltonian, we introduce the intrinsic coherent
+frame in which the ground state of a nucleus with N bosons can be expressed as a boson condensate. The
+bosonic intrinsic coherent state for the ground state band of a given even-even nucleus can be written in
+the form [47–49]
+|Nβγ⟩ =
+1
+√
+N!
+[b†(β, γ)]N|0⟩
+(48)
+where |0⟩ is the boson vacuum and b† is the boson creation operator which acts in the intrinsic system
+and is given by:
+b† =
+1
+�
+1 + β2 [s† + βcosγ(d†
+0) + 1
+√
+2βsinγ(d†
+2 + d†
+−2)]
+(49)
+where β is the quadrupole deformation parameter which measures the axial deviation from spherical
+symmetry and the parameter γ controls the departure from axial symmetries.
+The ground state PES is the expectation value of the Hamiltonian in the intrinsic coherent state
+PES = ⟨Nβγ|H|Nβγ⟩
+(50)
+The associated PES of the Hamiltonian (45) for x = −
+√
+7/2 reads
+E(N, β, γ) = ϵd
+Nβ2
+1 + β2 + a2
+�
+N
+1 + β2 (5 + 11
+4 β2) + N(N − 1)
+(1 + β2)2 (4β2 − 2
+√
+2β3cos3γ + 1
+2β4)
+�
+(51)
+Equation (51) can be written in another form as
+E(N, β, γ) = g1
+Nβ2
+1 + β2 + N(N − 1)
+(1 + β2)2 [g2β2 + g3β3cos3γ + g4β4] + c
+(52)
+7
+
+where the coefficients involve linear combination of the Hamiltonian parameters
+g1 = ϵd − 9
+4a2,
+g2 = 4a2
+g3 = 2
+√
+2a2,
+g4 = 1
+2a2,
+c = 5Na2
+Also, equation (51) can be rewritten in general form as
+E(N, β, γ) = A2β2 + A3β3cos3γ + A4β4
+(1 + β2)2
++ A0
+(53)
+where the coefficients read
+A2 =
+�
+ϵ +
+�
+4N − 25
+4
+�
+a2
+�
+N,
+A3 = 2
+√
+2a2(N − 1)N
+A4 =
+�
+ϵ +
+�2N + 5
+4
+− 4
+�
+a2
+�
+N,
+A0 = 5a2N
+For a2 = 0, we get the pure spherical vibrator U(5) limit and for ϵd = 0, we get the pure deformed
+rotational Su(3) limit.
+Another important quantity that tests the nature of the shape phase transition of low lying collective
+states the reduced electric quadrupole transition probabilities B(E2).
+In IBM, the general form of the electric quadrupole operator is written in the form [50]
+T(E2) = eQ(sdIBM)
+(54)
+The coefficient e is the boson’s effective charge.
+The reduced electric quadrupole transition probabilities are given by
+B[E2, Ii → If] =
+1
+2Ii + 1|⟨If||T(E2)||Ii⟩|2
+(55)
+For rotational SU(3), yield
+B(E2, I + 2 → I) = e2 3
+4
+(I + 2)(I + 1)
+(2I + 3)(2I + 5)(2N − 1)(2N + I + 3)
+(56)
+Q(I) = −e
+�
+16π
+40
+I
+2I + 3(4N + 3)
+(57)
+For the special case for I=0, we have
+B(E2, 2+
+1 → 0+
+1 ) = e2 1
+5N(2N + 3)
+(58)
+5
+Numerical Calculations and Discussion
+In this section, we applied our formalism to eight pairs of nuclei having identical bands (IB’s) in rare-
+earth region namely: (162Y b−166 Hf), (162Er−166 Y b), (162Dy −166 Er), (160Dy −168 Y b), (160Er−168 Hf),
+(158Er −170 W), (158Dy −170 Hf) and (156Dy −172 W).
+To calculate the ground state positive parity excitation energy E(I) for each nucleus, we suggested the
+CRF3.
+The parameters α, γ, σ of CRF3 have been determined by a fitting procedure using a computer-
+simulated search program to minimize the root mean square deviation of the calculated excitation ener-
+gies from the experimental ones. The quality of the fitting is indicated by the standard common definition
+of x
+x =
+�
+1
+N Σi
+�Eexp(Ii) − Ecal(Ii)
+δEexp(Ii)
+�2
+8
+
+where N is the number of experimental data points entering the fitting procedure and δEexp(Ii) is the
+experimental error in the excitation energies - The experimental excitation energies are taken from [51].
+The optimized best adopted values of parameters for each nucleus of our studied nuclei are listed in
+Table (1).
+Figure 1: Systematic of the calculated (solid curves) ground state energies for our selected even-even rare earth Dy,
+Er, YB, Hf, W isotopes versus neutron number N and comparison with the experimental ones (dashed curves). The
+spin-parity are labeled by Iπ.
+9
+
+68Er Exp
+6Dy Exp
+68Er Cal
+68Er Exp
+2500F
+2500F
+2500
+2500
+12+
+12*
+12t
+2000
+12+
+2000
+2000
+2000
+1500
+10*
+10+
+1500*
+1500Q
+10
+1500
+10
+ (KeV)
+Energies (KeV)
+01
+KeV
+KeV
+Energies
+Energies (
+Energies
+8+
+1000Q
+8+
+1000
+1000
+1000
+10
+6
+6
+500
+500
+500
+4+
+2
+G
+2
+2+
+92
+94
+96
+92
+t6
+96 
+90
+92
+t6
+96
+98
+90
+92
+t6
+96
+98
+N
+N
+70 Yb Cal
+70Yb Exp
+72Hf Cal
+72Hf Exp
+2500
+2500F
+2500
+2500
+12
+12
+12
+12
+2000*
+2000Q
+2000
+2000
+10*
+10*
+10+
+10
+1500
+1500
+1500
+1500
+Energies (KeV)
+Energies (KeV)
+Energies (KeV)
+8
+8
+1000
+1000
+1000
+1000
+6
+6
+6
+500*
+500G
+500
+4
+4
+21
+2
+10
+2
+G
+o2
+oL
+96
+94
+98
+92
+94
+96
+98
+95
+96
+97
+98
+94
+95
+86
+N
+N
+N
+74 W Cal
+74W Exp
+2500
+2500
+12
+12t
+2000
+2000
+10°
+10*
+500
+1500
+(KeV)
+(KeV)
+Energies
+8t
+1000
+1000
+6
+G
+61
+500
+4t
+G
+96
+96.5
+97
+97.5
+98
+96
+96.5
+97
+97.5
+ 98
+NTable 1: Values of optimized best parameters α, γ, σ of the collective rotational formula(CRF3) for ground state
+bands in our selected even-even rare-earth nuclei. Np and Nn are the number of valance protons and the number of
+valance neutrons respectively.
+Nuclide
+α (KeV)
+γ (10−3)
+σ (10−3)
+Np
+Nn
+Dy 156
+22.96
+6.964
+14.54
+16
+8
+158
+16.48
+2.163
+4.339
+16
+10
+160
+14.49
+0.8683
+2.021
+16
+12
+162
+13.49
+1.398
+2.233
+16
+14
+Er 158
+32.76
+9.699
+23.52
+14
+8
+160
+20.73
+3.017
+6.641
+14
+10
+162
+17.01
+1.440
+3.212
+14
+12
+166
+13.49
+0.2573
+1.188
+14
+16
+Yb 162
+27.87
+6.334
+14.27
+12
+10
+166
+17.08
+2.053
+3.95
+12
+14
+168
+14.72
+1.039
+2.425
+12
+16
+Hf 166
+26.60
+5.565
+12.67
+10
+12
+168
+20.58
+3.116
+6.849
+10
+14
+170
+15.92
+-0.00749
+1.391
+10
+16
+W 170
+26.44
+5.714
+13.55
+8
+14
+172
+20.68
+3.944
+9.279
+8
+16
+Figure 2: The calculated energy ratio R4/2 = E(4+
+1 )/E(2+
+1 ) versus neutron number N characterizes the low lying
+spectrum in Dy, Er, Yb, Hf, and W isotopes. The symbols o, ∗, �, △, and x denote 66Dy,68 Er,70 Y b,72 Hf, and
+74W respectively.
+The systematic of the excitation energies of the low spin states as a function of neutron number N
+in the considered even-even Dy, Er, Yb, Hf, W isotopes in the mass region A= 156 - 172 in the normally
+deformed nuclear are shown in Figure(1) and compared with the experimental ones. Only the ground
+state of positive parity and spin Iπ = 2+, 4+, 6+, 8+, 10+, 12+ has been indicated. We can see that the
+excitation energies decrease with increasing the neutron number. Also, Figure(2) illustrate the calculated
+10
+
+o Dy
+162
+Dy
+3.3
+* Er
+166
+3*
+ Yb
+168.
+Yb
+△ Hf
+162
+Er*
+166.
+Yb
+× W
+158
+Dyo
+170.
+3.2
+ZHf
+168
+160
+AHf
+3.1
+Er*
+172,
+W
+3
+R4/2
+166
+156
+aHf
+170.
+Dy
+162
+Ybo
+2.9
+2.8
+158
+Er
+2.7
+90
+92
+94
+96
+98
+Nenergy ratio R4/2 as a function of neutron number N for our studied nuclei. We observe that for each
+isotopic chain the value of R4/2 increases with increasing N (that is the deformation increased), and the
+difference in R4/2 for all pairs of IB’s is ranging from 0.4 % to 2.5 % except the two pairs including the
+two isotopes 170,172W (the difference is about 5%).
+Figure 3: The calculated results of kinematic J(1) (dashed curves) and dynamic J(2) (solid curves) moments of
+inertia plotted as a function of rotational frequency ¯hω for the studied eight pairs of identical bands in the rare-earth
+region. The ∗ and o correspond to the lighter and heavier nucleus respectively.
+For the eight pairs of IB’S, the kinematic J(1) and the dynamic J(2) moments of inertia derived from
+the transition energies are plotted versus the rotational frequency ¯hω as shown in Figure(3). It can be
+seen that for all bands J(1) is smaller than J(2) and a smooth gradual increase in both J(1) and J(2) with
+increasing ¯hω are seen and the similarities between each pair of IB’S are observed.
+11
+
+170W
+162Yb -_ 166Hf
+70 
+70 
+60 
+60
+J(), J(2) (h? MeV-1)
+50 
+40
+40 
+30
+30
+20
+G
+10 
+0
+0.1
+0.12
+0.14
+0.16
+0.18
+0.2
+0.22
+0.24
+0.26
+0.28
+0.1
+0.12
+0.14
+0.16
+0.18
+0.2
+0.22
+0.24
+0.26
+0.28
+ho(MeV)
+ho(MeV)
+156Dy _ 172W
+160Er -_ 168Hf
+90
+80
+J(M), J(2) (h? MeV-l)
+70
+J(I), J(2) (h? MeV-l)
+50
+50
+40
+40
+30
+20
+20
+0.08
+0.1
+0.12
+0.14
+0.16
+0.18
+0.2
+0.22
+0.24
+0.26
+0.28
+0.08
+0.1
+0.12
+0.14
+0.16
+0.18
+0.2
+0.22
+0.24
+0.26
+0.28
+ho(MeV)
+ho(MeV)
+158Dy _ 170Hf
+162Er - 166Yb
+100
+70 
+06
+65
+(h? MeV-l)
+J(), J(2) (h? MeV-1)
+80 
+60
+70 
+50
+J(I), J(2) (
+60 
+45
+50
+40
+40 
+30
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+0.18
+0.2
+0.22
+0.24
+0.26
+0.08
+0.1
+0.12
+0.14
+0.16
+0.18
+0.2
+0.22
+0.24
+0.26
+0.28
+ho(MeV)
+ho(MeV)
+160Dy
+168Yb
+162Dy
+166Er
+一
+70
+70
+65
+65
+J(I), J2) (h? MeV-1)
+60
+J(), J(2) (h? MeV-1)
+60
+55
+5
+50
+50
+45
+45
+40
+35 
+40
+30
+35
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+0.18
+0.2
+0.22
+0.24
+0.26
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+0.18
+0.2
+0.22
+0.24
+0.26
+ho(MeV)
+ho(MeV)The IB’s correlation quantities exist between the considered pairs of nuclei which exhibit the same
+identical excitation energies in their ground state bands are listed in Table (2). These quantities include
+the P. Factor, structure Factor SF, Saturation parameter SP, the F-Spin and its projection F0, pairing gaps
+△, and the deformation parameter β. The maximum structure factor for our region of nuclei is SF= 6720.
+It is seen that the ratio NpNn/△ rather than the product NpNn may be a better parameter for studying
+the IB’s. Note that nuclei with symmetric ±F0 values have identical NpNn values. For example the pair
+(160Er and 168Hf) have (Np, Nn) = (14, 10) and (10, 14) respectively, so that NpNn = 140 and F0 = ±1.
+Therefore if any F-spin multiplet has F0 =|Np − Nn|/4, those indicate that the pair of nuclei are similar in
+structure if they have identical (|F0|, NpNn).
+Table 2: The identical band quantities of our eight pairs of nuclei.
+NpNn
+P
+SF
+SP
+|δ|%
+|k|%
+(158Er − 170W )
+112
+5.090
+2464
+0.7317
+1.28
+1.27
+(162Y b − 166Hf)
+120
+5.4545
+2640
+0.7179
+2.94
+2.45
+(156Dy − 172W )
+128
+5.333
+3072
+0.6862
+6.73
+6.28
+(160Er − 168Hf)
+140
+5.833
+3360
+0.6666
+1.35
+1.22
+(158Dy − 170Hf)
+160
+6.1538
+4160
+0.6176
+1.28
+1.27
+(162Er − 166Y b)
+168
+6.6461
+4368
+0.6060
+0.22
+0.20
+(160Dy − 168Y b)
+192
+6.6857
+5376
+0.5555
+0.10
+0.30
+(162Dy − 166Er)
+224
+7.466
+6720
+0.5
+1.29
+1.26
+(Nπ, Nν)
+N
+Nν
+Nπ
+(F, F0)
+△ (MeV)
+NpNn
+△
+(MeV−1)
+βG
+158Er
+(7,4)
+11
+0.571
+(5.5,1.5)
+0.954
+117.4
+0.2173
+170W
+(4,7)
+11
+1.750
+(5.5,-1.5)
+0.920
+121.739
+0.2206
+162Y b
+(6,5)
+11
+0.833
+(5.5,0.5)
+0.942
+127.388
+0.2270
+166Hf
+(5,6)
+11
+1.2
+(5.5,-0.5)
+0.931
+128.893
+0.2254
+156Dy
+(8,4)
+12
+0.5
+(6,2)
+0.960
+133.333
+0.2601
+172W
+(4,8)
+12
+2.0
+(6,-2)
+0.914
+140.043
+0.2459
+160Er
+(7,5)
+12
+0.714
+(6,1)
+0.948
+147.679
+0.2643
+168Hf
+(5,7)
+12
+1.4
+(6,-1)
+0.925
+151.351
+0.2517
+158Dy
+(8,5)
+13
+0.625
+(6.5,1.5)
+0.954
+167.714
+0.3026
+170Hf
+(5,8)
+13
+1.6
+(6.5,-1.5)
+0.920
+173.913
+0.2754
+162Er
+(7,6)
+13
+0.857
+(6.5,0.5)
+0.942
+178.343
+0.2896
+166Y b
+(6,7)
+13
+1.166
+(6.5,-0.5)
+0.931
+180.451
+0.2814
+160Dy
+(8,6)
+14
+0.75
+(7,1)
+0.948
+202.531
+0.3181
+168Y b
+(6,8)
+14
+1.333
+(7,-1)
+0.925
+207.567
+0.2993
+162Dy
+(8,7)
+15
+0.875
+(7.5,0.5)
+0.942
+237.791
+0.3256
+166Er
+(7,8)
+15
+1.142
+(7.5,-0.5)
+0.931
+240.601
+0.3167
+The percentage differences ratios in transition energy δ and the rigid rotor ratio δR between pairs
+of levels in two nuclei are calculated and listed in Table(3) for our eight pairs of IB’s. In spite of the
+parameters NpNn, P, SF and SP are the same for the pairs (156Dy,172 W), this pair is not really identical
+according to their high average percentage differences in transition energies (approximately 6.7%).
+For each nucleus in isotopic chains of 66Dy,68 Er,70 Y b,72 Hf and 74W, the values of lowest dynamical
+moments of inertia J(2)
+lowest were calculated and displayed against the neutron number N in Figure(4) - It
+can be seen that J(2)
+lowest increases with increasing the neutron number N and the difference inJ(2)
+lowest for
+each pair of IB’s is very small ( approximately a horizontal line). As an example of two nuclei that exhibit
+good IB’s, the pair 162
+68 Er(J(2)
+lowest = 31.525¯h2MeV −1) and 166
+70 Y b(J(2)
+lowest = 31.519¯h2MeV −1), that is nearly
+the same J(2)
+lowest.
+12
+
+Table 3: The percentage differences ratios in transition energies δ, the fractional change in transition energies
+divided by the rigid rotor ratio δR and the ratio R = δ/δR for the eight pairs of identical bands.
+Identical pairs
+|δ| = △Eγ
+Eγ2
+%
+δR
+⟨Rδ⟩
+(162Y b − 166Hf)
+2.964
+4.149
+0.714
+(162Er − 166Y b)
+0.415
+4.149
+0.100
+(162Dy − 166Er)
+1.297
+4.149
+0.312
+(160Er − 168Hf)
+1.352
+8.471
+0.159
+(160Dy − 168Y b)
+1.131
+8.471
+0.133
+(158Er − 170W )
+10.826
+12.976
+0.834
+(158Dy − 170Hf)
+1.765
+12.976
+0.136
+(156Dy − 172W )
+7.410
+17.671
+0.419
+Figure 4: The lowest dynamical moment of inertia J(2)
+lowest against the neutron number N for the eight pairs of
+identical bands. The solid line connects each pair and symbols o, ∗, △, �, and ♦ denotes 66Dy,68 Er,70 Y b,72 Hf,
+and 74W respectively.
+We classified our selected pairs of IB’s into four multiplets = (A+4), Z+2), (A+B,Z+4), (A+12,Z+6), and
+(A+16,Z+8) and the percentage differences in transition energies δ = △Eγ/Eγ2 as a function of spin I (up
+to I=10) have been calculated and illustrated Figure (5). It is seen that the pairs of IB’s have approximately
+similar δ ( less than 2.5 %) except the two pairs which include the tungsten isotopes 170,172W where the
+value of δ reaches ∼ 6 − 10% in spite of they have the same NpNn value (NpNn = 112 for 158Er,170 W and
+NpNn = 128 for 156Dy,172 W).
+To further investigation for IB’s we used the SU(3) rotational limit of the IBM to extract the quadrupole
+deformation βIBM for each nucleus. The calculated βIBM is plotted against the ratio Nν/Nπ (where Nν
+and Nπ are the number of valence neutron and valence proton bosons respectively) in Figure(6). It is seen
+that βIBM is the same for each pair of IB’s (horizontal line).
+13
+
+o Dy
+162
+166
+米
+Er
+Dy
+Er
+38
+△Yb
+Hf
+160
+168
+Yb
+36
+170
+34
+158
+Hf
+Dy
+32
+162
+166
+Er
+Yb
+2 MeV-l)
+172
+30
+W
+156
+Dy
+168
+28
+160
+west
+Er*
+JHO
+26
+170
+M
+158 
+Er
+24
+米
+162.
+166.
+JH.
+Yb
+90
+92
+94
+96
+98
+NFigure 5: Percentage difference in transition energies δ = △Eγ/Eγ2 for the eight pairs of multiplet (A+4,Z+2),
+(A+8,Z+4), (A+12,Z+6), and (A+16,Z+8) for Dy, Er, Yb, Hf, and W isotopes. The dashed curve represents the
+ratio of the rigid rotor.
