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+y9FKT4oBgHgl3EQfMS25/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +z9AyT4oBgHgl3EQfPPZM/content/2301.00020v1.pdf filter=lfs diff=lfs merge=lfs -text diff --git a/19FQT4oBgHgl3EQfFDWe/content/tmp_files/2301.13240v1.pdf.txt b/19FQT4oBgHgl3EQfFDWe/content/tmp_files/2301.13240v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0771c750731a3dfc3929562da373237c56ee9f54 --- /dev/null +++ b/19FQT4oBgHgl3EQfFDWe/content/tmp_files/2301.13240v1.pdf.txt @@ -0,0 +1,4229 @@ +AdS super gluon scattering up to two loops: +A position space approach +Zhongjie Huanga,b, Bo Wanga,b, Ellis Ye Yuana,b, Xinan Zhouc +aZhejiang Institute of Modern Physics, School of Physics, Zhejiang University, +Hangzhou, Zhejiang 310058, China +bJoint Center for Quanta-to-Cosmos Physics, Zhejiang University, +Hangzhou, Zhejiang 310058, China +cKavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, +Beijing 100190, China. +E-mail: eyyuan@zju.edu.cn, b w@zju.edu.cn, zjhuang@zju.edu.cn, +xinan.zhou@ucas.ac.cn +Abstract: We carry out a bootstrap study of four-point correlators in 4d N = 2 SCFTs +which are dual to super Yang-Mills on AdS5×S3. We focus on the simplest 1 +2-BPS operators +which correspond to the super gluons in the massless current multiplet. Our computation +is based on an ansatz in position space which is inspired by a hidden symmetry structure +manifest in the leading terms of the Lorentzian singularities of the correlators. By using +other consistency conditions, we completely fix the super gluon correlators at one and two +loops in the bulk genus expansion, up to possible counterterms. Our results reveal a number +of interesting properties enriched by the color structures. In particular, the implication of +hidden conformal symmetry on the full super gluon reduced correlator exhibits an analogous +pattern as in the AdS5 × S5 supergravity correlators recently computed up to two loops. +arXiv:2301.13240v1 [hep-th] 30 Jan 2023 + +Contents +1 +Introduction +2 +2 +Preliminaries +5 +2.1 +Four-point correlators +6 +2.2 +Projectors and color decomposition +7 +2.3 +Spectrum and conformal block decomposition +9 +3 +Leading logarithmic singularities +11 +3.1 +Recursion by unitarity +12 +3.2 +Hidden conformal symmetry +14 +4 +One-loop correlator +16 +4.1 +Ansatz +16 +4.2 +Constraints +20 +4.3 +Results at one loop +21 +4.4 +Comparison with the Mellin space result +25 +5 +Two-loop correlator +26 +5.1 +Color structures at two loops +26 +5.2 +Ansatz and constraints +28 +5.3 +Results at two loops +32 +6 +Outlook +34 +A Single-valued multiple polylogarithms as basis functions +35 +B Analytic result of the one-loop reduced correlator +38 +C Bulk-point limit +40 +D Recursion of twist-4 data at log2 u +43 +– 1 – + +1 +Introduction +The AdS/CFT correspondence maps correlation functions of local operators in the CFT +to on-shell scattering amplitudes in AdS. In the holographic limit, these observables are +expanded in powers of 1/c with respect to the large central charge. At the leading order, the +holographic correlators are just given by the generalized free field theory due to the large N +factorization and they can be computed simply by Wick contractions. However, to extract +nontrivial dynamical information one needs to go to higher orders in 1/c . Computing these +subleading contributions is in general intractable from the CFT side alone as the theory +is strongly coupled. The weakly coupled dual description makes it possible, at least in +principle, as holographic correlators can be computed as amplitudes at various loop orders +by using the AdS generalization of the standard Feynman diagram expansion. However, +it should be noted that such a recipe is rather impractical to use beyond the few simplest +cases [1–5], due to the proliferation of diagrams and complicated AdS vertices [6]. In fact, +just at the tree level, i.e., at order 1/c, the computation of general four-point functions +remained an unsolved problem for almost two decades. +A much better strategy, initiated in [7, 8], is the bootstrap approach, which led to the +complete tree-level four-point functions of 1 +2-BPS operators with arbitrary Kaluza-Klein +(KK) levels for IIB supergravity in AdS5 × S5. The bootstrap approach exploits both the +amplitude intuition from the bulk and the superconformal constraints from the boundary, +and is currently the most efficient method