+Figure 6: The quadrupole deformation parameter βIBM was calculated from SU(3) limit of IBM as a function of
+Nν/Nπ for our eight pairs of identical bands.
+14
+
+162Yb - 166Hf
+162Er - 166Yb
+162Dy _ 166Er
+0.07
+0.07
+0.07
+8| = 2.94 %
+8/ = 0.22 %
+[8| = 1.29 %
+0.06
+0.06
+0.06
+0.05
+0.05
+900
+0.04
+ 0.04 
+0.04
+8
+8 0.03
+0.03
+0.03
+0.02
+0.02
+0.02
+0.01
+0.01 
+0.01
+0.01
+10
+10
+160Er_168Hf
+60Dy
+168Yb
+0.14
+[8|= 1.35 %
+0.12
+=.1 %
+d'
+0.1
+0.08
+0.08
+8 0.06
+0.06
+0.04
+0.04
+0.02
+0.02
+６
+10
+6
+10
+160Dy _ 168Yb
+158Dy
+_ 170Hf
+0.14 
+0.25
+0.12
+[8/ = .1 %
+0.2
+[8/ = 1.28 %
+0.1
+0.15
+0.08
+8 0.06
+8 0.1
+0.04
+0.05
+0.02
+0.05
+6
+10
+6
+156Dy - 172W
+[8| =6.73 %
+.25
+0.2
+8 0.150.355
+162Dy
+166Er
+N=15
+0.35
+0.345
+160Dy
+168Yb
+N-14
+0.34
+0.335
+162Er
+166Yb
+170Hf
+N=13
+βIBM
+0.33
+0.325
+156Dy
+160Er
+168Hf
+N=12172W
+0.32
+0.315
+158Er
+162Yb
+166Hf
+G
+0.31
+0.5
+A
+1.5
+2
+Nv/N元Figure 7: Sketch of the potential energy surface PES calculated from the U(5)-SU(3) shape phase transitions of
+IBM with intrinsic coherent state versus the deformation parameters β for the eight pairs of even-even nuclei
+having identical bands.
+For each nucleus, by using the IBM Hamiltonian equation (45) and its eigenvalues equation (53), the
+PES’s have been calculated as a function of deformation parameter β along the axial trajectory γ = 0°, 60°.
+The results are illustrated in Figure(7) and the corresponding calculated parameter of the PES’s A2, A3, A4
+and Ao which are linear combinations of the original parameters ϵ0 and a2 are listed in Table(4). From
+the graphs presented in Figure(7), we observe the similarity in PES’s for each pair of IB’s. All studied
+nuclei are deformed and have rotational characters, the prolate deformation is deeper than the oblate
+deformation.
+15
+
+162Dy  166Er
+162Er-166Yb
+2.5
+1.5k
+2
+1.5
+0.5
+ (KeV)
+(KeV)
+0
+PES
+0.5
+0
+-1
+-0.5
+-1.5
+-2
+-1
+-2
+-1
+0
+-1.5
+-0.50
+0.5
+1
+1.5
+β
+β
+162Yb _ 166Hf
+168Yb
+1.5
+1.5
+0.5
+ (KeV)
+0.5
+0
+0
+-1
+-1
+-1.5
+-1.5
+-2
+-1.5
+-1
+-0.5
+0
+0.5
+1
+1.5
+2
+-2
+-1
+0
+β
+β
+160Er - 168Hf
+158Dy - 170Hf
+2
+2.5
+1.5
+2
+1.5
+PES (KeV)
+(KeV)
+0.5
+0.5
+PES(
+0
+0
+-0.5
+0.5
+-1
+-1.5
+-1.5
+-2
+-1
+0
+1
+2
+-1.5
+-1
+0
+0.5
+1.5
+2
+β
+158Er-170W
+156Dy 172W
+2.5
+0.6
+2
+0.4
+(KeV)
+1.5
+0.2
+1
+0
+-0.4
+-0.5
+0.6
+-1
+-2-1.5
+-1
+-0.5
+0
+0.5
+1.5
+-1.5
+-1
+-0.5
+0.5
+1.5
+β
+βTable 4: Values of the adopted best (PES) parameters A2, A3, A4, A0 ( in KeV ) for the studied eight pairs of
+identical bands. NB is the total number of bosons.
+NB
+A2
+A3
+A4
+A0
+162Dy
+15
+-2.4667
+-0.5863
+1.6665
+-0.3265
+166Er
+15
+-1.6586
+-2.0341
+4.4739
+-0.7875
+162Er
+13
+-5.0526
+-2.5496
+3.7667
+-0.9375
+166Y b
+13
+-5.3088
+-3.1366
+4.0554
+-0.925
+162Y b
+11
+-4.84
+-1.6163
+3.6775
+-0.9
+166Hf
+11
+-2.8484
+-1.9547
+3.9131
+-0.8625
+160Dy
+14
+-1.9568
+-0.8838
+1.1005
+-0.3
+168Y b
+14
+-5.3088
+-3.1366
+4.0554
+-0.925
+160Er
+12
+-3.0403
+-2.3636
+4.1401
+-0.8625
+168Hf
+12
+-3.463
+-2.4694
+4.039
+-0.875
+158Dy
+13
+-1.6288
+-1.1822
+1.0095
+-0.288
+170Hf
+13
+-3.1845
+-3.395
+4.497
+-0.8375
+158Er
+11
+-1.6586
+-2.0541
+4.4739
+-0.7875
+170W
+11
+-0.9761
+-2.4841
+4.7606
+-0.7546
+156Dy
+12
+-1.5043
+-1.2135
+0.9961
+-0.3
+172W
+12
+-0.8852
+-1.4675
+1.0599
+-0.313
+6
+Conclusion
+By using a novel three parameters collective rotational formula (CRF3), the positive parity ground state
+excitation energies are calculated for sixteen nuclei in rare-earth region. The optimized three parameters
+are deduced by using a computer simulated search program in order to obtain a minimum root mean
+square deviation of the calculated excitation energies from the measured ones. The potential energy
+surfaces are calculated by using the sd-version of the interacting boson model.
+The problem of low-spin identical bands in normal deformed nuclei in rare-earth region is treated. We
+have exhibited identical bands in eight pairs of conjugate even-even nuclei of widely dispersed spanning
+as much as sixteen mass unit. Each pair with the same F-spin and projections ±F0 values have identical
+product of valence proton and neutron numbers NpNn values. Also, the values of dynamical moments
+of inertia for each identical band pair are approximately the same. We extracted all the identical band
+symmetry parameters like P-factor, saturation parameter, and structure factor which all depend on Np
+and Nn. The pairing interaction energy, the quadrupole transition probabilities, and the energy ratios are
+also treated.
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+19
+

UNAzT4oBgHgl3EQf0_5k/content/tmp_files/2301.01792v1.pdf.txt

@@ -0,0 +1,1048 @@
+Saturation of fishbone modes by self-generated zonal flows in
+tokamak plasmas
+G. Brochard, C. Liu, X. Wei, W. Heidbrink, Z. Lin, N. Gorelenkov,
+S.D. Pinches, P. Liu, J. H. Nicolau, H. L¨utjens
+Abstract
+Gyrokinetic and kinetic-MHD simulations of n=1 fishbone modes in DIII-D plasmas find that self-generated zonal flows
+can dominate the fishbone saturation. The saturation mechanism is identified in phase space, where the zonal flows
+prevent holes and clumps from persisting or drifting in phase space with mode down-chirping, reducing the wave-particle
+resonant drive. This saturation is confirmed by quantitative agreement with experimental measurements for both mode
+saturation amplitude and neutron emissivity. Zonal flows shearing rate exceeds the drift-wave growth rate, consistent
+with the ITB observed in DIII-D plasmas. The deliberate destabilization of fishbones for the development of high
+performance scenarios in ITER is then proposed.
+Introduction. - Energetic Particles (EPs) in tokamak plas-
+mas can destabilize a large spatial range of instabilities that
+may lead to their outward transport. This is a critical issue
+for burning plasmas as in ITER [1] since such a transport
+can degrade the fusion performances, the plasma confine-
+ment as well as threaten the reactor’s integrity. This trans-
+port therefore needs to be predicted for mitigation strategies
+to be incorporated in plasma scenarios.
+Fortunately, it was discovered theoretically [2][3][4][5] and
+shown numerically [6][7][8][9][10] that instabilities arising at
+the microscopic and mesoscopic scales such as drift-waves
+and Alfv´en eigenmodes (AEs) are able to excite zonal flows
+(ZFs), that can mitigate the saturation amplitudes of these
+modes, and therefore the associated EP transport. Besides
+this mitigation, the destabilisation of zonal flows can gener-
+ate strongly sheared poloidal flows that suppress turbulent
+transport by damping drift-waves turbulence [11], resulting
+in the formation of an internal transport barrier (ITB) that
+greatly enhances plasma confinement [12][13]. Macroscopic
+MHD modes triggered by energetic particles such as the fish-
+bone instability [14][15] however were not self-consistently
+observed to trigger n = m = 0 zonal flows so far.
+The
+mechanism dominating the fishbone saturation was identi-
+fied in nonlinear simulations [16][17][18][19] to be the res-
+onant wave-particle trapping due to kinetic nonlinearities,
+mode-mode nonlinearities playing a secondary role.
+In this Letter, we report the first self-consistent gyrokinetic
+simulations finding fishbone saturation by the self-generated
+zonal flows, in a DIII-D discharge. This discharge is chosen
+for validation purposes to predict the EP transport in a
+ITER baseline prefusion scenario [20]. The zonal flows are
+found to be force-driven by the fishbone and are the main
+mechanism for the fishbone saturation. This mechanism is
+observed for the first time in phase space, where zonal flows
+prevent hole and clump structures from persisting or drift-
+ing in the nonlinear phase, reducing the EP resonant drive.
+This saturation by zonal flows is confirmed by experimental
+measurements, as simulations including zonal flows are able
+to recover quantitatively, for the first time, the mode satura-
+tion amplitude and the neutron emissivity drop. Moreover,
+the shearing rate generated by the fishbone-induced zonal
+flows exceeds the linear growth rate of unstable drift-wave
+modes, similar to recent numerical work based on EAST
+discharges [21]. This strong E × B suppression is consistent
+with the ITB arising experimentally after fishbone bursts in
+the DIII-D discharge. It confirms the long suspected role of
+fishbones in ITB formation [22], fishbone bursts having been
+observed to precede ITBs in ASDEX [23], MAST [24][25],
+HL-2A [4] and EAST [26][27] plasmas. Finally, gyrokinetic
+simulations find that the fishbone-induced EP transport in
+the ITER scenario is marginal, 2% of the core EPs being re-
+distributed, similar to previous studies on the alpha fishbone
+in ITER DT scenarios [19]. The intentional destabilization
+of fishbone modes in ITER scenarios is therefore possibly a
+way to enhance fusion performances.
+Experimental setup.
+- The selected DIII-D discharge
+#178631 [28] has a nearly circular oval shape (elongation
+κ = 1.17, triangularity δ = 0.07) that is limited on the car-
+bon inner wall. The major radius is R0 = 1.74 m, the minor
+radius is a = 0.64 m, the toroidal field is 2.0 T, the plasma
+current is 0.88 MA, and the line-average electron density is
+∼ 2.0 × 1019 m−3. This discharge was chosen primarily be-
+cause it has an accurately known, weakly reversed, q profile
+with q0 = 1.2, qmin = 1.09, and q95 = 3.8 values that re-
+semble the profile predicted for the ITER baseline scenario.
+The deuterium, L-mode plasma is heated by 3.8 MW of 81
+1
+arXiv:2301.01792v1  [physics.plasm-ph]  4 Jan 2023
+
+keV deuterium beams that are injected in the midplane in
+the direction of the plasma current and by 1.0 MW of 2nd
+harmonic, central electron cyclotron heating.
+Numerical setups.
+- The DIII-D discharge #178631 is
+studied numerically mostly through gyrokinetic simulations
+with the GTC code [6][29][30][31], and with kinetic-MHD
+simulations using the M3D-C1 [32][33][34] and XTOR-K
+[35][36][37] codes.
+GTC capability at simulating MHD
+modes was recently verified and validated on DIII-D ex-
+periments [38]. The magnetic configuration is reproduced
+from the EFIT code at t=1580ms. Plasma profiles are ob-
+tained from TRANSP simulations.
+To simulate properly
+MHD modes, the sum of partial pressures need to add up
+to the total pressure in EFIT, which is not always the case
+using TRANSP profiles. To ensure it, the EP pressure is
+constrained as pf = ptot − pi − pe, given that the uncer-
+tainty on EP profiles in TRANSP is the highest. The exper-
+imental NBI distribution is reproduced from the NUBEAM
+code. Such a distribution is described in our first-principle
+simulations with an anisotropic slowing-down model, us-
+ing a zero-th order Legendre expansion [39].
+A superpo-
+sition of three slowing-downs is used to reproduce the in-
+jection energies at nominal, half and third energies.
+The
+critical velocity is artificially set to recover similar gradi-
+ents in the (E, v||/v) phase space. All nonlinear simulations
+cover the whole simulation domain, with an edge buffer after
+ρT =
+�
+ψT /ψT,edge = 0.8 in GTC suppressing equilibrium
+gradients. GTC retains only the n=1 mode in its simula-
+tions, with or without the n=m=0 zonal component, using
+kinetic thermal/fast ions and fluid electrons. M3D-C1 cov-
+ers low n modes n ∈ [0, 2] with both thermal and fast ions
+kinetic effects. Due to the anisotropic nature of the cho-
+sen configuration that has βf/βtot = 54% on axis, XTOR-K
+only evolves the n=1 mode, as the n=0 mode contains both
+equilibrium and perturbed fields in the code, contrarily to
+GTC and M3D-C1.
+XTOR-K treats kinetically only the
+fast ion specie. Convergence studies over spatial grid size,
+time step and number of particles per cell were successfully
+conducted.
+Fishbone mitigation by self-induced zonal flows - The im-
+pact of MHD nonlinearities on the n=1 fishbone were pre-
+viously examined numerically by keeping side-band n=0-4
+modes, highlighting reduction of initial saturation ampli-
+tude [18][21], and generation of n=m=0 sheared poloidal
+flows [19][21].
+The role played specifically by zonal flows
+in fishbone mitigation was however not identified. The ef-
+fects of zonal flows on the fishbone instability are studied
+here self-consistently for the first time with the gyrokinetic
+GTC code.
+A gyrokinetic treatment of zonal flows is es-
+sential as it takes into account their collisionless damping
+[40], which is absent in the kinetic-MHD formalism without
+kinetic thermal ions effects. For the considered DIII-D con-
+figuration, a n=1 fishbone mode is linearly unstable, close to
+marginal stability at pf,thres = 0.8pf, with a growth rate of
+γn=1 = 8.5×104 s−1 and a mode frequency of ω/2π = 17kHz
+in GTC simulations.
+(a)
+(b)
+(c)
+(d)
+Figure 1: Time evolution of (a) the volume-averaged per-
+turbed electrostatic potential eφ/Te (n=0,1), and (b) the
+the n=1 mode frequency ω, with and without zonal flows
+in GTC simulations. The linearly resonant precessional fre-
+quency plus the zonal E × B frequency is also displayed.
+(c) eφ/Te mode structure in the poloidal plane after mode
+saturation. (d) Zonal electric field eEr,00/Te after mode sat-
+uration.
+When the realistic beam is replaced by its equivalent
+Maxwellian distribution, this mode is fully stabilized, high-
+lighting the sensitivity of fishbone instabilities over EP dis-
+tributions.
+Nonlinear simulations are performed with and without the
+n=m=0 component, as illustrated in Fig.1. The time evo-
+lution of the volume-averaged electrostatic potential eφ/Te,
+displayed on Fig.1a, shows that the n=1 fishbone mode is
+able to force-drive the n=m=0 zonal flow, with a growth
+rate twice that of the n=1.
+As shown analytically in [5]
+for TAEs, the mechanism for this zonal flow generation is
+the charge separation induced by nonlinear EP redistribu-
+tion, as opposed to the usual one relying on Reynolds and
+Maxwell stresses [2][3][4][9]. As the n=0 amplitude exceeds
+the n=1 at t=0.13ms, the zonal mode forces the fishbone
+to
+2
+
+Mode amplitude
+n=0, ZFs
+-n=1. without ZFs
+10-1
+-n=1, with ZFs
+e
+10°
+e
+10~3
+n=1
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+0.18
+0.2
+Time (ms)Mode frequency
+24
+22
+一w/2π without ZFs
+-w/2π with ZFs
+20
+d.res
+18
+(ZH) / 
+16
+14
+12
+10
+8
+6
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+0.18
+0.2
+Time (ms)ed
+-/T-, t=0.19ms
+n=1
+ed
+n=1
+e
+0.1
+0.6
+-q=2
+....q=3
+0.4
+0.05
+0.2
+0
+0
+N
+-0.2
+-0.05
+-0.4
+-0.6
+-0.1
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+R (m)eE
+T
+at t=0.19ms and q
+r,00°
+e
+-0.2
+e
+3
+-0.4
+eE.
+2
+-0.6
+-0.8
+0
+0.2
+0.4
+0.6
+0.8
+ld(a)
+(b)
+(c)
+(d)
+Figure 2: Radial envelope of δTe after saturation without
+(a) and with (b) zonal flows in GTC, M3D-C1 and XTOR-
+K simulations, compared to the ECE measurement for the
+DIII-D #178631 discharge. (c) Time evolution of the sim-
+ulated neutron drop, with and without zonal flows. (d) EP
+density profiles in GTC simulations before and after fishbone
+burst.
+saturate at δB/B0 ∼ 2 × 10−3, with a saturation amplitude
+lower by a factor of 4 compared to the case without zonal
+flows. The zonal flows saturates at an even larger amplitude,
+about six times larger than the n=1 mode when including
+zonal flows, with a spontaneous growth after t=0.15ms when
+the n=1 is fully saturated. Such mitigation by zonal flows
+have been theoretically predicted [2][3][5] and numerically
+observed [7][8][9][10] for Alfv´en eigenmodes, but never so
+far for the fishbone instability.
+The zonal flows inclusion
+also lowers significantly the EP diffusivity at saturation,
+from 30 to 4 m2.s−1.
+As shown in Figure 1b, the mode
+frequency down-chirps after the n=1 mode saturation with
+and without zonal flows, which is a typical fishbone signa-
+ture, with similar chirping rates. Just before saturation, the
+case without zonal flows experiences a notable up-chirping
+of the mode frequency, that stops when the mode starts
+saturating. This increase may be attributed to the larger
+mode amplitude near saturation. The n=1 electrostatic po-
+tential and the n=0 radial electric field after saturation at
+t=0.19ms are displayed on Fig.1c-d.
+The n=1 mode fea-
+tures a dominant m=1 harmonic centered around qmin, as
+well as a significant m=2 side-band that vanishes after q = 2.
+The zonal electric field exhibits a macroscopic structure cen-
+tered near qmin as well, which differs from the usual mi-
+croscopic/mesoscopic scale observed with drift-waves/AEs-
+induced zonal flows. This large structure can be attributed
+to the charge separation provoked by the outward drift of
+resonant EPs within the n=1 mode. It leads to a strongly
+sheared poloidal rotation in the electron direction, which is
+opposite to the n=1 fishbone rotation, and a weak toroidal
+rotation.
+This fishbone mitigation by self-generated zonal flows is ex-
+perimentally confirmed by DIII-D measurements as can be
+seen in Fig.2. The δTe envelope obtained from GTC, M3D-
+C1 and XTOR-K nonlinear simulations at saturation are
+compared with the ECE measurements on Fig.2 (a-b), with
+and without zonal flows inclusion. The δTe envelope is de-
+fined here as the n=1 sum of all poloidal harmonics. With-
+out zonal flows, XTOR-K and GTC results have compa-
+rable saturation amplitudes with δTe,max ∼ 500 − 600 eV,
+which are three time larger than the experimental satura-
+tion. The simulated envelopes differ however, GTC results
+having a dominant m=2 harmonic after ρ = 0.34. When
+including zonal flows however, M3D-C1 and GTC satura-
+tion amplitudes at δTe,max ∼ 200 eV match very well with
+the experimental one.
+The significant m=2 harmonic in
+GTC simulations leads to a quantitative agreement with
+the ECE measurement, which provides a nonlinear valida-
+tion for GTC regarding fishbone instabilities, completing the
+linear one obtained in [38] for kink instabilities.
+Nonlin-
+ear scans for the fishbone saturation amplitude performed
+over the radial position and amplitude of qmin recover the
+same significant mitigation by zonal flows. This nonlinear
+validation is further demonstrated by comparing the sim-
+ulated and experimental volume-averaged neutron emissiv-
+ity. In GTC the volume-averaged neutron flux is defined as
+ΓN = ni
+�N
+k δ(x − xf,k)δ(v − vf,k)σ(vf,k)vf,k with ni the
+thermal ion density profile, xk and vk the position and ve-
+locity of EPs and σ the D-D nuclear fusion cross section,
+assuming reasonably that vi ≪ vf.
+As shown on Fig.2c,
+without zonal flows GTC recovers a neutron drop at satura-
+tion of about 6%, much higher than the experimental one at
+δΓN = 0.9% ± 0.3%. When including zonal flows however,
+the neutron drop yields δΓN ∼ 1.1%, which falls within the
+experimental interval. As expected from these neutron drop
+values, the fishbone-induced EP transport with zonal flows
+is rather weak as shown on Fig. 2d, with about 3% of EPs
+inside of the qmin volume redistributed outward. The redis-
+tribution is more significant without zonal flows, as it affects
+15% of EPs in the core plasma.
+Mechanism for fishbone mitigation by zonal flows - Beyond
+the additional dissipation brought by the inclusion of the
+n=0 toroidal mode [7], phase-space analysis reveals that
+zonal flows influence the time evolution of coherent phase
+space structures, impacting the n=1 fishbone mode satu-
+ration. On Fig.3, the instantaneous EP transport ∂tδf is
+displayed in the invariants phase space diagram (Pζ, λ =
+µB0/E) at fixed magnetic momentum µB0 = 45keV before
+3
+
+ T.(eV), without ZFs
+600
+=1.09
+q=2
+min
+500
+-XTOR-K n=1
+-GTC n=1
++ECE
+400
+(eV)
+300
+OS
+200
+100
+0
+0
+0.2
+0.4
+0.6
+0.8
+PT T.(eV), with ZFs
+600
+.
+=1.09
+q=2
+min
+M3D-C1, n=0,1,2
+500
+GTC n=0,1
+ECE
+400
+(eV)
+e
+300
+OS
+200
+100
+0
+0
+0.2
+0.4
+0.6
+0.8
+PTNeutron drop
+0
+Experimental
+-1
+neutron drop
+-2
+Neutron drop (%)
+-Without ZFs
+3
+_With ZFs
+-6
+0.08
+0.1
+0.12
+0.14
+0.16
+0.18
+0.2
+Time (ms)X1018
+EP density profiles
+10
+-t = Oms
+...t = 0.19ms with ZFs
+-t = 0.19ms without ZFs
+8
+6
+EP
+=1.09
+9
+min
+q=2
+n
+4
+:
+2
+0
+0
+0.2
+0.4
+0.6
+0.8
+1
+ldand after the fishbone saturation, with and without zonal
+flows. The instantaneous transport is chosen rather than the
+usual perturbed EP distribution δf as the fishbone mode
+frequency is chirping in the nonlinear phase, which leads
+phase space structure to drift in time. In the linear phase,
+the mode is driven by two resonances, the precessional one
+ω = ωd linked to trapped particles, and a drift-transit one
+ω = ωζ −ωb due to passing particles, with ωζ = qωb +ωd the
+drift frequency and ωb the bounce/transit frequency. The
+passing and trapped phase space zones are separated by a
+black line on the diagrams.
+(a)
+(b)
+(c)
+(d)
+Figure 3: Time evolution of the instantaneous EP transport
+∂tδf without (left) and with (right) zonal flows, in the invari-
+ants (Pζ, λ) phase space diagram at fixed µ (µB0 = 45keV ).
+As can be observed on Fig.3 (a-b), a hole and clump struc-
+ture develops around each resonances in the linear phase,
+indicating a resonant outward EP redistribution. In the non-
+linear phase, the dynamical evolution of these phase space
+structures differ significantly with and without zonals flows.
+In their absence, the hole and clump in the trapped region
+drifts to higher ψ positions, under the influence of the mode
+down-chirping as ωd ∝ 1/ψ, while the one in the passing part
+does not move. However with zonal flows, the phase space
+structure in the trapped region remains static, even thought
+the mode is chirping down, and the hole and clump in the
+passing part vanishes. Such behaviours prevent the fishbone
+mode from affecting resonantly new EPs, which leads to its
+weaker saturation due to the absence of drive.
+These differences in dynamical evolution can be explained
+by the influence of the zonal flows on the EPs wave-
+particle resonance. The perturbed radial electric field as-
+sociated with zonal flows generates an additional drift ve-
+locity δvE,00 = δE00 ×B/B2. This additional velocity leads
+to an E × B drift frequency defined in general geometry
+as ωE,00 = ⟨vE,00 · (∇ζ − q∇θ)⟩ with ⟨· · ·⟩ the bounce-
+average operator, which yields δωE,00 = δEψ = −∇φ00 us-
+ing a thin-orbit width approximation for simplicity. This
+is similar to the so-called ”orbit-squeezing” effects in neo-
+classical theory [41], EPs have an overall decrease of their
+precessional frequency due to their large orbit width over a
+strongly sheared radial electric field. As can be observed on
+Fig.1b, the time evolution of the precessional frequency of
+linearly resonant EPs plus the perturbed E×B frequency at
+ρ = ρqmin matches almost exactly the time evolution of the
+fishbone frequency with zonal flows, which explains why the
+phase space structure in the trapped region remains static.
+The strongly sheared E × B poloidal flow can also perturb
+the EPs transit frequencies due to their large orbit width,
+leading to a resonance detuning and the disappearance of
+the ω = ωζ − ωb hole and clump. Zonal flows are therefore
+able to dominate the fishbone saturation by strongly reduc-
+ing the resonant wave-particle trapping.
+Fishbone-induced ion ITB formation - On top of affecting
+the fishbone mode mitigation, the zonal flows also generate a
+strong shearing rate within ρT ∈ [0.1, 0.5] with γE ∼ 3×105
+s−1.
+High-n electrostatic GTC simulations with kinetic
+trapped electrons were performed for this DIII-D configu-
+ration, finding that the most unstable drift-wave is a TEM
+mode at ρ = 0.4, shown on Fig.4a, with a linear growth rate
+of γT EM = 1.38 × 105 s−1. The shearing rate being larger
+than the TEM growth rate, as displayed on Fig.4b, the sim-
+ulated fishbone mode could then lead to turbulence modu-
+lation by suppression the TEM growth through zonal flows
+[11], confirming the speculated role of fishbones in the emer-
+gence of ITBs [22]. This modulation is supported experi-
+mentally in DIII-D by the charge exchange recombination
+spectroscopy diagnostic. The formation of an ion ITB after
+fishbone bursts occurring at t=1581,1594,1607 and 1615ms
+can indeed be observed on Fig.4 c. The core-increase of Ti
+cannot be explained by additional heating from the beam, as
+it was at constant power since t=300ms, multiple slowing-
+down times before the onset of fishbones. Fishbone bursts
+were also observed to precede ion-ITB in four others DIII-D
+discharges with similar heating power, density, current and
+qmin parameters. Electrons are not affected by the ITB, as
+zonal flows are only able to mitigate ion-scale turbulence
+[42].
+EP transport in ITER prefusion baseline - The GTC code
+having been nonlinearly validated for fishbone simulations,
+it can now be applied to the selected ITER scenario to pre-
+dict the fishbone-induced EP transport. Similar to the DIII-
+D simulations, the NBI beam is reproduced from an analyt-
+ical anisotropic slowing-down distribution.
+4
+
+0,of, with ZFs, t=0.13ms
+3
+1.1
+3
+2
+1
+1
+4
+0.9
+,=1.09
+入=μ/B。
+min
+0
+T
+0.8
+P
+-1
+0.7
+-2
+0.6
+-3
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+0.3
+P 
+wall0,of, no ZFs, t=0.2ms
+60
+..
+1.1
+3
+3
+40
+1
+20
+0.9
+in=1.09
+入=μ/B。
+min
+0
+T
+0.8
+p
+-20
+0.7
+Va
+-40
+0.6
+-60
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+0.3
+P
+wall0,of, with ZFs, t=0.2ms
+...
+20
+1.1
+d
+3=3:
+3
+15
+1
+10
+5
+4
+0.9
+入=μ/B。
+0
+T
+0.8
+P
+-5
+-10
+0.7
+V
+-15
+0.6
+-20
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+0.3
+P
+wall0,of, no ZFs, t=0.13ms
+...8
+3
+1.1
+d
+3
+3
+3
+2
+1
+1
+0.9
+n =1.09
+入=μ/B。
+min
+0
+T
+0.8
+P
+-1
+0.7
+Vab
+-2
+0.6
+-3
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+0.3
+P
+b
+wall(a)
+(b)
+(c)
+Figure 4: a) Electrostatic potential φ of unstable TEM mode
+in the poloidal plane b) Fishbone-induced shearing rate pro-
+file after saturation c) Ti profiles in eV before and after
+fishbone bursts from charge exchange recombination spec-
+troscopy, exhibiting an ion-ITB.
+Linear GTC simulations show that the configuration is un-
+stable to the n=1 fishbone with the realistic beam, with
+a mode growth rate and frequency of γ = 4.4 × 104 s−1
+and ω/2π = 48 kHz, while simulations with equivalent
+Maxwellian distributions find a configuration stable to n=1
+modes.
+Similarly to DIII-D based simulations, the zonal flows inclu-
+sion lowers the n=1 mode saturation. The zonal electric field
+also peaks with negative values close to the qmin surface,
+with a subdominant positive layer further in the plasma.
+Electrostatic GTC simulations were also performed for this
+ITER scenario, finding an unstable TEM mode at ρ = 0.71
+with γT EM = 3 × 104 s−1. At that location, the fishbone-
+induced shearing rate is three times larger than the TEM
+linear growth rate, suggesting that an ion-ITB can also be
+triggered for this ITER scenario.
+However after saturation with zonal flows, the n=1 mode
+abruptly explodes. This numerical instability is due to how
+zonal flows are computed in GTC. The flux-surface averaged
+potential φ00 is computed over the equilibrium flux surfaces,
+which can be a strong assumption in the nonlinear fishbone
+phase as δB/B grows. This computation will soon be modi-
+fied to include the perturbed flux surface to enable long time
+cross-scale simulation between microturbulence and MHD
+modes with GTC. The study of the fishbone-induced EP
+transport for that scenario is then conducted without the
+inclusion of zonal flows to achieve a long nonlinear phase.
+The transport level will then represent the upper-bound as
+zonal flows decrease it significantly.
+After the end of the fishbone burst, the overall redistribu-
+tion within qmin is of order 2% of the initial distribution,
+with both inward and outward EP fluxes due to positive
+and negative EP equilibrium pressure gradient. Such a re-
+distribution tends to marginally flatten the initial pressure
+gradient, the NBI pressure drive being too low to cause large
+redistribution. Overall, the NBI fishbone should not impact
+significantly the plasma heating of this ITER baseline pre-
+fusion, similar to what was shown for the alpha-fishbone in
+the ITER 15MA baseline DT scenario [19].
+Conclusion - Since fishbone oscillations may not cause signif-
+icant EP redistribution in ITER plasmas, it can be of great
+interest to design ITER scenarios to trigger them on purpose
+rather than avoiding them. As was shown in this Letter,
+fishbones can generate zonal flows which present two ad-
+vantages : 1) mitigating the fishbone saturation and its im-
+pact on EP transport and 2) creating strong shearing rates
+that can damp drift-wave instabilities and hence reducing
+the turbulent transport. While it was observed here and in
+several tokamak discharges [23][24][43][26][27][25] that fish-
+bone oscillations led to ITBs formation, that was not the
+case in some others such as JET [44][45][42], despite efforts
+to reproduce the fishbone-induced ITB formation observed
+in ASDEX plasmas [23].
+It may therefore exist a para-
+metric dependency for the fishbone instability that controls
+the emergence of strongly sheared fishbone-induced zonal
+flows. The numerical identification and experimental obser-
+vation of such a dependency could enable the creation of
+high-performance scenarios, of crucial importance for ITER
+burning plasmas.
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+7
+

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+Thermal curvature perturbations
+in thermal inflation
+Mar Bastero-Gil,a Joaquim M. Gomes,b and Jo˜ao G. Rosac
+aDepartamento de F´ısica Te´orica y del Cosmos, Universidad de Granada,
+Granada-18071, Spain
+bDepartment of Mathematical Sciences, University of Liverpool,
+Liverpool L69 7ZL, United Kingdom
+cUniv Coimbra, Faculdade de Ciˆencias e Tecnologia da Universidade de Coimbra and CFisUC,
+Rua Larga, 3004-516 Coimbra, Portugal
+E-mail: mbg@ugr.es, j.m.gomes@liverpool.ac.uk, jgrosa@uc.pt
+Abstract. We compute the power spectrum of super-horizon curvature perturbations gen-
+erated during a late period of thermal inflation, taking into account fluctuation-dissipation
+effects resulting from the scalar flaton field’s interactions with the ambient radiation bath.
+We find that, at the onset of thermal inflation, the flaton field may reach an equilibrium
+with the radiation bath even for relatively small coupling constants, maintaining a spectrum
+of thermal fluctuations until the critical temperature Tc, below which thermal effects stop
+holding the field at the false potential minimum. This enhances the field variance compared
+to purely quantum fluctuations, therefore increasing the average energy density during ther-
+mal inflation and damping the induced curvature perturbations. In particular, we find that
+this inhibits the later formation of primordial black holes, at least on scales that leave the
+horizon for T > Tc. The larger thermal field variance also reduces the duration of a period
+of fast-roll inflation below Tc, as the field rolls to the true potential minimum, which should
+also affect the generation of (large) curvature perturbations on even smaller scales.
+arXiv:2301.11666v1  [hep-ph]  27 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+Thermal inflation
+2
+3
+Curvature Perturbations
+5
+4
+Comparison between the thermal and quantum power spectra
+10
+5
+Conclusion
+12
+A Evolution of the temperature during thermal inflation
+13
+B Field correlation functions
+14
+1
+Introduction
+It is widely believed that the universe went through a period of inflation in its early stages
+[1–4], thus explaining its observed homogeneity and isotropy on large scales, as well as its
+apparently small spatial curvature.
+Most importantly, inflation in principle provided the
+seeds for the small curvature perturbations that grew into the large-scale structure that we
+observe in the Universe.
+Although the simplest models postulate a single period of slow-roll inflation lasting for at
+least 50-60 e-folds after the largest presently observable scales became super-horizon, there is
+a priori no reason to exclude scenarios with multiple inflation periods with different dynamics.
+In particular, it is well known that reheating after inflation may lead to the production of e.g.
+topological defects if the associated reheating temperature exceeds the grand unification scale
+(∼ 1016 GeV) [5] or other unwanted relics such as moduli or gravitinos in supersymmetric
+(SUSY) models [6–8]. Such relics could have overclosed the Universe or spoiled the successful
+predictions of primordial nucleosynthesis through their late decay [9]. This and the fact that
+currently there is no evidence for such relics motivates considering scenarios with additional
+inflationary stages that could have diluted their abundances [10–14].
+One of the most appealing possibilities is a late period of thermal inflation, where a
+scalar flaton field is trapped in a false vacuum by thermal effects above a certain critical
+temperature. Candidates to drive such a secondary inflation period are ubiquitous in SUSY
+and supergravity theories, in particular given the many flat directions in the scalar potential
+that characterize such models at the renormalizable level [15]. The spectrum of curvature
+perturbations generated during such a period (or possibly multiple periods) need not be
+nearly as scale-invariant as the one generated by the first period of slow-roll inflation, during
+which the large-scale perturbations observable in the Cosmic Microwave Background (CMB)
+anisotropies became super-horizon. In fact, this spectrum was recently computed in [16],
+where it was shown that large curvature perturbations could have been generated (on small
+scales) during a period of thermal inflation and a fast roll inflation period [17] that potentially
+followed it once thermal effects stopped trapping the field in the false vacuum state. These
+large curvature/density perturbations could have then collapsed into a significant population
+of primordial black holes upon horizon-reentry later in the radiation-dominated epoch. Such a
+– 1 –
+
+possibility has attracted a substantial interest in the recent literature given the latter’s appeal
+as dark matter candidates and the possibility that these may explain the recent LIGO/Virgo
+detections of heavy black hole binaries (see e.g. [18]).
+The analysis in [16] considered, however, only the part of the curvature spectrum gen-
+erated by quantum fluctuations of the flaton scalar field. Since thermal effects are a crucial
+aspect in the dynamics of thermal inflation, one should investigate whether thermal fluctua-
+tions also play an important role, which is our goal with this work. We note, in particular,
+that the flaton field is trapped in a false vacuum at temperatures above a certain critical tem-
+perature, as we review in the next section, due to the large thermal mass resulting from its
+interactions with the ambient thermal bath. It is well-known that such interactions also lead
+to fluctuation-dissipation effects, resulting in an effective Langevin-like equation describing
+the dynamics of the scalar field. Such effects have been thoroughly analyzed in the context
+of warm inflation scenarios [19–36], in setting initial conditions for slow-roll inflation in a
+pre-inflationary radiation epoch [37], and in cosmological phase transitions both after and
+during (warm) inflation [38,39]. Our objective is then to apply the techniques developed in
+these contexts to the case of thermal inflation, and investigate their role in the generation of
+curvature perturbations during this period.
+Surprisingly, we find that for thermal flaton fluctuations the amplitude of the curvature
+power spectrum is suppressed with respect to the purely quantum case analyzed in [16], at
+least for scales exiting the horizon before the temperature decreases below the critical value.
+This is essentially due to the fact that, as we will show, thermal effects, by enhancing flaton
+density fluctuations, also increase the time-dependent part of the average energy density
+during thermal inflation. This effect overcomes the enhancement of individual perturbation
+modes, therefore suppressing the corresponding power spectrum.
+This work is organized as follows. We will start by constructing a generic model for
+thermal inflation in Section 2. The curvature perturbation spectrum induced by the thermal
+flaton fluctuations is computed in Section 3. In Section 4 we compare our result with the
+purely quantum computation performed in [16], discussing and summarizing our conclusions
+in Section 5. We use natural units throughout this work, ℏ = c = kB = 1 and the reduced
+Planck mass MP = 2.435 × 1018 GeV.
+2
+Thermal inflation
+Let us consider a scalar field φ interacting with a thermal radiation bath at temperature
+T, with energy density ρR = π2
+30g∗T 4, where g∗ denotes the number of relativistic degrees
+of freedom. For concreteness, we consider a radiation bath made up of NF Dirac fermion
+species ψi, which interact with the scalar field through Yukawa interactions with universal
+coupling constant g:
+LY = −gφ
+NF
+�
+i=1
+¯ψiψi .
+(2.1)
+We take the mass of the fermions mψi ≪ T, so that they can be treated as relativistic degrees
+of freedom, but such that mψi > H so that flat quantum field theory calculations for the
+decay width of scalars into fermions are valid [37].
+We assume that the scalar field φ corresponds to a renormalizable flat direction, or flaton
+field, common in several SUSY/supergravity scenarios [11,12,14,40,41], such that its potential
+is only lifted by soft terms such as a mass term from SUSY breaking, and non-renormalizable
+– 2 –
+
+terms. We are interested in the case where the squared mass term is negative, such that the
+field acquires a large expectation value M0 at zero temperature from the latter’s interplay
+with the non-renormalizable operators.