for computing holographic correlators. At the +moment, there is already a wealth of results at tree level. +For example, general four- +point functions of arbitrary 1 +2-BPS operators have been computed in closed forms in all +maximally superconformal theories [9, 10], as well as in theories with half the amount of +maximal superconformal symmetry [11–13].1 By contrast, our understanding for loop level +correlators is much more limited, even in the paradigmatic example of IIB supergravity +on AdS5 × S5. The first one-loop correlator was computed in [15, 16] for the stress tensor +multiplet in position space and later in Mellin space [17]. The calculation was generalized +to four-point functions with higher KK levels in [18–20]. However, explicit one-loop results +are still case-by-case with the exception for the ⟨22pp⟩ family in [20]. At two loops and +higher, the situation is more difficult. The strategy at one loop, which is based on the AdS +unitarity method [21], now requires the additional input of multi-trace operators. Such +information is not yet available in the literature.2 +Therefore, one can in principle only +compute a part of the correlator that corresponds to the iterated s-channel cuts in flat +space [24, 25]. However, it turns out that this difficulty can be overcome at two loops +by formulating an ansatz that is structured by an observed extra hidden symmetry in the +leading Lorentzian singularities, together with additional physical constraints such as the +behavior in the flat-space limit [26]. In this way, the four-point two-loop correlator of stress +tensor multiplets has also been bootstrapped [26, 27]. +1See [14] for a recent review. +2For example, at two loops there are exchange contributions from triple-trace operators. These can be +in principle extracted from tree-level five-point functions. However, only five-point functions of the form +⟨pp222⟩ have been computed [22, 23] while extracting the data requires all ⟨pqr22⟩ five-point functions. +– 2 – + +In this paper, we continue to explore the loop-level calculation of holographic correla- +tors. However, instead of considering correlators of super gravitons, we will focus on super +gluons of SYM in AdS. More precisely, we consider a decoupling sector of certain 4d N = 2 +SCFTs in the holographic limit. These SCFTs can be engineered by using either a stack +of N D3-branes probing F-theory singularities [28, 29] or D3-branes with probe D7-branes +[30]. The near horizon geometries in both cases include an AdS5 ×S3 subspace which hosts +localized degrees of freedom corresponding to the gluons. In the limit of N → ∞, the gluon +degrees of freedom effectively decouple from the graviton degrees of freedom living in the +full 10d bulk via 1/N suppressions in the vertices [13]. The resulting physics in 8d is the +same regardless of the model we choose. Strictly speaking, the decoupling happens only +at the leading order and correlators at subleading orders include gravity contributions as +well. However, in this paper we will choose to turn off gravity to all orders in 1/N and our +goal is to compute the super gluon four-point correlators in this SYM theory in AdS5 × S3 +to two loops. +The motivations for considering super gluon correlators in such a setup are two fold. +First, as we already mentioned, holographic correlators are on-shell scattering amplitudes +in AdS. It is natural to wonder if various remarkable properties of flat-space amplitudes +admit generalizations in curved backgrounds. In particular, does the double copy relation +[31], which famously states gravity is the “square” of YM, still holds in AdS? To this end, +it makes sense to decouple gravity and study the amplitudes of just SYM in AdS. In fact, +analysis of this model at tree level already showed evidence for such a generalization at +four points [32]. Here we will compute the loop corrections of the super gluon four-point +functions which will serve as the starting point for exploring further generalizations of +double copy at higher genus. Second, the super gluon case also provides a useful playground +for acquiring deeper understandings of various results from the supergravity setup. The +position space method for computing loop-level correlators so far have only been tested +in AdS5 × S5 and it is a priori unclear whether it can be applied to other backgrounds. +In this paper, we will show that such a method can