+The interaction with the radiation bath induces,
+however, a thermal mass correction such that the field’s effective mass is of the form [42]:
+m2
+eff = α2T 2 − m2 ,
+(2.2)
+where m corresponds to the zero temperature (tachyonic) mass and α is the effective coupling
+to the thermal bath. For the Yukawa interactions described above we have α2 = g2NF /6 at
+one-loop order. This implies, in particular, that for temperatures above the critical value,
+Tc ≡ m/α, the origin is a stable minimum of the scalar potential, whereas for lower tempera-
+tures the minimum is non-trivial and asymptotes to M0 in the limit of vanishing temperature.
+The origin thus constitutes a false vacuum state, near which we may write the scalar potential
+as:
+V (φ) = 1
+3M2
+0 m2 + 1
+2m2
+effφ2 + · · · ,
+(2.3)
+where for concreteness we have chosen the constant term such that, if the leading non-
+renormalizable term is ∼ φ6 the cosmological constant vanishes at the minimum, V (φ =
+M0) = 0, although this is not crucial to our analysis. For typical flat directions, M0 ≫ m,
+since the scale at which the non-renormalizable operators become relevant is generically large
+(around the grand unification or even the Planck scale).
+If, after the first period of slow-roll inflation, the Universe is reheated to attain a tem-
+perature T > Tc, the flaton field will thus be driven to the false minimum at the origin by
+Hubble friction, where it is trapped and gives a contribution V0 = M2
+0 m2/3 to the vacuum
+energy. Since the temperature drops as the universe expands, i.e. ρR ∝ a−4, eventually this
+vacuum energy may become dominant, thus triggering a new period of inflation, with expan-
+sion rate H ≃ mM0/3MP ≲ m. Thermal inflation thus begins when the temperature drops
+below:
+Ti =
+� 10
+g∗π2
+� 1
+4 �
+M0m .
+(2.4)
+Assuming that there is no significant entropy production during thermal inflation, as we
+confirm in Appendix A, the temperature of the radiation bath drops as T ∝ a−1 during
+thermal inflation, eventually reaching the critical value Tc below which the minimum at the
+origin is destabilized. The nature of the phase transition (or smooth crossover) that ensues is
+model-dependent and irrelevant to our discussion (see e.g. [43]), since we are mostly interested
+in what happens for temperatures Tc < T < Ti.
+We note that thermal inflation is only possible if the flaton field has a non-negligible
+interaction with the thermal bath, and in particular Ti > Tc imposes:
+α >
+�g∗π2
+10
+� 1
+4 � m
+M0
+.
+(2.5)
+For instance, for m ∼ 10 TeV and M0 ∼ MP , this imposes the lower bound α ≳ 10−7 for
+g∗ = 10−100. Although this may not seem too restrictive, we note that the number of e-folds
+of thermal inflation is given by:
+N(TI)
+e
+= ln
+�Ti
+Tc
+�
+= 1
+2 ln
+�M0
+m
+�
++ 1
+4 ln
+� 10
+π2g∗
+�
++ ln(α) .
+(2.6)
+– 3 –
+
+For the reference values given above, we see that a period of thermal inflation lasting more
+than 10 e-folds is only possible for α ≳ 0.01, with even larger effective couplings required for
+scenarios with a smaller hierarchy between the mass scales m and M0.
+We note that inflation does not necessarily end when the temperature falls below Tc,
+since expansion only stops accelerating once the flaton’s kinetic energy surpasses its potential
+energy. Below Tc the field develops a tachyonic instability, since m2
+eff ≃ −m2 < 0 once T ≪ Tc,
+and its value moves away from the origin as ∼ emt ∼ e
+m
+H Ne for H ≲ m, and there may be
+a period of fast-roll inflation [17] until the field gets close to the minimum at M0 and its
+kinetic energy takes over. Note that, in the opposite regime m ≲ H, thermal inflation would
+be followed by an additional period of slow-roll inflation, but we will not consider this regime
+in our discussion. The duration of the fast-roll period is, of course, model dependent and,
+moreover, dependent on the mean field value at the critical temperature.
+In [16,17] it was shown that this period may last for as much as, or even longer than, the
+thermal inflation period for H/m ≲ 1, depending on the flaton’s mass value. This assumed,
+however, that the mean field value at the critical temperature is set by quantum fluctuations,
+which as we will see is not necessarily the case. In particular, thermal fluctuations typically
+enhance the field’s variance at Tc, therefore reducing the duration of the subsequent fast-
+roll period.
+For this reason, we will restrict our analysis to the thermal inflation period
+(Tc < T < Ti), discussing the implications of our results to the subsequent cosmological
+evolution at the end of our discussion.
+Independently of whether or not there is a significant period of inflation below Tc, the
+field will eventually begin oscillating about the minimum of its potential and decay away
+through the Yukawa interactions in Eq. (2.1) [44]. Although we do not specify the exact
+nature of the fermion fields in the thermal bath, since we are only modelling the interactions
+between the flaton and the ambient radiation and our discussion is largely independent of the
+particular interactions considered, it is implicit that such interactions will eventually lead to
+the reheating of the Standard Model degrees of freedom at temperatures exceeding at least
+a few MeV to ensure the correct conditions for primordial nucleosynthesis.
+We note that having late thermal inflation and fast-roll inflation periods alters the
+predictions of inflationary cosmology [45], since the largest CMB scales leave the horizon
+50-60 e-folds before the end of the full inflationary epoch, including the primary slow-roll
+inflation period, which therefore must necessarily be shorter.
+Although the leading effect of the interactions between the flaton and the thermal bath
+is the thermal mass correction responsible for its trapping at the origin, it also induces
+fluctuation-dissipation effects in the flaton’s dynamics that, as we will see, can play an im-
+portant role in the evolution of field perturbations during thermal inflation. These have been
+considered in [46] to analyze the nature of the phase transition at Tc, but their effects on
+the associated spectrum of curvature perturbations have so far been overlooked. To study
+them, we consider the full Langevin-like equation for the flaton field modes φk of comoving
+momentum k, which can be obtained through standard techniques in linear response theory
+assuming the ambient radiation bath is close to an equilibrium state, and is given by (see
+e.g. [25,47]):
+¨φk + (3H + Γφ) ˙φk + ω2
+kφk = ξk ,
+(2.7)
+where ω2
+k = k2/a2 +m2
+eff and Γφ is the dissipation coefficient, which for a field oscillating near
+a local minimum of its potential (in this case the false minimum at the origin for T > Tc)
+coincides with its finite-temperature decay width [48]. On the right hand side of (2.7), ξk
+is a stochastic noise term which encodes the randomness of the field’s interactions with the
+– 4 –
+
+thermal bath. For modes with physical momentum p = k/a ≲ πT it is well approximated by
+a gaussian white noise term with a two-point correlator given by the fluctuation-dissipation
+relation [46,49].:
+⟨ξk(t)ξk′(t′)⟩ = 2ΓφT (2π)3
+a3
+δ3(k + k′)δ(t − t′) .
+(2.8)
+We note that physically this is reminiscent of the Brownian motion of a heavy particle in an
+gas, for which random collisions with the gas molecules induce an effective friction that damps
+its motion. However, the particle never actually comes to rest due to the very same random
+collisions, eventually reaching an equilibrium with the gas. We expect something very similar
+to occur to the flaton field modes, with the combined effects of dissipation (Γφ) and thermal
+fluctuations (ξk) driving the field towards a thermal equilibrium with the radiation bath.
+This behaviour has been observed for scalar fields interacting with a radiation bath both
+in an inflationary and non-inflationary context [37, 39], so we anticipate that the same will
+occur in the case of thermal inflation.
+At finite temperature the flaton decay width into relativistic fermions is given by [27,37]:
+Γφ(p) = 3m2
+effα2
+4πωp
+�
+1 + 2T
+p ln
+�1 + exp(− ω+
+T )
+1 + exp(− ω−
+T )
+��
+,
+(2.9)
+where ω± = |ωp±p|
+2
+and we have neglected the mass of the fermions, T ≫ mψi. Note that
+fermions acquire a mass through their interaction with the flaton field but, as we will obtain
+bellow,
+�
+⟨φ2⟩ ≲ T for perturbative couplings.
+Since the thermal bath will excite field modes p ≲ T and meff ≲ T, the decay width can
+be well approximated by:
+Γφ ≃ 3m2
+effα2
+16πT
+≃ 3α4
+16πT ,
+(2.10)
+where in the last step we have used meff ≃ αT for T ≳ Tc. At the onset of thermal inflation,
+we then have:
+Γφ
+H
+����
+Ti
+≃
+9
+16π
+� 10
+g∗π2 ,
+�1/4
+α4
+MP
+√M0m
+≃ 2.3g−1/4
+∗
+� α
+0.03
+�4 �MP
+M0
+�1/2 �
+m
+10 TeV
+�−1/2
+,
+(2.11)
+so that we expect dissipative effects to play an important role in the field’s dynamics roughly
+for the same range of the effective coupling α leading to a period of thermal inflation lasting
+for more than 10 e-folds, as we have seen above. In the next section we compute the thermal
+field correlators and associated curvature perturbation power spectrum to better quantify
+this statement.
+3
+Curvature Perturbations
+Let us consider the gauge-invariant curvature perturbation on uniform density hypersurfaces,
+which in the flat gauge can be written as [50,51]:
+ζ = − H
+˙⟨ρ⟩
+δρ ,
+(3.1)
+– 5 –
+
+where the perturbation of a generic function is given by δf(t, x) ≡ f(t, x) − ⟨f(t, x)⟩, and
+brackets denote its thermal averaged value. The dimensionless power spectrum of ζ is defined
+as [16],
+∆2
+ζ(k) = k3
+2π2
+�
+d3x exp(−ik · x) ⟨ζ(0)ζ(x)⟩ ,
+=
+2k3
+(2π)2
+� H
+˙⟨ρ⟩
+�2 �
+d3x exp(−ik · x) ⟨δρ(0)δρ(x)⟩ .
+(3.2)
+The total energy density ρ during thermal inflation includes the contributions from both the
+flaton field and the radiation fluid [52]:
+ρ = ρφ + ρR = 1
+2
+˙φ2 + V (φ) + 1
+2a−2(t)∂iφ∂iφ + π2
+30g∗T 4 ,
+(3.3)
+and so we have
+⟨ρ⟩ = π2
+30g∗T 4 + 1
+3m2M2
+0 + 1
+2m2
+eff ⟨φ2⟩ + 1
+2 ⟨ ˙φ2⟩ + 1
+2a−2 ⟨∂iφ∂iφ⟩ ,
+(3.4a)
+δρ = 1
+2m2
+effδ(φ2) + 1
+2δ( ˙φ2) + 1
+2a−2δ(∂iφ∂iφ) .
+(3.4b)
+Since density perturbations involve perturbations of quadratic functions of the field and its
+derivatives, the power spectrum, Eq. (3.2), involves contributions of the form:
+⟨δ(Xi(0)2)δ(Xj(x)2)⟩ = ⟨Xi(0)2Xj(x)2⟩ − ⟨Xi(0)2⟩ ⟨Xj(x)2⟩ ,
+(3.5)
+where Xi generically denotes the field perturbations and their derivatives. The first term
+on the right-hand side corresponds to 4th moments involving the gaussian variables Xi.
+According to Isserlis’ theorem [53] it is possible to write a kth moment of zero-average
+gaussian variables in terms of their variances. Thus, the correlators can be simply written
+as [54]:
+⟨δ(Xi(0)2)δ(Xj(x)2)⟩ = 2 ⟨Xi(0)Xj(x)⟩2 .
+(3.6)
+The two-point correlation function for the energy density is then:
+⟨δρ(0)δρ(x)⟩ = m4
+eff
+2
+⟨φ(0)φ(x)⟩2 + m2
+eff ⟨φ(0) ˙φ(x)⟩
+2 + a−2m2
+eff ⟨φ(0)∂iφ(x)⟩2 ,
++ 1
+2 ⟨ ˙φ(0) ˙φ(x)⟩
+2 + a−2 ⟨ ˙φ(0)∂iφ(x)⟩
+2 + a−4
+2
+⟨∂iφ(0)∂jφ(x)⟩2 ,
+(3.7)
+that is, contributions from all possible correlation functions involving φ, ˙φ and ∂iφ.
+We note that we are interested in computing the curvature perturbation power spectrum
+on super-horizon scales, k ≪ aH. To do this we need to compute the field variance ⟨φ2⟩ and
+the average kinetic and gradient energies appearing in Eq. (3.4a), which involve integrating
+over all thermally excited field modes. Since the noise term correlator is exponentially sup-
+pressed for physical momentum scales p ≳ πT [46], we use this value as a hard cutoff. This
+can be translated into a comoving momentum cutoff kc = πTc if we set a(Tc) = 1, following
+the conventions of [16] to allow for a better comparison with the purely quantum calculation.
+To compute the power spectrum we need the three combinations of the correlations
+between φk and ˙φk, i.e. ⟨φkφk⟩, ⟨φk ˙φk⟩ and ⟨ ˙φk ˙φk⟩. These are the building blocks of all the
+remaining correlation functions involved in the power spectrum. We will explicitly compute
+– 6 –
+
+the correlator of the field modes and list all others in Appendix B as their computation
+follows similar steps.
+The equal-time two-point correlation function of the field modes can be written in terms
+of the Green’s function associated with (2.7) and the noise correlator:
+⟨φk(z)φk′(z)⟩ = H−4
+� z
+zi
+ds1
+� z
+zi
+ds2 s−2
+1 s−2
+2 Gs(z, s1)Gs(z, s2) ⟨ξk(s1)ξk′(s2)⟩ ,
+(3.8)
+where we have traded the time-dependence for a dependence on the variable z = T/H, with
+zi = Ti/H. Note that z ∝ a−1 during thermal inflation, so that it is a decreasing function
+of time. We have ignored the contributions from the homogeneous solutions of (2.7) since,
+as we will see bellow, they quickly become subdominant. These are required, however, to
+compute the Green’s function, which is given by the usual expression:
+Gs(z, s) = φ(1)
+k (s)φ(2)
+k (z) − φ(1)
+k (z)φ(2)
+k (s)
+W(φ(1)
+k , φ(2)
+k )(s)
+,
+(3.9)
+where φ(1)
+k
+and φ(2)
+k
+are the homogeneous solutions of equation (2.7) and W denotes their
+Wronskian.
+During most of thermal inflation, except for temperatures close to the critical value,
+the thermal mass dominates over the field’s zero temperature mass, αT ≫ m. This allows
+us to compute analytically the field modes, and thus obtain the field’s two-point correlation
+function with a decay width of the form (2.10).
+The homogeneous equation of motion for the flaton field modes (2.7) can be written in
+terms of the z variable as:
+z2φ′′
+k − z (2 + γz) φ′
+k + z2¯ω2
+kφk = 0 ,
+(3.10)
+where ¯ω2
+k ≡ ω2
+k/T 2 ≃ k2/T 2
+c + α2 and γ ≡ 3α4/16π, such that Γφ/H = γz. Let us define
+φk = zeγz/2χk, such that:
+χ′′
+k +
+�
+¯ω2
+k − γ2
+4 − γ
+z − 2
+z2
+�
+χk = 0 .
+(3.11)
+Even though we can express the exact solutions of the above equation in terms of Whittaker
+functions [55], it is more instructive to note that, since γ ≪ ¯ω2
+k for α ≲ 1 and z > zc =
+m/αH > α−1 > 1, we may neglect all the terms inside the brackets in Eq. (3.11) except for
+the one involving ¯ω2
+k to a good approximation. This means that the homogeneous solutions
+are approximately given by:
+φ(1)
+k (z) ≃ ze
+γ
+2 z sin(¯ωkz) ,
+φ(2)
+k (z) ≃ ze
+γ
+2 z cos(¯ωkz) ,
+(3.12)
+thus constituting oscillatory functions in the z variable with an amplitude decreasing due to
+both Hubble expansion (z ∝ a−1) and the field’s decay into the light fermions. This yields
+the Green’s function:
+Gs(z, s) = 1
+¯ωk
+z
+s exp
+�γ
+2(z − s)
+�
+sin
+�
+¯ωk(z − s)
+�
+.
+(3.13)
+– 7 –
+
+The noise correlation function can be written in terms of the z variable as:
+⟨ξk(z1)ξk′(z2)⟩ = 2Hz1ΓφT (2π)3
+a3
+δ3(k + k′)δ(z1 − z2) ,
+≃ 2γH3z6
+1
+(2π)3
+z3c
+δ3(k + k′)δ(z1 − z2) ,
+(3.14)
+where in the second line we used the dominance of the thermal mass for T > Tc.
+We may now substitute Eqs. (3.13) and (3.14) into Eq. (3.8) to obtain the field’s two-
+point correlation function:
+⟨φk(z)φk′(z)⟩ = (2π)3δ3(k + k′) T
+a3ω2
+k
+(1 − δ) ,
+δ = exp
+�
+− 3α4
+16π
+Ti
+H
+�
+1 − T
+Ti
+��
+,
+(3.15)
+where again we used that ¯ωk ≫ γ. Note that for Γφ/H(Ti) ≳ 1, we have δ ≪ 1 for all
+temperatures below Ti (but above Tc), thus yielding a thermal equilibrium distribution for
+the field modes that is independent of the decay width. This means that if the field decays
+efficiently at the onset of thermal inflation it will attain an equilibrium distribution that
+simplify redshifts with expansion (with corresponding decrease in temperature).
+This is
+a generic result obtained in other cosmological contexts [37, 39] that we now recover also
+within thermal inflation – it simply states that if the field interacts significantly with the
+thermal bath at some point during its evolution it reaches a near-thermal configuration that
+is subsequently maintained unless there is some significant change in the field’s properties
+(in our case the tachyonic instability just below the critical temperature).
+We note that the two-point correlation function vanishes at the onset of thermal inflation
+by construction, since the integral Eq. (3.8) is zero at z = zi.
+This assumes that field
+modes were not excited when thermal inflation begins, which need not be the case since
+interactions with the thermal bath are present in the prior radiation-dominated epoch. If field
+modes thermalize before its vacuum energy becomes dominant, Eq. (3.15) will nevertheless
+hold (with δ ≃ 0), since this result is also valid for a radiation-dominated cosmological
+background [37]. However, we note that during the radiation era Γφ/H ∝ T/H ∝ a, while
+Γφ/H ∝ a−1 during thermal inflation, so that this ratio attains its maximum value at the
+onset of thermal inflation. Recalling Eq. (2.11), we conclude that α ≳ 0.01 is required for
+field thermalization if the zero temperature mass m is not far from the TeV scale at which
+new physics may be expected. As discussed in the previous section, this is exactly the regime
+where a period of thermal inflation lasting more than 10 e-folds (and which can in particular
+sufficiently dilute unwanted relics of the first reheating process) can occur. We will thus
+henceforth focus our analysis on this parametric regime, in which the field thermalizes either
+before or at the onset of the thermal inflation epoch.
+We may now use Eq. (3.15) to compute the field variance and related correlation func-
+tions, as we detail in Appendix B. We obtain for the total average energy density:
+⟨ρ⟩ = π2
+30
+�
+g∗ + 5
+π(1 − δ)
+�
+T 4 + 1
+3m2M2
+0 ,
+(3.16)
+where we note that the field contributes essentially as an additional bosonic degree of freedom
+to the radiation energy density if thermalization is efficient (δ ≪ 1). Its contribution is not
+exactly one degree of freedom since we have considered a hard-cutoff on the momentum of
+the modes that are excited by interactions with the thermal bath at kc = πTc.
+This is
+– 8 –
+
+only an approximation to the smooth cutoff associated with the noise correlator [46], which
+nevertheless captures the essential physics of the problem.
+Using the values of each component of the power spectrum (3.5) given in Appendix B,
+the density perturbations are:
+�
+d3x exp(−ik · x) ⟨δρ(0)δρ(x)⟩ ≈ πT 5
+6a3
+�
+1 + 3
+� 3α4
+32π2
+�2�
+1 − α
+π arctan
+�π
+α
+� ��
+(1 − δ)2 ,
+(3.17)
+to leading order on super-horizon scales k < aH < αTc. We note that the first term within
+the square brackets dominates over the second one. This then yields for the power spectrum
+on super-horizon scales:
+∆2
+ζ
+(therm)(k) =
+150
+(2π)5
+k3
+T 3c
+(1 − δ)2
+�
+g∗ + 5
+π(1 − δ) − 5
+π
+3α4
+64π
+T
+H δ
+�2 ,
+≃
+150
+(2π)5
+α3
+g2
+∗,f
+�H
+m
+�3� k
+kc
+�3
+,
+(3.18)
+where in the second line we have taken the prompt thermalization limit, i.e. δ ≪ 1, in
+which case the flaton field contributes to the total number of relativistic degrees of freedom,
+given by g∗,f ≃ g∗ + 5/π. Note that this result is time-independent, reflecting the freeze-out
+of curvature perturbations on super-horizon scales and thus the single-fluid nature of the
+dynamics, i.e. the fact that the flaton field thermalized with the radiation bath.
+The power spectrum is blue-tilted so its maximum value is attained for the last scale to
+leave the horizon during thermal inflation, i.e. kc = H which leaves at T = Tc. Although our
+calculation assumes the dominance of the thermal piece of the flaton’s mass, an approximation
+that breaks down close to the critical temperature, we may extrapolate our results with a
+reasonable accuracy to kc, thus yielding an upper bound on the power spectrum of scales
+leaving the horizon before the phase transition, in the thermal equilibrium limit:
+∆2
+ζ
+(therm, max)(k) ≃ 150
+(2π)5
+α3
+g2
+∗,f
+�H
+m
+�3
+.
+(3.19)
+The power spectrum would, thus, be maximized for g∗,f ∼ α ∼ H
+m ∼ 1, yielding ∆2
+ζ
+(therm, max) ∼
+10−2, but in realistic scenarios with perturbative couplings and at least one fermionic degree
+of freedom in the ambient thermal bath the power spectrum should have a parametrically
+smaller amplitude.
+Hence, if the flaton field has significant interactions with the radiation bath, α ≳ 0.01 (as
+expected in scenarios with a significant number of e-folds of thermal inflation), the thermal
+nature of its fluctuations suppresses the amplitude of the induced curvature perturbations
+on super-horizon scales, which is the main result of this work. While this may seem sur-
+prising, given that thermal fluctuations generically have a larger amplitude than quantum
+vacuum fluctuations (as considered in [16]), it has a simple physical explanation: fluctuation-
+dissipation effects increase not only the density fluctuations on super-horizon scales but also
+the field variance and the average gradient and kinetic energies, thus, the average energy den-
+sity. The latter effect turns out to be more significant and, hence, decreases the amplitude
+of the associated curvature power spectrum with respect to the quantum case.
+– 9 –
+
+A relevant consequence of our analysis is that, in realistic scenarios, we do not expect the
+amplitude of the curvature power spectrum to be sufficiently large to lead to the formation of
+primordial black holes, which would require ∆2
+ζ ≳ 10−2 [56–58], at least on scales that become
+super-horizon above the critical temperature. This motivates a better comparison with the
+results obtained in [16] for quantum flaton fluctuations, where larger curvature perturbations
+were obtained. We pursue this comparison in the next Section.
+4
+Comparison between the thermal and quantum power spectra
+The linear approximation to the quantum power spectrum is given in [16] by:
+∆2
+ζ
+(quan)(k) =
+4
+√π
+Γ(ν)
+ν2Γ
+�
+ν − 3
+2
+�
+�H
+m
+�3−2ν� k
+kc
+�3�� k
+kc
+�2
++ m2
+H2
+�−ν
+,
+(4.1)
+where ν =
+�
+m2/H2 + 9/4. To better compare our results with those obtained assuming
+purely quantum flaton fluctuations in [16], we plot both power spectra as a function of
+comoving momentum in Figure 1. We show the case of H/m = 0.3 (which according to the
+analysis in [16] yields all dark matter in the form of primordial black holes) and taking α = 1,
+NF = 1 and δ = 0 to maximize the thermal power spectrum. We note that the thermal power
+spectrum is only shown up to k = kc, since our calculation is only valid for modes that exit
+the horizon before the phase transition; whereas the quantum calculation can be extended to
+larger momentum, assuming a subsequent period of fast-roll inflation as mentioned earlier.
+quantum
+thermal
+0.5
+1
+5
+10
+10-5
+10-4
+10-3
+10-2
+k / kc
+Δζ
+2
+Figure 1. The quantum power spectrum (blue) and the thermal power spectrum (red) as a function
+of k for H/m = 0.3, α = 1, mNF = 1 and δ = 0.
+As one can clearly see in this figure, thermal fluctuations significantly suppress the cur-
+vature perturbation spectrum with respect to the quantum case, for the reasons explained
+in the above section. Furthermore, whereas quantum vacuum fluctuations may yield a suffi-
+ciently large amplitude to lead to primordial black hole formation, a thermalized flaton field
+induces much smaller perturbations, although they may nevertheless exceed the even smaller
+fluctuations observed on large scales in the CMB anisotropies spectrum.
+We should note that the quantum power spectrum peaks at scales that leave the horizon
+for T < Tc, where our approximations break down. Extending our calculation to this regime
+– 10 –
+
+would involve a different form of the dissipation coefficient, since as the field experiences
+a tachyonic instability the latter no longer corresponds to the perturbative decay width
+at finite temperature. Let us note, however, that fluctuation-dissipation effects are more
+pronounced at the start of thermal inflation as discussed earlier, so that they no longer
+play a significant role near Tc. If the field thermalizes at the onset of thermal inflation, it
+will nevertheless maintain an equilibrium distribution with a decreasing temperature due to
+inflationary expansion. Let us then compare the magnitude of field fluctuations at Tc in both
+the quantum vacuum and thermal cases. The thermal variance is obtained by expanding the
+field in terms of its modes
+⟨φ(x)φ(y)⟩ =
+�
+d3k
+(2π)3
+d3k′
+(2π)3 ⟨φkφk′⟩ exp(ik · x) exp(ik · y) ,
+(4.2)
+and using the field modes correlator (3.15), we obtain for the field variance in the thermalized
+limit:
+⟨φ2⟩therm =
+2
+(2π)2
+T
+a
+� kcutoff
+0
+dk
+k2
+k2 + α2T 2c
+= T 2
+2π
+�
+1 − α
+π arctan
+�π
+α
+��
+,
+(4.3)
+which we note is only mildly dependent on the effective coupling α, while the quantum one
+is given by [16]:
+⟨φ2⟩quan =
+� H
+2π
+�2 Γ2(ν)22ν
+6π
+�aH
+m
+�2ν
+F
+�
+ν, 3
+2; 5
+2; −
+�aH
+m
+�2�
+,
+(4.4)
+where F(a, b, c, z) denotes the Hypergeometric function. The field variance in both cases is
+shown in Figure 2, where we extrapolate the thermal variance beyond the phase transition
+purely for comparison purposes.
+quantum
+thermal
+0.1
+0.5
+1
+5
+10
+10-8
+10-4
+1
+a / ac
+ϕ2 / H2
+Figure 2. Quantum (blue) and thermal (red) field variance as a function of the scale factor for
+H/m = 0.3, α = 1 and δ = 0. The critical temperature corresponds to the dashed vertical line, below
+which the thermal variance is extrapolated, as indicated by the dashed red line.
+As one can clearly observe in this figure, the quantum field variance is several orders of
+magnitude smaller than the thermal variance before the phase transition, which validates our
+calculation in neglecting vacuum fluctuations in the thermalized flaton scenario. While at
+the critical temperature this is still true, if one extrapolates the thermal variance for T < Tc
+– 11 –
+
+(a > ac = 1), we see that quantum fluctuations become dominant less than one e-fold after
+the critical temperature is attained.
+While this extrapolation is non-trivial, since the fluctuation-dissipation effects would
+have to be re-computed, it may suggest that vacuum perturbations may become dominant
+after the phase transition, in which case the computation in [16] would hold. In fact, the peak
+in the quantum power spectrum is obtained for modes with k = H
+2
+�
+3(2ν + 3) > kc = H,
+which leave the horizon for temperatures below the critical value and thus, in the example
+shown above, already in the regime where the quantum variance is dominant.
+This would, in fact, suggest that large enough curvature perturbations leading to pri-
+mordial black hole formation may be generated after thermal inflation (from quantum fluc-
+tuations), but it is not clear that quantum and thermal fluctuations may be examined in-
+dependently nor that the thermal variance maintains its form below Tc. In addition, and
+perhaps most importantly, the fact that the thermal variance is still typically a few orders of
+magnitude larger than the quantum one at the critical temperature indicates that the flaton
+field should reach the minimum of its potential much more quickly if it thermalizes, therefore
+considerably shortening, or even possibly, precluding an ensuing period of fast-roll inflation.
+A more complete analysis of the problem including both thermal and quantum fluctua-
+tions in the analysis, potentially along the lines of [59], is required to compute the spectrum
+of curvature perturbations on scales that leave the horizon at temperatures below Tc, and is
+left for future work.
+5
+Conclusion
+We have computed the spectrum of curvature perturbations generated during thermal in-
+flation taking into account the thermal fluctuations of the flaton field driving this period.
+These are associated with fluctuation-dissipation effects driven by the flaton’s interactions
+with the ambient radiation bath. Our analysis involved solving the Langevin-like equation
+effectively describing the evolution of the flaton’s Fourier modes. We computed the associ-
+ated correlation functions in the approximation of a gaussian white noise and a dominant
+thermal contribution to the flaton’s mass, for temperatures above the critical value at which
+the flaton is held at the false vacuum at the origin.
+We have concluded that, if the flaton’s (finite-temperature) decay width exceeds the
+Hubble parameter at the onset of thermal inflation, the field essentially thermalizes with
+the ambient radiation bath, contributing approximately as an extra relativistic degree of
+freedom. This occurs when the effective coupling between the flaton and the thermalized
+degrees of freedom α ≳ 0.01, which roughly corresponds to the parametric regime where over
+10 e-folds of thermal inflation (above Tc) occur. We found that the consequent increase in
+the field variance and the average gradient and kinetic energies enhances the background
+energy density (namely its time-dependent part that determines curvature perturbations)
+with respect to a field with purely quantum vacuum fluctuations analyzed in [16]. Despite the
+enhancement of super-horizon density fluctuations in the thermal case, the overall amplitude
+of the curvature power spectrum is significantly reduced with respect to the quantum case, so
+that thermal fluctuations behave very differently compared to their quantum counterparts,
+regarding the generation of curvature perturbations during periods of thermal inflation.
+While our analysis is not applicable for modes that leave the horizon once the tempera-
+ture has fallen below the critical value and the field starts rolling towards the true minimum
+of its potential, we expect thermal effects to become less relevant in this regime and quantum
+– 12 –
+
+fluctuations to become dominant, potentially yielding large curvature perturbations at such
+scales as computed in [16]. However, a full analysis including both quantum and thermal
+fluctuations in the dynamics of the flaton field is required to accurately describe the puta-
+tive fast-roll inflation phase below the critical temperature. It must be noted, in any case,
+that such a phase is necessarily shortened by the fact that the field variance at the critical
+temperature, which sets the typical field value at this stage, is much larger if the field ther-
+malizes with the radiation bath. It is therefore unclear whether super-horizon fluctuations
+with k > kc can be generated in this phase.
+We have modelled the thermal bath through a set of fermion species coupled to the
+flaton field, but we expect our main conclusions to hold with the inclusion of other bosonic
+fields, like scalars or vector bosons: if Γφ > H at some stage during thermal inflation, the
+field will be driven towards a thermal fluctuation spectrum. Only the details of how and when
+this equilibrium is attained may depend on the types of light fields that interact with the flat
+direction. Our analysis shows that thermalization does not require large coupling constants
+describing the interaction between the flaton and the radiation bath.
+In any case such
+couplings cannot be too suppressed to sustain a sufficiently long period of thermal inflation
+that may, in particular, dilute any unwanted relics generated after the primary slow-roll
+inflation period. Hence, fluctuation-dissipation effects cannot in general be neglected in the
+dynamics of the flaton field and on the curvature perturbations they induce during thermal
+inflation. This is particularly relevant if one wishes to understand whether thermal inflation
+periods may leave behind a sizeable population of primordial black holes, and we hope that
+our work motivates further exploration of these and related issues, including other potential
+implications for structure formation in our Universe [60].
+Acknowledgements
+M.B.G. work has been partially supported by MICINN (PID2019-105943GB-I00/AEI/10.130
+39/501100011033) and “Junta de Andaluc´ıa” grant P18-FR-4314. JMG acknowledges the
+support from the Funda¸c˜ao para a Ciˆencia e a Tecnologia, I.P. (FCT) through the Research
+Fellowship No.
+2021.05180.BD derived from Portuguese national funds.
+This work was
+supported by the CFisUC project No. UID/FIS/04564/2020 and by the FCT-CERN grant
+No. CERN/FIS-PAR/0027/2021.
+A
+Evolution of the temperature during thermal inflation
+In our calculation we assumed that no significant entropy is produced during thermal inflation
+as a result of fluctuation-dissipation effects, i.e. that T ∝ a−1. In this appendix we aim to
+verify this assumption. The flaton field satisfies the Langevin-like equation [25]:
+¨φ + (3H + Γφ) ˙φ − a−2∇2φ + m2
+effφ = ξ ,
+(A.1)
+and by multiplying both sides by ˙φ we obtain:
+˙ρφ + 3H(ρφ + pφ) = ξ ˙φ − Γφ ˙φ2 + α2T ˙Tφ2 + a−2∂i( ˙φ∂iφ) ,
+(A.2)
+where the field’s energy density and pressure are given by:
+ρφ = 1
+2
+˙φ2 + 1
+2a−2∂iφ∂iφ + V (φ) ,
+pφ = 1
+2
+˙φ2 − 1
+6a−2∂iφ∂iφ − V (φ) .
+(A.3)
+– 13 –
+
+Conservation of the full energy-momentum tensor then yields the following continuity equa-
+tion for the radiation energy density:
+˙ρR + 4HρR = − ⟨ξ ˙φ⟩ + Γφ ⟨ ˙φ2⟩ − α2T ˙T ⟨φ2⟩ − a−2 ⟨∂i( ˙φ∂iφ⟩) .
+(A.4)
+We note that the radiation energy density is an ensemble average over the energy density
+of the relativistic degrees of freedom, which justifies considering also the thermal average
+of the terms on the right-hand side of the above equation. Here we have also neglected the
+sub-leading corrections to the radiation energy and entropy densities from the fermions’ finite
+mass, ∼ g ⟨
+�
+φ2⟩ ∼ gT ≪ T.
+Using the field solutions we obtained for the correlators1:
+⟨ξ ˙φ⟩ = π
+6 ΓφT 4 ,
+Γφ ⟨ ˙φ2⟩ = π
+6 ΓφT 4(1 − δ) ,
+α2T ˙T ⟨φ2⟩ = α2
+2πT 3 ˙T
+�
+1 − α
+π arctan
+�π
+α
+��
+(1 − δ) ≈ 15α2
+4π3g∗
+(1 − δ) ˙ρR ,
+⟨∂i( ˙φ∂iφ⟩ = 0 .
+(A.5)
+Note that the third term is related to the time-dependence of the thermal flaton mass, and
+yields a contribution to the variation of the radiation energy density comparable to the above-
+mentioned sub-leading corrections from the fermions’ non-vanishing mass. For consistency,
+we thus neglect this term, and obtain:
+˙ρR + 4HρR = −5/π
+g∗
+ΓφρRδ .
+(A.6)
+From this we immediately see that the right-hand side can only be significant if Γφ ≳ H,
+but this implies a quick thermalization of the flaton field such that δ → 0 exponentially
+fast, thus making this term negligible. This simply reflects the balance between the effects
+of fluctuations and dissipation as the flaton field reaches an equilibrium with the radiation
+bath. Note, furthermore, that the term on the right-hand side is suppressed by the relative
+contribution of the flaton to the number of relativistic species in equilibrium, (g∗,f − g∗)/g∗,
+as obtained in Section 3. We therefore conclude that one may consistently assume ρR ∝ a−4
+and hence that T ∝ a−1 during thermal inflation.
+B
+Field correlation functions
+Here we list the field correlation functions used to compute the curvature perturbation power
+spectrum. As we mentioned above, when integrating over momentum modes we consider
+a sharp cut-off at k = πTc, which constitutes a good approximation to the behaviour of
+the noise correlation function [46].
+To compute the curvature perturbation power spec-
+trum on super-horizon scales, k ≪ aH, we consider the leading order results in k/αTc ∼
+(k/aH)(M0/MP )a ≪ 1 considering M0 < MP and noting that in our convention a < ac = 1
+above the critical temperature.
+1⟨∇( ˙φ∇φ⟩) = −
+�
+d3k1
+(2π)3
+d3k2
+(2π)3 ⟨ ˙φk1φk2⟩ k2 · (k1 + k2) exp [ix · (k1 + k2)] = 0 , since the integral of this delta
+function is non-zero if and only if k1 = −k2.
+– 14 –
+
+Mode correlators
+The building blocks of all field correlators are the equal time correlators between the field
+modes and their time derivatives. Writing φk and ˙φk in terms of the Green’s function (3.13):
+φk = H−2
+� z
+zi
+ds s−2Gs(z, s)ξk(s) ,
+˙φk = H−1z
+� z
+zi
+ds s−2∂zGs(z, s)ξk(s) ,
+(B.1)
+one finds:
+⟨φkφk′⟩ = (2π)3δ3(k + k′) T
+a3ω2
+k
+(1 − δ) ,
+⟨ ˙φk ˙φk′⟩ = (2π)3δ3(k + k′) T
+a3 (1 − δ) ,
+⟨φk ˙φk′⟩ = −Γφ
+2 ⟨φkφk′⟩ = −(2π)3δ3(k + k′) TΓφ
+2a3ω2
+k
+(1 − δ) .
+(B.2)
+Field correlators
+To compute the total energy density (3.4a) one needs to determine the field variance and the
+average kinetic and gradient energies. Expanding the field in terms of comoving momentum
+modes, these are given by:
+⟨φ2⟩ =
+�
+d3k
+(2π)3
+d3k′
+(2π)3 ⟨φkφk′⟩ exp(ix · (k + k′)) ,
+⟨ ˙φ2⟩ =
+�
+d3k
+(2π)3
+d3k′
+(2π)3 ⟨ ˙φk ˙φk′⟩ exp(ix · (k + k′)) ,
+⟨∂iφ∂iφ⟩ =
+�
+d3k
+(2π)3
+d3k′
+(2π)3 ⟨φkφk′⟩ k · k′ exp(ix · (k + k′)) .
+(B.3)
+Inserting the mode correlation functions (B.2) and integrating over comoving momenta up
+to πTc one obtains:
+⟨φ2⟩ = T 2
+2π (1 − δ)
+�
+1 − α
+π arctan
+�π
+α
+��
+,
+⟨ ˙φ2⟩ = πT 4
+6 (1 − δ) ,
+⟨∂iφ∂iφ⟩ = π
+2 a2T 4(1 − δ)
+�1
+3 −
+�α
+π
+�2
++
+�α
+π
+�3
+arctan
+�π
+α
+��
+.
+(B.4)
+Contributions to the power spectrum
+Consider the power spectrum of a generic correlator ⟨Xi(0)Xj(x)⟩, for example X1 = φ,
+X2 = ˙φ and X3 = ∂iφ, that appears in (3.7):
+�
+d3x exp(−ik · x) ⟨Xi(0)Xj(x)⟩2 .