be successfully applied to AdS5 × S3 +and leads to similar results to the supergravity case. In the process, we also provide a +nontrivial consistency check of the one-loop result which was previously obtained in [33] +using Mellin space techniques. Moreover, the various different color structures allow us to +have a more refined understanding of the dynamical structures of the correlators which are +similar in the two cases, whereas in the supergravity case all structures are mixed up due +to the absence of colors. +Let us briefly outline our strategy and the key results of the paper. Our approach is +similar to that of [15, 19, 26]. We first make an ansatz in position space which requires a set +of building block functions. Due to the similarity with the supergravity case at tree level, we +assume that single-valued multiple polylogarithms (SVMPLs) continue to be a good basis +for the super gluon correlators at one and two loops. In other words, the correlators are +assumed to be linear combinations of SVMPLs with rational functions of the cross ratios as +coefficients. However, this turns out to be a bit too general. In the supergravity case, the +existence of a tree-level 10d superconformal symmetry [34] highlights a special eighth-order +differential operator ∆(8) which relates the correlators of the top and bottom components +– 3 – + +of the super graviton multiplet. By unitarity this symmetry extends to the leading part +of the Lorentzian singularities at arbitrary loops. Using this operator at loop levels, the +supergravity correlators can be more succinctly written in terms of the pre-correlators L +[19, 26, 27] +H1-loop +sugra = ∆(8)L1-loop +sugra + 1 +4Htree +sugra , +(1.1a) +H2-loop +sugra = +� +∆(8)�2 +L2-loop +sugra + 5 +4H1-loop +sugra − 1 +16Htree +sugra , +(1.1b) +together with additional lower-order correlators. A similar 8d superconformal symmetry +also appears in the tree-level super gluon correlators [13] and the role of ∆(8) is replaced +by a fourth-order operator ∆(4). In analogy with the supergravity case, we assume that +similar pre-correlators can also be defined for super gluons +H1-loop +SYM = ∆(4)L1-loop +SYM + ¯Htree +SYM , +(1.2a) +H2-loop +SYM = +� +∆(4)�2 +L2-loop +SYM + � +H1-loop +SYM + � +Htree +SYM , +(1.2b) +where ¯Htree +SYM and � +Htree +SYM are “tree-like” correlators and � +H1-loop +SYM +is a “one-loop-like” corre- +lator. We will be more precise about the meaning of “tree-like” and “one-loop-like”. But +for the moment it suffices to say they are characterized by the transcendental degrees of +SVMPLs expected at each loop order. Then the position space ansatz in terms of SVMPLs +is formulated in terms of the pre-correlators L and the lower-order objects Hi, in parallel +with the supergravity story. Note that, unlike supergravity, super gluon correlators have +different color structures. Therefore, we make such an ansatz for each independent color +structure and assume the correlator to be a linear combination of all these structures. To +perform the bootstrap, we impose a number of consistency conditions. These are +• Leading logarithmic singularities +• Crossing symmetry +together with a few other constraints. Here the leading logarithmic singularities rely only +on the tree-level data and can be computed at any loop order. At two loops, the additional +constraints further include comparison with the scattering amplitude in a proper flat-space +limit that can be computed independently using flat-space techniques, and the data of +twist-4 operators which can be extracted from the tree and one-loop correlators. Imposing +these constraints, we find that all parameters in the ansatz are fixed except for those +corresponding to the counterterms needed for the UV divergences. Moreover, the tree-like +and one-loop-like terms turn out to be exactly the tree-level and one-loop correlators except +for simple replacements for the color structures. +The rest of the paper is organized as follows. We review in Section 2 some preliminaries +of super gluon four-point functions which include the superconformal kinematics, color +structure and superconformal block decomposition. In Section 3 we review how the leading +logarithmic singularities can be constructed from the tree-level data and compute them in +closed forms using hidden conformal symmetry. In Section 4 we introduce the position +– 4 – + +space method and demonstrate it