+(B.5)
+Note that, upon expanding each quantity Xj(x) in terms of comoving momentum modes, this
+yields four momentum integrals and a volume integral. Two of the momentum integrals can
+– 15 –
+
+be performed using the two delta functions appearing in the mode correlators (B.2). Then,
+the volume integral will generate a delta function with the two surviving momentum modes:
+�
+d3x exp[−ix · (k1 + k2 + k)] = (2π)3δ3(k1 + k2 + k) .
+(B.6)
+After integrating this delta function over another of the 3-momentum variables, we are left
+with a single 3-dimensional integral over k that we need to compute in each case. In the
+following table we give the different contributions to the power spectrum in terms of their
+corresponding momentum integrals:
+Table 1. Contributions to the power spectrum in Eq. (3.17).
+field-field
+m4
+eff
+2
+�
+d3x exp(−ik · x) ⟨φ(0)φ(x)⟩2
+(1 − δ)2
+α3
+2(2π)3 T 5
+a3 I1(k)
+field-kinetic
+m2
+eff
+�
+d3x exp(−ik · x) ⟨φ(0) ˙φ(x)⟩
+2
+(1 − δ)2
+α
+(2π)3
+� 3α4
+32π
+�2 T 5
+a3 I1(k)
+field-gradient
+a−2m2
+eff
+�
+d3x exp(−ik · x) ⟨φ(0)∂iφ(x)⟩2
+(1 − δ)2
+α3
+(2π)3 T 5
+a3 I2(k)
+kinetic-kinetic
+1
+2
+�
+d3x exp(−ik · x) ⟨ ˙φ(0) ˙φ(x)⟩
+2
+(1 − δ)2
+α3
+2(2π)3 T 5
+a3 I3(k)
+kinetic-gradient
+a−2 �
+d3x exp(−ik · x) ⟨ ˙φ(0)∂iφ(x)⟩
+2
+(1 − δ)2
+α
+(2π)3
+� 3α4
+32π
+�2 T 5
+a3 I2(k)
+gradient-gradient
+a−4
+2
+�
+d3x exp(−ik · x) ⟨∂iφ(0)∂jφ(x)⟩2
+(1 − δ)2
+α3
+2(2π)3 T 5
+a3 I4(k)
+The momentum integrals can be expressed in terms of the normalized comoving mo-
+mentum y = k/αTc with norm 0 < y < π/α. These are given by:
+I1(k) =
+�
+dy dΩ
+y2
+(y2 + 1)[(y + k/(αTc))2 + 1] ,
+I2(k) =
+�
+dy dΩ
+y2y · (y + k/(αTc))
+(y2 + 1)[(y + k/(αTc))2 + 1] ,
+I3(k) =
+�
+dy dΩ y2 = 4π4
+3α3 ,
+I4(k) =
+�
+dy dΩ
+y2[y · (y + k/(αTc))]2
+(y2 + 1)[(y + k/(αTc))2 + 1] ,
+(B.7)
+where dΩ denotes integration over the solid angle in momentum space. Except for I3(k), all
+integrals above depend non-trivially on k. To leading order in k/αTc these integrals are given
+by:
+I1(k) ≃ 4π
+�
+− 1
+2
+πα
+α2 + π2 + 1
+2 arctan(π/α)
+�
+,
+I2(k) ≃ 4π
+�π
+α + 1
+2
+απ
+α2 + π2 − 3
+2 arctan(π/α)
+�
+,
+I4(k) ≃ 4π
+�
+− 2π
+α + 1
+3
+π3
+α3 − 1
+2
+απ
+α2 + π2 + 5
+2 arctan(π/α)
+�
+,
+(B.8)
+which are the expressions used to compute the curvature perturbation power spectrum (3.17)
+given in the main body of this article.
+– 16 –
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@@ -0,0 +1,959 @@
+arXiv:2301.13693v1  [math.NA]  31 Jan 2023
+Application of dimension truncation error analysis to
+high-dimensional function approximation
+Philipp A. Guth†
+Vesa Kaarnioja‡
+February 1, 2023
+Abstract
+Parametric mathematical models such as partial differential equations with random
+coefficients have received a lot of attention within the field of uncertainty quantifica-
+tion. The model uncertainties are often represented via a series expansion in terms of
+the parametric variables. In practice, this series expansion needs to be truncated to
+a finite number of terms, introducing a dimension truncation error to the numerical
+simulation of a parametric mathematical model. There have been several studies of
+the dimension truncation error corresponding to different models of the input random
+field in recent years, but many of these analyses have been carried out within the
+context of numerical integration. In this paper, we study the L2 dimension truncation
+error of the parametric model problem. Estimates of this kind arise in the assessment
+of the dimension truncation error for function approximation in high dimensions. In
+addition, we show that the dimension truncation error rate is invariant with respect to
+certain transformations of the parametric variables. Numerical results are presented
+which showcase the sharpness of the theoretical results.
+1
+Introduction
+In the field of uncertainty quantification it is common to study mathematical models with
+uncertain influences parameterized by countably infinite sequences of random variables.
+Consider, for instance, an abstract model M : X × U → Y such that
+M(g(y), y) = 0,
+(1)
+where X and Y are separable Hilbert spaces and U is a nonempty subset of the infinite-
+dimensional sequence space of parameters RN. The solution g(y) ∈ X to (1) for y ∈ U, if
+it exists, may be computationally expensive to evaluate. To this end, it may be preferable
+to instead approximate g using a surrogate which is cheap to evaluate and hence enables,
+e.g., efficient sampling of the (approximated) solution.
+Some possible surrogate models include, but are not limited to, Gaussian process
+regression [3], reduced basis approaches [1, 21], generalized polynomial chaos expansions
+[4, 23], neural network approximations [2, 7, 9, 22], and kernel interpolation based on
+lattice point sets [16, 25, 26]. The results presented in this manuscript are particularly
+well-suited to the analysis of kernel methods used in conjunction with the so-called periodic
+model discussed in [13, 16, 17], and we will devote a section of this manuscript to explore
+the application of our dimension truncation results within this framework.
+†Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences,
+Altenbergerstraße 69, A-4040 Linz, Austria, philipp.guth@ricam.oeaw.ac.at
+‡Department of Mathematics and Computer Science, Free University of Berlin, Arnimallee 6, 14195
+Berlin, Germany, vesa.kaarnioja@fu-berlin.de
+1
+
+Integration
+Function approximation
+Affine parametric
+[6, 20]
+[16]
+operator equation setting
+rate O(s− 2
+p +1)
+rate O(s− 1
+p + 1
+2)
+Non-affine parametric
+[8, 12]
+this paper
+operator equation setting
+rate O(s− 2
+p +1)
+rate O(s− 1
+p + 1
+2)
+Table 1: An overview of various dimension truncation results.
+A natural first step for the numerical treatment of (1) is the approximation by a
+dimensionally-truncated model Ms : X × Us → Y such that
+Ms(gs(y≤s), y≤s) = 0,
+where ∅ ̸= Us ⊆ Rs and gs(y≤s) ∈ X for all y≤s ∈ Us. Consider the problem of finding
+a surrogate solution gs,n := An(gs) using an algorithm An which uses n point evaluations
+of the s-dimensional function gs, where the surrogate belongs to X such that
+∥gs − gs,n∥L2µ(U;X)
+n→∞
+−−−→ 0
+with some known convergence rate and µ indicating a probability measure on U. The total
+error of the approximation obtained in this fashion can be estimated using the triangle
+inequality
+∥g − gs,n∥L2µ(U;X) ≤ ∥g − gs∥L2µ(U;X) + ∥gs − gs,n∥L2µ(U;X).
+In this manuscript we focus on the first term—the dimension truncation error—which is
+independent of the chosen approximation scheme An.
+Dimension truncation error rates are typically studied for problems involving partial
+differential equations (PDEs) with random inputs. For integration problems a dimension
+truncation rate is derived in [20] for the source problem with an affine parameterization
+of the diffusion coefficient. This rate was then improved by [6] in the generalized context
+of affine parametric operator equations. Dimension truncation has also been studied for
+coupled PDE systems arising in optimal control problems under uncertainty [10], in the
+context of the periodic model of uncertainty quantification for both numerical integra-
+tion [17] and kernel interpolation [16], as well as for Bayesian inverse problems governed
+by PDEs [5, 15]. The results in these papers have been proved using Neumann series,
+which is known to work well in the affine parametric setting, but may lead to suboptimal
+results if the problem depends nonlinearly on the parameters.
+In the non-affine setting, using Taylor series makes it possible to derive dimension
+truncation error rates by exploiting the parametric regularity of the problem, whereas the
+Neumann series approach relies fundamentally on the parametric structure of the model.
+The Taylor series approach was first applied in [8], and motivated the authors in [11]
+and [12] to derive dimension truncation error rates for sufficiently smooth, Banach space
+valued integrands, and with parameters following a generalized β-Gaussian distribution.
+An overview of the various dimension truncation error bounds studied in the literature is
+given in Table 1.
+Our manuscript is structured as follows. Subsection 1.1 introduces the multi-index
+notation used throughout the paper.
+The problem setting is introduced in Section 2,
+including the central assumptions for the ensuing dimension truncation analysis. Section 3
+contains the L2 dimension truncation theorem for Hilbert space valued functions, and
+in Section 4 we discuss the invariance of the dimension truncation rate under certain
+transformations of the variables. Numerical experiments assessing the sharpness of our
+2
+
+theoretical results are presented in Section 5. The paper ends with some conclusions in
+Section 6.
+1.1
+Notations and preliminaries
+Throughout this manuscript, boldfaced symbols are used to denote multi-indices while the
+subscript notation mj is used to refer to the j-th component of multi-index m. Let
+F := {m ∈ NN
+0 : |m| < ∞}
+denote the set of finitely supported multi-indices, where the order of multi-index m is
+defined as
+|m| :=
+�
+j≥1
+mj.
+Moreover, we denote
+|m|∞ := max
+j≥1 mj,
+and, for any sequence x := (xj)∞
+j=1 of real numbers and m ∈ F, we define
+xm :=
+�
+j≥1
+xmj
+j ,
+where we use the convention 00 := 1.
+2
+Problem setting
+Let X be a real separable Hilbert space, U := [− 1
+2, 1
+2]N a set of parameters, and suppose
+that g(y) ∈ X is a parameterized family of functions with smooth dependence on y ∈
+U.
+We define gs(y) := g(y≤s, 0) := g(y1, . . . , ys, 0, 0, . . .) and assume that µ(dy) :=
+�
+j≥1 µ(dyj) is a countable product probability measure, i.e., µ(U) = 1. We suppose that
+1. For µ-a.e. y ∈ U, there holds
+∥g(y) − gs(y)∥X
+s→∞
+−−−→ 0.
+2. Let (Θk)k≥0 and b := (bj)j≥1 be sequences of nonnegative numbers such that b ∈
+ℓp(N) for some p ∈ (0, 1) and b1 ≥ b2 ≥ · · · .
+Suppose that g is continuously
+differentiable up to order k + 1, with
+∥∂νg(y)∥X ≤ Θ|ν|bν
+for all y ∈ U and for all ν ∈ Fk := {ν ∈ NN
+0 : |ν| ≤ k + 1}, where k := ⌈
+1
+1−p⌉.
+3. There holds
+� 1/2
+−1/2 yj µ(dyj) = 0 and there exists a constant Cµ ≥ 0 such that
+� 1/2
+−1/2 |yj|k µ(dyj) ≤ Cµ for all k ≥ 2.
+If Assumption 2 holds, then we infer that y �→ G(g(y)) for all G ∈ X′ is continuous as
+a composition of continuous mappings. Hence y �→ G(g(y)) is measurable for all G ∈ X′,
+i.e., y �→ g(y) is weakly measurable.
+Since X is assumed to be a separable Hilbert
+space, by Pettis’ theorem (cf., e.g., [24, Chapter 4]) we obtain that y �→ g(y) is strongly
+measurable. The upper bound in Assumption 2 is µ-integrable. Thus we conclude from
+Bochner’s theorem (cf., e.g., [24, Chapter 5]) and Assumption 2 that g is µ-integrable over
+U.
+3
+
+Further, µ-a.e. equality defines an equivalence relation among strongly µ-measurable
+functions. By L2
+µ(U; X) we denote the Hilbert space of equivalence classes of strongly
+µ-measurable functions f : U → X with norm
+∥f∥L2µ(U;X) :=
+� �
+U
+∥f(y)∥2
+X µ(dy)
+� 1
+2
+< ∞.
+Moreover, under the Assumptions 1 and 2 it can be shown that g, gs ∈ L2
+µ(U; X) and
+lim
+s→∞ ∥g(y) − g(y≤s, 0)∥L2µ(U;X) = lim
+s→∞
+� �
+U
+∥g(y) − g(y≤s, 0)∥2
+X µ(dy)
+� 1
+2
+= 0,
+by applying Lebesgue’s dominated convergence theorem (see, e.g., [18, Theorem 1] and
+[14, Section 26]) to
+F s(y) := ∥g(y) − g(y≤s, 0)∥2
+X,
+which converges µ-a.e. to zero by Assumption 1, and can be bounded by (2Θ0)2 by As-
+sumption 2. We use the superscript to avoid confusion with the notation used to denote
+dimensionally-truncated functions elsewhere in the document.
+3
+Dimension truncation error
+We will require the following parametric regularity bound for the main dimension trunca-
+tion result.
+Lemma 1. Under Assumption 2, there holds
+|∂ν∥g(y) − gs(y)∥2
+X| ≤
+�
+max
+0≤ℓ≤|ν|
+2Θℓ
+ℓ!
+�2
+(|ν| + 1)!bν
+for all ν ∈ Fk and y ∈ U.
+Proof. Let ν ∈ Fk. We apply the Leibniz product rule with respect to the inner product
+of the Hilbert space X to obtain
+∂ν∥g(y) − gs(y)∥2
+X = ∂ν⟨g(y) − gs(y), g(y) − gs(y)⟩X
+=
+�
+m≤ν
+� ν
+m
+�
+⟨∂m(g(y) − gs(y)), ∂ν−m(g(y) − gs(y))⟩X.
+Using the Cauchy–Schwarz inequality together with Assumption 2 yields
+|∂ν∥g(y) − gs(y)∥2
+X| ≤
+�
+m≤ν
+� ν
+m
+�
+∥∂m(g(y) − gs(y))∥X∥∂ν−m(g(y) − gs(y))∥X
+≤ 4
+�
+m≤ν
+� ν
+m
+�
+Θ|m|bmΘ|ν|−|m|bν−m
+= 4bν
+|ν|
+�
+ℓ=0
+ΘℓΘ|ν|−ℓ
+�
+|m|=ℓ
+m≤ν
+� ν
+m
+�
+= 4bν
+|ν|
+�
+ℓ=0
+ΘℓΘ|ν|−ℓ
+|ν|!
+ℓ!(|ν| − ℓ)!
+≤ 4
+�
+max
+0≤ℓ≤|ν|
+Θℓ
+ℓ!
+�2
+(|ν| + 1)!bν,
+where we used the Vandermonde convolution �
+|m|=ℓ
+m≤ν
+� ν
+m
+�
+=
+�|ν|
+ℓ
+�
+=
+|ν|!
+ℓ!(|ν|−ℓ)!.
+4
+
+The main result of this document is stated below.
+Theorem 1. Let g(y) ∈ X, y ∈ U, satisfy Assumptions 1–3. Then
+∥g − gs∥L2µ(U;X) = O(s− 1
+p + 1
+2),
+where the implied coefficient is independent of s.
+Proof. Let s ≥ 1 and define
+F s(y) := ∥g(y) − gs(y)∥2
+X
+for y ∈ U.
+In the special case of the uniform distribution µ(dy) = dy, we can apply [12, Theorem 4.2]
+to obtain
+∥g − gs∥2
+L2(U;X) =
+����
+�
+U
+(F s(y) − F s(y≤s, 0)) dy
+���� = O(s− 2
+p +1),
+from which the claim follows. For completeness, we present the proof below for the prob-
+ability measure µ and because parts of the argument will also be useful to establish the
+invariance of the dimension truncation rate in Section 4.
+Developing the Taylor expansion of F s about (y≤s, 0) and observing that F s(y≤s, 0) =
+0, we obtain
+F s(y) =
+k
+�
+ℓ=1
+�
+|ν|=ℓ
+νj=0 ∀j≤s
+yν
+ν! ∂νF s(y≤s, 0)
++
+�
+|ν|=k+1
+νj=0 ∀j≤s
+k + 1
+ν! yν
+� 1
+0
+(1 − t)k∂νF s(y≤s, ty>s) dt,
+(2)
+where y>s := (yj)j>s. Integrating both sides over y ∈ U yields
+�
+U
+F s(y) µ(dy) =
+k
+�
+ℓ=1
+�
+|ν|=ℓ
+νj=0 ∀j≤s
+1
+ν!
+�
+U
+yν∂νF s(y≤s, 0) µ(dy)
++
+�
+|ν|=k+1
+νj=0 ∀j≤s
+k + 1
+ν!
+�
+U
+� 1
+0
+(1 − t)kyν∂νF s(y≤s, ty>s) dt µ(dy).
+If ν ∈ Fk is such that νj = 1 for any j > s, then Fubini’s theorem together with Assump-
+tion 3 imply for the summands appearing in the first term that
+�
+U
+yν∂νF s(y≤s, 0) µ(dy) =
+� �
+j>s
+�
+1
+2
+− 1
+2
+yνj
+j µ(dyj)
+�
+�
+��
+�
+=0
+�
+[− 1
+2, 1
+2]s ∂νF s(y≤s, 0) µ(dy>s).
+Therefore all multi-indices with any component equal to 1 can be removed from the first
+sum (especially, we can omit all multi-indices with |ν| = 1). Further, applying the regu-
+larity bound proved in Lemma 1 and writing open the definition of F s yields
+�
+U
+∥g(y) − gs(y)∥2
+X µ(dy) ≤ Ck
+µ
+�
+max
+0≤ℓ≤k
+2Θℓ
+ℓ!
+�2
+(k + 1)!
+k
+�
+ℓ=2
+�
+|ν|=ℓ
+νj=0 ∀j≤s
+νj̸=1 ∀j>s
+bν
++ Ck+1
+µ
+�
+max
+0≤ℓ≤k+1
+2Θℓ
+ℓ!
+�2
+(k + 2)!
+�
+|ν|=k+1
+νj=0 ∀j≤s
+1
+ν!bν,
+(3)
+5
+
+where we used
+� 1
+0 (1 − t)k dt =
+1
+k+1 and Assumption 3.
+The second term in (3) can
+be estimated from above using the multinomial theorem in conjunction with Stechkin’s
+lemma:
+�
+|ν|=k+1
+νj=0 ∀j≤s
+1
+ν!bν ≤
+�
+|ν|=k+1
+νj=0 ∀j≤s
+|ν|!
+ν! bν =
+� �
+j>s
+bj
+�k+1
+≤ s(k+1)(− 1
+p +1)
+� �
+j≥1
+bp
+j
+� k+1
+p
+.
+On the other hand, the first term in (3) can be estimated similarly to [6]:
+�
+2≤|ν|≤k
+νj=0 ∀j≤s
+νj̸=1 ∀j>s
+bν ≤
+�
+0̸=|ν|∞≤k
+νj=0 ∀j≤s
+νj̸=1 ∀j>s
+bν = −1 +
+�
+j>s
+�
+1 +
+k
+�
+ℓ=2
+bℓ
+j
+�
+= −1 +
+�
+j>s
+�
+1 + b2
+j
+k−2
+�
+ℓ=0
+bℓ
+j
+�
+≤ −1 +
+�
+j>s
+�
+1 + b2
+j
+k−2
+�
+ℓ=0
+bℓ
+1
+� �� �
+=:βk
+�
+≤ −1 + exp
+�
+βk
+�
+j>s
+b2
+j
+�
+=
+�
+ℓ≥1
+1
+ℓ!
+�
+βk
+�
+j>s
+b2
+j
+�ℓ
+.
+Using �
+j>s b2
+j ≤ s− 2
+p +1(�
+j≥1 bp
+j)
+2
+p , which follows from Stechkin’s lemma, we further
+estimate
+�
+ℓ≥1
+1
+ℓ!
+�
+βk
+�
+j>s
+b2
+j
+�ℓ
+≤ s− 2
+p +1 �
+ℓ≥1
+1
+ℓ!(βk∥b∥2
+p)ℓ = s− 2
+p +1(−1 + exp(βk∥b∥2
+p)
+since s− 2
+p +1 ≥ (s− 2
+p +1)ℓ for all ℓ ≥ 1.
+Altogether, the above discussion yields the bound
+∥g(y) − gs(y)∥2
+L2µ(U;X) =
+�
+U
+∥g(y) − gs(y)∥2
+X µ(dy) = O(s− 2
+p +1 + s(k+1)(− 1
+p +1)),
+where the implied coefficient is independent of s. Since we assumed that k = ⌈
+1
+1−p⌉, the
+assertion follows by taking the square root on both sides.
+4
+Invariance of the dimension truncation rate under trans-
+formations of variables
+An interesting consequence of the Taylor series argument used in Theorem 1 is that the di-
+mension truncation rate remains invariant under certain transformations of the variables.
+This has been previously observed in the context of dimension truncation for integration
+problems under the periodic model [13]. To make this notion precise, let us consider a
+mapping ξ: U → U, ξ(y) := (ξ(y1), ξ(y2), . . .), which satisfies the following conditions:
+4. There hold ξ(0) = 0 and
+� 1/2
+−1/2 ξ(y) dy = 0.
+5. There exists Cξ ≥ 0 such that
+� 1/2
+−1/2 |ξ(y)|k dy ≤ Cξ for all k ≥ 2.
+Then we obtain the following as a consequence of Theorem 1.
+Corollary 1. Let g(y) ∈ X, y ∈ U, satisfy Assumptions 1–3 and let ξ : U → U satisfy
+Assumptions 4–5. Define the ξ-transformed function gξ by
+gξ(y) := g(ξ(y)),
+y ∈ U,
+6
+
+and its dimension truncation by gξ,s(y) := gξ(y≤s, 0) for y ∈ U. Then
+∥gξ − gξ,s∥L2µ(U;X) = O(s− 1
+p + 1
+2 ),
+where the implied coefficient is independent of s.
+Proof. We introduce F s
+ξ (y) := ∥gξ(y) − gξ,s(y)∥2
+X for y ∈ U. By carrying out the change
+of variable y ← ξ(y) in (2), we obtain
+F s
+ξ (y) =
+k
+�
+ℓ=1
+�
+|ν|=ℓ
+νj=0 ∀j≤s
+ξ(y)ν
+ν!
+∂νF s(ξ(y≤s, 0))
++
+�
+|ν|=k+1
+νj=0 ∀j≤s
+k + 1
+ν! ξ(y)ν
+� 1
+0
+(1 − t)k∂νF s(ξ(y≤s, ty>s)) dt.
+Integrating the above formula on both sides over y ∈ U and utilizing Lemma 1 as well as
+Assumption 5, we obtain—in complete analogy with the proof of Theorem 1—that
+�
+U
+∥gξ(y) − gξ,s(y)∥2
+X dy ≤ Ck
+ξ
+�
+max
+0≤ℓ≤k
+2Θℓ
+ℓ!
+�2
+(k + 1)!
+k
+�
+ℓ=2
+�
+|ν|=ℓ
+νj=0 ∀j≤s
+νj̸=1 ∀j>s
+bν
++ Ck+1
+ξ
+�
+max
+0≤ℓ≤k+1
+2Θℓ
+ℓ!
+�2
+(k + 2)!
+�
+|ν|=k+1
+νj=0 ∀j≤s
+1
+ν!bν.
+The desired result follows by exactly the same argument as in the proof of Theorem 1.
+As an application, with U := [− 1
+2, 1
+2]N, let ξ: U → U satisfy the Assumptions 4 and 5,
+let D ⊂ Rd, d ∈ {1, 2, 3}, be a bounded Lipschitz domain, and let f : D → R be a fixed
+source term. Consider the parametric PDE problem
+�
+−∇ · (aξ(x, y)∇uξ(x, y)) = f(x),
+x ∈ D, y ∈ U,
+uξ(x, y) = 0,
+x ∈ ∂D, y ∈ U,
+(4)
+endowed with the ξ-transformed diffusion coefficient
+aξ(x, y) := a0(x) +
+∞
+�
+i=1
+ξ(yi)ψi(x),
+x ∈ D, y ∈ U,
+which is assumed to satisfy the following:
+6. There exist amin, amax > 0 such that 0 < amin ≤ aξ(x, y) ≤ amax < ∞ for all x ∈ D
+and y ∈ U.
+7. a0 ∈ L∞(D) and ψi ∈ L∞(D) for all i ∈ N.
+8. �∞
+i=1 ∥ψi∥p
+L∞(D) < ∞ for some p ∈ (0, 1).
+In this case, the transformation ξ(y) := ( 1
+√
+6 sin(2πyj))j≥1 corresponds to the so-called
+periodic model studied in [13, 16, 17]. Let X := H1
+0(D) be equipped with the norm ∥v∥X :=
+7
+
+�
+D ∥∇v(x)∥2
+Rd dx. In this special case, it is known that the weak solution u(·, y) ∈ X to (4)
+for y ∈ U satisfies the parametric regularity bound
+∥∂νuξ(·, y)∥X ≤ (2π)|ν|∥f∥X′
+amin
+�
+m≤ν
+|m|!bm �
+j≥1
+S(νj, mj)
+for all ν ∈ F and y ∈ U, where the source term f ∈ X′, S(·, ·) denotes the Stirling number
+of the second kind, and b := (bj)j≥1 is defined by setting bj :=
+∥ψj∥L∞(D)
+√
+6amin
+for all j ≥ 1.
+Let µ(dµ) = dy. Then Corollary 1 can be used to deduce that
+∥uξ − uξ,s∥L2µ(U;X) = O(s− 1
+p + 1
+2 ),
+where the constant is independent of the dimension s.
+In fact, if Xh is a conforming
+finite element subspace of X, uξ,h(·, y) ∈ Xh denotes the finite element discretization of
+uξ(·, y) ∈ X for all y ∈ U, and uξ,h,s(·, y) ∈ Xh denotes the dimension truncation of
+uξ,h(·, y) for all y ∈ U, then we have
+∥uξ,h − uξ,h,s∥L2µ(U;X) = O(s− 1
+p + 1
+2),
+independently of s.
+Finally, we present an example illustrating how our results can be applied to nonlinear
+quantities of interest.
+Example. Let X := H1
+0(D) as above. Consider the nonlinear quantity of interest
+Gnl(v) := ∥v∥2
+X :=
+�
+D
+∥∇v(x)∥2
+Rd dx,
+v ∈ X.
+(5)
+If u(·, y) ∈ X is the solution to (4) with U = [− 1
+2, 1
+2]N, µ(dy) := dy, and ξ(y) := y, then
+it is known to satisfy Assumptions 1–3 with the regularity bound
+∥∂νu(·, y)∥X ≤ C|ν|!bν,
+where the constant C > 0 only depends on ∥f∥X′ and amin. By the Leibniz product rule,
+there holds
+∂νGnl(u(·, y)) =
+�
+D
+�
+m≤ν
+� ν
+m
+�
+∇∂mu(x, y) · ∇∂ν−mu(x, y) dx
+≤
+�
+m≤ν
+� ν
+m
+�
+∥∂mu(·, y)∥X∥∂ν−mu(·, y)∥X
+≤ C2 �
+m≤ν
+� ν
+m
+�
+|m|!bm|ν − m|!bν−m
+= C2bν
+|ν|
+�
+ℓ=0
+ℓ!(|ν| − ℓ)!
+�
+m≤ν
+|m|=ℓ
+� ν
+m
+�
+= C2bν(|ν| + 1)!,
+where we used the Vandermonde convolution �
+|m|=ℓ
+m≤ν
+� ν
+m
+�
+=
+�|ν|
+ℓ
+�
+=
+|ν|!
+ℓ!(|ν|−ℓ)!.
+It follows from Theorem 1 that
+∥Gnl(u) − Gnl(us)∥L2µ(U;R) = O(s− 1
+p + 1
+2),
+independently of s. Moreover, if ξ(y) := ( 1
+√
+6 sin(2πyj))j≥1, then it follows from Corollary 1
+that
+∥Gnl(uξ) − Gnl(uξ,s)∥L2µ(U;R) = O(s− 1
+p + 1
+2 ),
+independently of s.
+8
+
+5
+Numerical experiments
+Let D = (0, 1)2 be a spatial domain, U = [− 1
+2, 1
+2]N, and f(x) := x1 a fixed source term.
+Let ξ: U → U, ξ(y) = ( 1
+√
+6 sin(2πyj))j≥1. We consider the PDE problem
+�
+−∇ · (aξ(x, y)∇uξ(x, y)) = f(x),
+x ∈ D, y ∈ U,
+uξ(x, y) = 0,
+x ∈ ∂D, y ∈ U,
+(6)
+equipped with the diffusion coefficient
+aξ(x, y) = 3
+2 +
+�
+j≥1
+ξ(yj)j−ϑ sin(jπx1) sin(jπx2),
+x ∈ D, y ∈ U, ϑ > 1.
+The PDE (6) is spatially discretized using a first-order conforming finite element method
+with mesh size h = 2−5.
+We consider the dimension truncation error for the full PDE solution using the formula
+∥uξ − uξ,s∥L2(U;L2(D)) ≈
+� �
+[− 1
+2 , 1
+2 ]s′ ∥uξ,s′(·, y) − uξ,s(·, y)∥2
+L2(D) dy
+� 1
+2
+,
+and we also consider the nonlinear quantity of interest (5), estimating the dimension
+truncation error using the formula
+∥Gnl(uξ) − Gnl(uξ,s)∥L2(U) ≈
+� �
+[− 1
+2, 1
+2]s′ |Gnl(uξ,s′(·, y)) − Gnl(uξ,s(·, y))|2 dy
+� 1
+2
+.
+In both cases, we choose s′ ≫ s and the high-dimensional integrals are approximated using
+a randomly shifted rank-1 lattice rule with 220 cubature nodes and a single random shift.
+As the integration lattice, we use in both cases an off-the-shelf rank-1 lattice rule [19,
+lattice-39101-1024-1048576.3600] and use the same random shift for each value of ϑ. As
+the reference solution, we use the PDE solution corresponding to s′ = 211.
+The numerical results for dimensions s ∈ {2k : k = 1, . . . , 9} and decay rates ϑ ∈
+{1.5, 2.0, 3.0} corresponding to the full PDE solution and the nonlinear quantity of interest
+are displayed in Figures 1 and 2, respectively. The theoretical convergence rates in each
+case are −1.0, −1.5, and −2.5, respectively, and they are displayed alongside the numerical
+results.
+The convergence graphs corresponding to the full PDE solution in Figure 1 display
+an aliasing behavior between 10 ≤ s ≤ 100, which may be explained by the contributions
+of the finite element discretization error as well as the use of an “off-the-shelf” lattice
+rule (in contrast to a “tailored” lattice rule). This behavior appear to be exacerbated in
+the convergence graphs corresponding to the nonlinear quantity of interest in Figure 2.
+Nonetheless, in all cases the theoretically anticipated convergence rates are easily observed
+in practice.
+We remark that the convergence graphs corresponding to the affine and
+uniform model with ξ(y) := (yj)j≥1 are extremely similar to the results corresponding to
+the periodic model, and have thus been omitted.
+9
+
+Figure 1: The dimension truncation errors of the full PDE solution corresponding to a periodically parameterized
+input random field with decay parameters ϑ ∈ {1.5, 2.0, 3.0}. The expected dimension truncation error rates are
+−1.0, −1.5, and −2.5, respectively.
+Figure 2: The dimension truncation errors of the nonlinear quantity of interest corresponding to a periodically
+parameterized input random field with decay parameters ϑ ∈ {1.5, 2.0, 3.0}. The expected dimension truncation
+error rates are −1.0, −1.5, and −2.5, respectively.
+6
+Conclusions
+Unlike many studies which have considered the dimension truncation error rate within
+the context of high-dimensional numerical integration, we considered the L2 dimension
+truncation error rate for parametric Hilbert space valued functions. Our theory covers
+both affine parametric as well as non-affine parametric problems with sufficiently smooth
+dependence on a sequence of bounded, parametric variables. The main dimension trun-
+cation results presented in this work can be applied to nonlinear quantities of interest of
+parametric model problems, provided that they satisfy the conditions of our framework.
+10
+
+In addition, the Hilbert space can be chosen to be a finite element subspace, indicating
+that our dimension truncation results are also valid for conforming finite element approx-
+imations of parametric PDEs.
+The L2 dimension truncation error rates considered in this work arise, e.g., in the
+study of high-dimensional function approximation of parametric PDEs. An example of
+such an approximation scheme is the kernel method over lattice point sets considered
+in [16]. The kernel method was analyzed in the context of the so-called periodic model, in
+which a countable number of independent random variables enter the input random field
+of the PDE as periodic functions. Our second main result shows that the L2 dimension
+truncation error rate remains invariant under certain transformations of the parametric
+variables: especially, the L2 dimension truncation rate considered in this work holds for
+periodically parametrized model problems such as those studied in [13, 16, 17].
+References
+[1] Bachmayr, M., Cohen, A., Dahmen, W.: Parametric PDEs: sparse or low-rank
+approximations? IMA J. Numer. Anal., 38(4):1661–1708 (2017)
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+12
+

X9FRT4oBgHgl3EQf_DhA/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf,len=486
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='13693v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='NA]  31 Jan 2023  Application of dimension truncation error analysis to  high-dimensional function approximation  Philipp A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Guth†  Vesa Kaarnioja‡  February 1, 2023  Abstract  Parametric mathematical models such as partial differential equations with random  coefficients have received a lot of attention within the field of uncertainty quantifica-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' The model uncertainties are often represented via a series expansion in terms of  the parametric variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' In practice, this series expansion needs to be truncated to  a finite number of terms, introducing a dimension truncation error to the numerical  simulation of a parametric mathematical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' There have been several studies of  the dimension truncation error corresponding to different models of the input random  field in recent years, but many of these analyses have been carried out within the  context of numerical integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' In this paper, we study the L2 dimension truncation  error of the parametric model problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Estimates of this kind arise in the assessment  of the dimension truncation error for function approximation in high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' In  addition, we show that the dimension truncation error rate is invariant with respect to  certain transformations of the parametric variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Numerical results are presented  which showcase the sharpness of the theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  1  Introduction  In the field of uncertainty quantification it is common to study mathematical models with  uncertain influences parameterized by countably infinite sequences of random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  Consider, for instance, an abstract model M : X × U → Y such that  M(g(y), y) = 0,  (1)  where X and Y are separable Hilbert spaces and U is a nonempty subset of the infinite-  dimensional sequence space of parameters RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' The solution g(y) ∈ X to (1) for y ∈ U, if  it exists, may be computationally expensive to evaluate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' To this end, it may be preferable  to instead approximate g using a surrogate which is cheap to evaluate and hence enables,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=', efficient sampling of the (approximated) solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  Some possible surrogate models include, but are not limited to, Gaussian process  regression [3], reduced basis approaches [1, 21], generalized polynomial chaos expansions  [4, 23], neural network approximations [2, 7, 9, 22], and kernel interpolation based on  lattice point sets [16, 25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' The results presented in this manuscript are particularly  well-suited to the analysis of kernel methods used in conjunction with the so-called periodic  model discussed in [13, 16, 17], and we will devote a section of this manuscript to explore  the application of our dimension truncation results within this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  †Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences,  Altenbergerstraße 69, A-4040 Linz, Austria, philipp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='guth@ricam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='oeaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='at  ‡Department of Mathematics and Computer Science, Free University of Berlin, Arnimallee 6, 14195  Berlin, Germany, vesa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='kaarnioja@fu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='de  1  Integration  Function approximation  Affine parametric  [6, 20]  [16]  operator equation setting  rate O(s− 2  p +1)  rate O(s− 1  p + 1  2)  Non-affine parametric  [8, 12]  this paper  operator equation setting  rate O(s− 2  p +1)  rate O(s− 1  p + 1  2)  Table 1: An overview of various dimension truncation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  A natural first step for the numerical treatment of (1) is the approximation by a  dimensionally-truncated model Ms : X × Us → Y such that  Ms(gs(y≤s), y≤s) = 0,  where ∅ ̸= Us ⊆ Rs and gs(y≤s) ∈ X for all y≤s ∈ Us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Consider the problem of finding  a surrogate solution gs,n := An(gs) using an algorithm An which uses n point evaluations  of the s-dimensional function gs, where the surrogate belongs to X such that  ∥gs − gs,n∥L2µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='X)  n→∞  −−−→ 0  with some known convergence rate and µ indicating a probability measure on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' The total  error of the approximation obtained in this fashion can be estimated using the triangle  inequality  ∥g − gs,n∥L2µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='X) ≤ ∥g − gs∥L2µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='X) + ∥gs − gs,n∥L2µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  In this manuscript we focus on the first term—the dimension truncation error—which is  independent of the chosen approximation scheme An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  Dimension truncation error rates are typically studied for problems involving partial  differential equations (PDEs) with random inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' For integration problems a dimension  truncation rate is derived in [20] for the source problem with an affine parameterization  of the diffusion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' This rate was then improved by [6] in the generalized context  of affine parametric operator equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Dimension truncation has also been studied for  coupled PDE systems arising in optimal control problems under uncertainty [10], in the  context of the periodic model of uncertainty quantification for both numerical integra-  tion [17] and kernel interpolation [16], as well as for Bayesian inverse problems governed  by PDEs [5, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' The results in these papers have been proved using Neumann series,  which is known to work well in the affine parametric setting, but may lead to suboptimal  results if the problem depends nonlinearly on the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  In the non-affine setting, using Taylor series makes it possible to derive dimension  truncation error rates by exploiting the parametric regularity of the problem, whereas the  Neumann series approach relies fundamentally on the parametric structure of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  The Taylor series approach was first applied in [8], and motivated the authors in [11]  and [12] to derive dimension truncation error rates for sufficiently smooth, Banach space  valued integrands, and with parameters following a generalized β-Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  An overview of the various dimension truncation error bounds studied in the literature is  given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  Our manuscript is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='1 introduces the multi-index  notation used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  The problem setting is introduced in Section 2,  including the central assumptions for the ensuing dimension truncation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Section 3  contains the L2 dimension truncation theorem for Hilbert space valued functions, and  in Section 4 we discuss the invariance of the dimension truncation rate under certain  transformations of the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Numerical experiments assessing the sharpness of our  2  theoretical results are presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' The paper ends with some conclusions in  Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='1  Notations and preliminaries  Throughout this manuscript, boldfaced symbols are used to denote multi-indices while the  subscript notation mj is used to refer to the j-th component of multi-index m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Let  F := {m ∈ NN  0 : |m| < ∞}  denote the set of finitely supported multi-indices, where the order of multi-index m is  defined as  |m| :=  �  j≥1  mj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  Moreover, we denote  |m|∞ := max  j≥1 mj,  and, for any sequence x := (xj)∞  j=1 of real numbers and m ∈ F, we define  xm :=  �  j≥1  xmj  j ,  where we use the convention 00 := 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  2  Problem setting  Let X be a real separable Hilbert space, U := [− 1  2, 1  2]N a set of parameters, and suppose  that g(y) ∈ X is a parameterized family of functions with smooth dependence on y ∈  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  We define gs(y) := g(y≤s, 0) := g(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' , ys, 0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=') and assume that µ(dy) :=  �  j≥1 µ(dyj) is a countable product probability measure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=', µ(U) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' We suppose that  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' For µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' y ∈ U, there holds  ∥g(y) − gs(y)∥X  s→∞  −−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Let (Θk)k≥0 and b := (bj)j≥1 be sequences of nonnegative numbers such that b ∈  ℓp(N) for some p ∈ (0, 1) and b1 ≥ b2 ≥ · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  Suppose that g is continuously  differentiable up to order k + 1, with  ∥∂νg(y)∥X ≤ Θ|ν|bν  for all y ∈ U and for all ν ∈ Fk := {ν ∈ NN  0 : |ν| ≤ k + 1}, where k := ⌈  1  1−p⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' There holds  � 1/2  −1/2 yj µ(dyj) = 0 and there exists a constant Cµ ≥ 0 such that  � 1/2  −1/2 |yj|k µ(dyj) ≤ Cµ for all k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  If Assumption 2 holds, then we infer that y �→ G(g(y)) for all G ∈ X′ is continuous as  a composition of continuous mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Hence y �→ G(g(y)) is measurable for all G ∈ X′,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=', y �→ g(y) is weakly measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  Since X is assumed to be a separable Hilbert  space, by Pettis’ theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=', [24, Chapter 4]) we obtain that y �→ g(y) is strongly  measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' The upper bound in Assumption 2 is µ-integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Thus we conclude from  Bochner’s theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=', [24, Chapter 5]) and Assumption 2 that g is µ-integrable over  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  3  Further, µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' equality de���nes an equivalence relation among strongly µ-measurable  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' By L2  µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' X) we denote the Hilbert space of equivalence classes of strongly  µ-measurable functions f : U → X with norm  ∥f∥L2µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='X) :=  � �  U  ∥f(y)∥2  X µ(dy)  � 1  2  < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  Moreover, under the Assumptions 1 and 2 it can be shown that g, gs ∈ L2  µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' X) and  lim  s→∞ ∥g(y) − g(y≤s, 0)∥L2µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='X) = lim  s→∞  � �  U  ∥g(y) − g(y≤s, 0)∥2  X µ(dy)  � 1  2  = 0,  by applying Lebesgue’s dominated convergence theorem (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=', [18, Theorem 1] and  [14, Section 26]) to  F s(y) := ∥g(y) − g(y≤s, 0)∥2  X,  which converges µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' to zero by Assumption 1, and can be bounded by (2Θ0)2 by As-  sumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' We use the superscript to avoid confusion with the notation used to denote  dimensionally-truncated functions elsewhere in the document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  3  Dimension truncation error  We will require the following parametric regularity bound for the main dimension trunca-  tion result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Under Assumption 2, there holds  |∂ν∥g(y) − gs(y)∥2  X| ≤  �  max  0≤ℓ≤|ν|  2Θℓ  ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  �2  (|ν| + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='bν  for all ν ∈ Fk and y ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Let ν ∈ Fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' We apply the Leibniz product rule with respect to the inner product  of the Hilbert space X to obtain  ∂ν∥g(y) − gs(y)∥2  X = ∂ν⟨g(y) − gs(y), g(y) − gs(y)⟩X  =  �  m≤ν  � ν  m  �  ⟨∂m(g(y) − gs(y)), ∂ν−m(g(y) − gs(y))⟩X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  Using the Cauchy–Schwarz inequality together with Assumption 2 yields  |∂ν∥g(y) − gs(y)∥2  X| ≤  �  m≤ν  � ν  m  �  ∥∂m(g(y) − gs(y))∥X∥∂ν−m(g(y) − gs(y))∥X  ≤ 4  �  m≤ν  � ν  m  �  Θ|m|bmΘ|ν|−|m|bν−m  = 4bν  |ν|  �  ℓ=0  ΘℓΘ|ν|−ℓ  �  |m|=ℓ  m≤ν  � ν  m  �  = 4bν  |ν|  �  ℓ=0  ΘℓΘ|ν|−ℓ  |ν|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' (|ν| − ℓ)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  ≤ 4  �  max  0≤ℓ≤|ν|  Θℓ  ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  �2  (|ν| + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='bν,  where we used the Vandermonde convolution �  |m|=ℓ  m≤ν  � ν  m  �  =  �|ν|  ℓ  �  =  |ν|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' (|ν|−ℓ)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='.  4  The main result of this document is stated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Let g(y) ∈ X, y ∈ U, satisfy Assumptions 1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Then  ∥g − gs∥L2µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='X) = O(s− 1  p + 1  2),  where the implied coefficient is independent of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Let s ≥ 1 and define  F s(y) := ∥g(y) − gs(y)∥2  X  for y ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  In the special case of the uniform distribution µ(dy) = dy, we can apply [12, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='2]  to obtain  ∥g − gs∥2  L2(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='X) =  ����  �  U  (F s(y) − F s(y≤s, 0)) dy  ���� = O(s− 2  p +1),  from which the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' For completeness, we present the proof below for the prob-  ability measure µ and because parts of the argument will also be useful to establish the  invariance of the dimension truncation rate in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  Developing the Taylor expansion of F s about (y≤s, 0) and observing that F s(y≤s, 0) =  0, we obtain  F s(y) =  k  �  ℓ=1  �  |ν|=ℓ  νj=0 ∀j≤s  yν  ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' ∂νF s(y≤s, 0)  +  �  |ν|=k+1  νj=0 ∀j≤s  k + 1  ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' yν  � 1  0  (1 − t)k∂νF s(y≤s, ty>s) dt,  (2)  where y>s := (yj)j>s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Integrating both sides over y ∈ U yields  �  U  F s(y) µ(dy) =  k  �  ℓ=1  �  |ν|=ℓ  νj=0 ∀j≤s  1  ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  �  U  yν∂νF s(y≤s, 0) µ(dy)  +  �  |ν|=k+1  νj=0 ∀j≤s  k + 1  ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  �  U  � 1  0  (1 − t)kyν∂νF s(y≤s, ty>s) dt µ(dy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  If ν ∈ Fk is such that νj = 1 for any j > s, then Fubini’s theorem together with Assump-  tion 3 imply for the summands appearing in the first term that  �  U  yν∂νF s(y≤s, 0) µ(dy) =  � �  j>s  �  1  2  − 1  2  yνj  j µ(dyj)  �  �  ��  �  =0  �  [− 1  2, 1  2]s ∂νF s(y≤s, 0) µ(dy>s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  Therefore all multi-indices with any component equal to 1 can be removed from the first  sum (especially, we can omit all multi-indices with |ν| = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Further, applying the regu-  larity bound proved in Lemma 1 and writing open the definition of F s yields  �  U  ∥g(y) − gs(y)∥2  X µ(dy) ≤ Ck  µ  �  max  0≤ℓ≤k  2Θℓ  ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  �2  (k + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  k  �  ℓ=2  �  |ν|=ℓ  νj=0 ∀j≤s  νj̸=1 ∀j>s  bν  + Ck+1  µ  �  max  0≤ℓ≤k+1  2Θℓ  ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  �2  (k + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  �  |ν|=k+1  νj=0 ∀j≤s  1  ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='bν,  (3)  5  where we used  � 1  0 (1 − t)k dt =  1  k+1 and Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  The second term in (3) can  be estimated from above using the multinomial theorem in conjunction with Stechkin’s  lemma:  �  |ν|=k+1  νj=0 ∀j≤s  1  ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='bν ≤  �  |ν|=k+1  νj=0 ∀j≤s  |ν|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' bν =  � �  j>s  bj  �k+1  ≤ s(k+1)(− 1  p +1)  � �  j≥1  bp  j  � k+1  p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  On the other hand, the first term in (3) can be estimated similarly to [6]:  �  2≤|ν|≤k  νj=0 ∀j≤s  νj̸=1 ∀j>s  bν ≤  �  0̸=|ν|∞≤k  νj=0 ∀j≤s  νj̸=1 ∀j>s  bν = −1 +  �  j>s  �  1 +  k  �  ℓ=2  bℓ  j  �  = −1 +  �  j>s  �  1 + b2  j  k−2  �  ℓ=0  bℓ  j  �  ≤ −1 +  �  j>s  �  1 + b2  j  k−2  �  ℓ=0  bℓ  1  � �� �  =:βk  �  ≤ −1 + exp  �  βk  �  j>s  b2  j  �  =  �  ℓ≥1  1  ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  �  βk  �  j>s  b2  j  �ℓ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  Using �  j>s b2  j ≤ s− 2  p +1(�  j≥1 bp  j)  2  p , which follows from Stechkin’s lemma, we further  estimate  �  ℓ≥1  1  ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  �  βk  �  j>s  b2  j  �ℓ  ≤ s− 2  p +1 �  ℓ≥1  1  ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' (βk∥b∥2  p)ℓ = s− 2  p +1(−1 + exp(βk∥b∥2  p)  since s− 2  p +1 ≥ (s− 2  p +1)ℓ for all ℓ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  Altogether, the above discussion yields the bound  ∥g(y) − gs(y)∥2  L2µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='X) =  �  U  ∥g(y) − gs(y)∥2  X µ(dy) = O(s− 2  p +1 + s(k+1)(− 1  p +1)),  where the implied coefficient is independent of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Since we assumed that k = ⌈  1  1−p⌉, the  assertion follows by taking the square root on both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  4  Invariance of the dimension truncation rate under trans-  formations of variables  An interesting consequence of the Taylor series argument used in Theorem 1 is that the di-  mension truncation rate remains invariant under certain transformations of the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  This has been previously observed in the context of dimension truncation for integration  problems under the periodic model [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' To make this notion precise, let us consider a  mapping ξ: U → U, ξ(y) := (ξ(y1), ξ(y2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' ), which satisfies the following conditions:  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' There hold ξ(0) = 0 and  � 1/2  −1/2 ξ(y) dy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' There exists Cξ ≥ 0 such that  � 1/2  −1/2 |ξ(y)|k dy ≤ Cξ for all k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  Then we obtain the following as a consequence of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Let g(y) ∈ X, y ∈ U, satisfy Assumptions 1–3 and let ξ : U → U satisfy  Assumptions 4–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Define the ξ-transformed function gξ by  gξ(y) := g(ξ(y)),  y ∈ U,  6  and its dimension truncation by gξ,s(y) := gξ(y≤s, 0) for y ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Then  ∥gξ − gξ,s∥L2µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='X) = O(s− 1  p + 1  2 ),  where the implied coefficient is independent of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' We introduce F s  ξ (y) := ∥gξ(y) − gξ,s(y)∥2  X for y ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' By carrying out the change  of variable y ← ξ(y) in (2), we obtain  F s  ξ (y) =  k  �  ℓ=1  �  |ν|=ℓ  νj=0 ∀j≤s  ξ(y)ν  ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  ∂νF s(ξ(y≤s, 0))  +  �  |ν|=k+1  νj=0 ∀j≤s  k + 1  ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' ξ(y)ν  � 1  0  (1 − t)k∂νF s(ξ(y≤s, ty>s)) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  Integrating the above formula on both sides over y ∈ U and utilizing Lemma 1 as well as  Assumption 5, we obtain—in complete analogy with the proof of Theorem 1—that  �  U  ∥gξ(y) − gξ,s(y)∥2  X dy ≤ Ck  ξ  �  max  0≤ℓ≤k  2Θℓ  ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  �2  (k + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  k  �  ℓ=2  �  |ν|=ℓ  νj=0 ∀j≤s  νj̸=1 ∀j>s  bν  + Ck+1  ξ  �  max  0≤ℓ≤k+1  2Θℓ  ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  �2  (k + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  �  |ν|=k+1  νj=0 ∀j≤s  1  ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='bν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  The desired result follows by exactly the same argument as in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  As an application, with U := [− 1  2, 1  2]N, let ξ: U → U satisfy the Assumptions 4 and 5,  let D ⊂ Rd, d ∈ {1, 2, 3}, be a bounded Lipschitz domain, and let f : D → R be a fixed  source term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Consider the parametric PDE problem  �  −∇ · (aξ(x, y)∇uξ(x, y)) = f(x),  x ∈ D, y ∈ U,  uξ(x, y) = 0,  x ∈ ∂D, y ∈ U,  (4)  endowed with the ξ-transformed diffusion coefficient  aξ(x, y) := a0(x) +  ∞  �  i=1  ξ(yi)ψi(x),  x ∈ D, y ∈ U,  which is assumed to satisfy the following:  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' There exist amin, amax > 0 such that 0 < amin ≤ aξ(x, y) ≤ amax < ∞ for all x ∈ D  and y ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' a0 ∈ L∞(D) and ψi ∈ L∞(D) for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' �∞  i=1 ∥ψi∥p  L∞(D) < ∞ for some p ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  In this case, the transformation ξ(y) := ( 1  √  6 sin(2πyj))j≥1 corresponds to the so-called  periodic model studied in [13, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Let X := H1  0(D) be equipped with the norm ∥v∥X :=  7  �  D ∥∇v(x)∥2  Rd dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' In this special case, it is known that the weak solution u(·, y) ∈ X to (4)  for y ∈ U satisfies the parametric regularity bound  ∥∂νuξ(·, y)∥X ≤ (2π)|ν|∥f∥X′  amin  �  m≤ν  |m|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='bm �  j≥1  S(νj, mj)  for all ν ∈ F and y ∈ U, where the source term f ∈ X′, S(·, ·) denotes the Stirling number  of the second kind, and b := (bj)j≥1 is defined by setting bj :=  ∥ψj∥L∞(D)  √  6amin  for all j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  Let µ(dµ) = dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Then Corollary 1 can be used to deduce that  ∥uξ − uξ,s∥L2µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='X) = O(s− 1  p + 1  2 ),  where the constant is independent of the dimension s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  In fact, if Xh is a conforming  finite element subspace of X, uξ,h(·, y) ∈ Xh denotes the finite element discretization of  uξ(·, y) ∈ X for all y ∈ U, and uξ,h,s(·, y) ∈ Xh denotes the dimension truncation of  uξ,h(·, y) for all y ∈ U, then we have  ∥uξ,h − uξ,h,s∥L2µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='X) = O(s− 1  p + 1  2),  independently of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  Finally, we present an example illustrating how our results can be applied to nonlinear  quantities of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Let X := H1  0(D) as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Consider the nonlinear quantity of interest  Gnl(v) := ∥v∥2  X :=  �  D  ∥∇v(x)∥2  Rd dx,  v ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  (5)  If u(·, y) ∈ X is the solution to (4) with U = [− 1  2, 1  2]N, µ(dy) := dy, and ξ(y) := y, then  it is known to satisfy Assumptions 1–3 with the regularity bound  ∥∂νu(·, y)∥X ≤ C|ν|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='bν,  where the constant C > 0 only depends on ∥f∥X′ and amin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' By the Leibniz product rule,  there holds  ∂νGnl(u(·, y)) =  �  D  �  m≤ν  � ν  m  �  ∇∂mu(x, y) · ∇∂ν−mu(x, y) dx  ≤  �  m≤ν  � ν  m  �  ∥∂mu(·, y)∥X∥∂ν−mu(·, y)∥X  ≤ C2 �  m≤ν  � ν  m  �  |m|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='bm|ν − m|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='bν−m  = C2bν  |ν|  �  ℓ=0  ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' (|ν| − ℓ)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  �  m≤ν  |m|=ℓ  � ν  m  �  = C2bν(|ν| + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=',  where we used the Vandermonde convolution �  |m|=ℓ  m≤ν  � ν  m  �  =  �|ν|  ℓ  �  =  |ν|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' (|ν|−ℓ)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='.  It follows from Theorem 1 that  ∥Gnl(u) − Gnl(us)∥L2µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='R) = O(s− 1  p + 1  2),  independently of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Moreover, if ξ(y) := ( 1  √  6 sin(2πyj))j≥1, then it follows from Corollary 1  that  ∥Gnl(uξ) − Gnl(uξ,s)∥L2µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='R) = O(s− 1  p + 1  2 ),  independently of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  8  5  Numerical experiments  Let D = (0, 1)2 be a spatial domain, U = [− 1  2, 1  2]N, and f(x) := x1 a fixed source term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  Let ξ: U → U, ξ(y) = ( 1  √  6 sin(2πyj))j≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' We consider the PDE problem  �  −∇ · (aξ(x, y)∇uξ(x, y)) = f(x),  x ∈ D, y ∈ U,  uξ(x, y) = 0,  x ∈ ∂D, y ∈ U,  (6)  equipped with the diffusion coefficient  aξ(x, y) = 3  2 +  �  j≥1  ξ(yj)j−ϑ sin(jπx1) sin(jπx2),  x ∈ D, y ∈ U, ϑ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  The PDE (6) is spatially discretized using a first-order conforming finite element method  with mesh size h = 2−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  We consider the dimension truncation error for the full PDE solution using the formula  ∥uξ − uξ,s∥L2(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='L2(D)) ≈  � �  [− 1  2 , 1  2 ]s′ ∥uξ,s′(·, y) − uξ,s(·, y)∥2  L2(D) dy  � 1  2  ,  and we also consider the nonlinear quantity of interest (5), estimating the dimension  truncation error using the formula  ∥Gnl(uξ) − Gnl(uξ,s)∥L2(U) ≈  � �  [− 1  2, 1  2]s′ |Gnl(uξ,s′(·, y)) − Gnl(uξ,s(·, y))|2 dy  � 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  In both cases, we choose s′ ≫ s and the high-dimensional integrals are approximated using  a randomly shifted rank-1 lattice rule with 220 cubature nodes and a single random shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  As the integration lattice, we use in both cases an off-the-shelf rank-1 lattice rule [19,  lattice-39101-1024-1048576.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='3600] and use the same random shift for each value of ϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' As  the reference solution, we use the PDE solution corresponding to s′ = 211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  The numerical results for dimensions s ∈ {2k : k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' , 9} and decay rates ϑ ∈  {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='0} corresponding to the full PDE solution and the nonlinear quantity of interest  are displayed in Figures 1 and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' The theoretical convergence rates in each  case are −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='0, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='5, and −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='5, respectively, and they are displayed alongside the numerical  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  The convergence graphs corresponding to the full PDE solution in Figure 1 display  an aliasing behavior between 10 ≤ s ≤ 100, which may be explained by the contributions  of the finite element discretization error as well as the use of an “off-the-shelf” lattice  rule (in contrast to a “tailored” lattice rule).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' This behavior appear to be exacerbated in  the convergence graphs corresponding to the nonlinear quantity of interest in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  Nonetheless, in all cases the theoretically anticipated convergence rates are easily observed  in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  We remark that the convergence graphs corresponding to the affine and  uniform model with ξ(y) := (yj)j≥1 are extremely similar to the results corresponding to  the periodic model, and have thus been omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  9  Figure 1: The dimension truncation errors of the full PDE solution corresponding to a periodically parameterized  input random field with decay parameters ϑ ∈ {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' The expected dimension truncation error rates are  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='0, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='5, and −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  Figure 2: The dimension truncation errors of the nonlinear quantity of interest corresponding to a periodically  parameterized input random field with decay parameters ϑ ∈ {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' The expected dimension truncation  error rates are −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='0, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='5, and −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  6  Conclusions  Unlike many studies which have considered the dimension truncation error rate within  the context of high-dimensional numerical integration, we considered the L2 dimension  truncation error rate for parametric Hilbert space valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Our theory covers  both affine parametric as well as non-affine parametric problems with sufficiently smooth  dependence on a sequence of bounded, parametric variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' The main dimension trun-  cation results presented in this work can be applied to nonlinear quantities of interest of  parametric model problems, provided that they satisfy the conditions of our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  10  In addition, the Hilbert space can be chosen to be a finite element subspace, indicating  that our dimension truncation results are also valid for conforming finite element approx-  imations of parametric PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='  The L2 dimension truncation error rates considered in this work arise, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=', in the  study of high-dimensional function approximation of parametric PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' An example of  such an approximation scheme is the kernel method over lattice point sets considered  in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' The kernel method was analyzed in the context of the so-called periodic model, in  which a countable number of independent random variables enter the input random field  of the PDE as periodic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
+page_content=' Our second main result shows that the L2 dimension  truncation error rate remains invariant under certain transformations of the parametric  variables: especially, the L2 dimension truncation rate considered in this work holds for  periodically parametrized model problems such as those studied in [13, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
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+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'}
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