by bootstrapping the one-loop correlator. In Section 5 +we apply the method to the two-loop correlator and obtain the full answer by imposing +constraints. In Section 6 we outline a few future directions. The paper also has several +appendices where we include further technical details. +In Appendix A we give a brief +review of the properties of SVMPLs. Appendix B contains the complete analytic result +for the reduced correlator at one loop. The details of the flat-space two-loop amplitude +are presented in Appendix C. In Appendix D we discuss the computations related to the +twist-4 data. +2 +Preliminaries +In this paper, we consider holographic correlators corresponding to super gluon scattering +in AdS. To be concrete, we consider SYM in AdS5 ×S3 which arises as a decoupling sector +of certain 4d N = 2 SCFTs. One can construct these SCFTs from a stack of N D3-branes, +by either using them to probe F-theory singularities [28, 29] or by adding a few probe +D7-branes [30]. In either case, in the near horizon limit there is an AdS5 × S3 subspace in +the total ten dimensional spacetime which is locally AdS5 ×S5. On this subspace there are +localized degrees of freedom transforming as an N = 1 vector multiplet and in the adjoint +representation of certain flavor group GF of the boundary CFT. Here GF depends on the +theory and is a gauge group from the bulk perspective.3 Since the N = 1 vector multiplet +contains fields with Lorentz spin at most 1, its KK reduction with respect to S3 also leads +to fields with the same maximal spin. These are the massless and massive AdS gluons +and their super partners. Because of the bound on spins all the KK modes have to reside +in 1 +2-BPS multiplets by the 4d N = 2 representation theory. Their conformal dimensions +are fully fixed by R-symmetry and therefore are independent of the bulk theory. More +precisely, the superconformal primaries of these 1 +2-BPS are scalar operators Ok labelled by +an integer k = 2, 3, . . .. They have conformal dimension ∆ = k and transform in the spin-k +2 +representation of the SU(2)R R-symmetry group. Moreover, they transform in the adjoint +representaiton of the flavor group GF . We will call the fields dual to these superprimaries +the super gluons. The real spinning gluon fields are superconformal descendants in the +multiplets. +By contrast, the super gravitons and their super partners live in the full ten dimen- +sional spacetime. +Unlike the supergluons, their KK spectrum depends on the specific +theory. However, an interesting fact of all these 4d N = 2 SCFTs is that there is a hier- +archy in the couplings at large N. For example, the cubic coupling of three super gluons +(or their superconformal descendants) is of order 1/ +√ +N, while the coupling involving two +super gluons and one super graviton is of order 1/N. Therefore, for large N the lead- +ing contribution to the super gluon correlators comes only from the 8d SYM. Subleading +corrections in 1/N will in general contain graviton contributions as well. +As mentioned in the introduction, we will continue to study loop corrections of the +four-point correlator of the super gluon operator O2 in the AdS5 × S3 SYM. Although this +does not give the full answer for this correlator in N = 2 SCFTs, it makes sense from +3Therefore, in the following we will use “flavor”, “color” and “gauge” interchangeably. +– 5 – + +the perspective of exploring curved space generalizations of gauge theory amplitudes. This +section serves to provide some preliminary features of this correlator, which will be used +in our bootstrap computation. +2.1 +Four-point correlators +With all indices restored, the super gluon operator O2 has the form +OI;a1a2 +2 +(x) , +(2.1) +where I = 1, . . . , dim(GF ) is the flavor symmetry index and ai = 1, 2 are the SU(2)R R- +symmetry indices. It is convenient to contract the R-symmetry indices with two-dimensional +polarization spinors +OI +2(x; v) = OI;a1a2 +2 +va1va2 . +(2.2) +The four-point function +GI1I2I3I4(xi; vi) = ⟨OI1 +2 (x1; v1)OI2 +2 (x2; v2)OI3 +2 (x3; v3)OI4 +2 (x4; v4)⟩ +(2.3) +is therefore a function of both the spacetime coordinates xi and the internal space spinors +vi. Exploiting the bosonic symmetries, i.e. conformal symmetry and R-symmetry, we can +write the correlator as a function of the cross ratios +GI1I2I3I4 = (v1 · v2)2 (v3 · v4)2 +x4 +12x4 +34 +GI1I2I3I4(u, v; α) +(2.4) +where xij = xi − xj, vi · vj = va +i vb +jϵab (ϵab being the 2d Levi–Civita symbol), and the cross +ratios are +u = x2 +12x2 +34 +x2 +13x2 +24 += z¯z , +v = x2 +23x2 +14 +x2 +13x2 +24 += (1 − z)(1 − ¯z) , +α = (v1 · v3) (v2 · v4) +(v1 · v2) (v3 · v4) . +(2.5) +In addition, the ferminonic generators in the superconformal algebra impose further con- +straints known as the superconformal Ward identities [35] +(x∂x − α∂α) GI1I2I3I4(z, ¯z; α) +�� +α=1/x = 0 , +x = z or ¯z. +(2.6) +Solving these identities, we can decompose the correlator into two parts +GI1I2I3I4(z, ¯z; α) = GI1I2I3I4 +0 +(z, ¯z; α) + RHI1I2I3I4(z, ¯z) , +(2.7) +where +R = (1 − zα)(1 − ¯zα) +z¯z +. +(2.8) +Note that our definiton of R is different from that of [13] by a z¯z in the denominator. The +first term GI1I2I3I4 +0 +is protected, while dynamical information of the correlator is encoded in +the reduced correlator HI1I2I3I4. In our case of O2 correlator, HI1I2I3I4 is simply a function +of the spacetime cross ratios {z, ¯z} (or equivalently {u, v}) and is free of the R-symmetry +cross ratio α. +– 6 – + +We study the expansion of the correlator with respect to the large flavor central charge +CJ 4. For convenience, we use the small parameter aF = 6/CJ , with respect to which the +expansion reads +HI1I2I3I4 +2222 +≡ H = H(0) + aF H(1) + a2 +F H(2) + a3 +F H(3) + · · · . +(2.9) +This expansion has a nice interpretation from the bulk point of view. The leading con- +tribution H(0) is associated with the disconnected part of scattering in AdS and can be +evaluated by generalized free field theory. The first correction H(1) is the tree-level scat- +tering of the super gluons, which has been obtained in [13]. The higher-order correction +H(L+1) corresponds to scattering at L loops, where the one-loop case has been computed +in [33] using Mellin space techniques. +2.2 +Projectors and color decomposition +Since we are studying gluon scattering, as usual the correlator H(L+1) at each perturbative +order splits into various color factors and their corresponding dynamical factors 5 +� +H(L+1)�I1I2I3I4 = +� +C +CI1I2I3I4H(L+1) +C +. +(2.10) +A color factor CI1I2I3I4 is constructed out of the structure constants fIJK of the gauge +group according to the topology of a diagram that may arise at the given loop order +according to Feynman rules (or Witten rules in AdS), and so the summation above carries +over all possible topologies at L loops. +The dynamical factors H(L+1) +C +are functions of +kinematic variables {z, ¯z}, and with the above decomposition they only rely on diagram +topologies as well, regardless of any specific choice of the gauge group GF . +The decomposition (2.10) is not the most convenient for practical computations as the +color factors CI1I2I3I4 are highly redundant. So instead one often seeks for other types +of color decompositions. Because our computation requires the input from CFT data of +the spectrum and the coefficients arising in OPEs, it is preferable to decompose the color +factors in a way that resembles the conformal block expansion. This can be fulfilled by +specifying a particular channel (say the s-channel) and introduce an operation P I1I2|I3I4 +a +that picks out irreducible represetation a of the flavor group from the tensor products of +two adjoints adj ⊗ adj. This is called an s-channel projector, and by definition it satisfies +the symmetry properties +P I1I2|I3I4 +a += (−1)|Ra|P I2I1|I3I4 +a +, +P I1I2|I3I4 +a += P I3I4|I1I2 +a +, +(2.11) +where |Ra| stands for the parity of representation a, and the idempotency condition +P I1I2|I5I6 +a +P I6I5|I3I4 +b += δabP I1I2|I3I4 +a +. +(2.12) +4The flavor central charge CJ appears in the flavor current two-point functions as ⟨J I +µ (x)J J +ν (0)⟩ = +CJ +2π2 +δIJ (δµν−2 xµxν +x2 +) +x6 +. Moreover, via supersymmetry, it is related to the three-point function coefficient C2 +2,2,2 +of ⟨O2O2O2⟩ by CJ = 1 +6C2 +2,2,2. +5In the context of scattering amplitudes these coefficients of color factors are more frequently called kine- +matic factors (when referring to numerators in Feynman diagrams). In this paper we call them dynamical +factors to remind the readers that they contains the dynamical information of the theory. +– 7 – + +In particular from (2.12) we also have P I1I2|I3I4 +a +P I1I2|I3I4 +b += δabdim(Ra). Therefore ev- +ery color factor appearing in (2.10) receives a unique decomposition onto the s-channel +projectors +CI1I2I3I4 = +� +a∈adj⊗adj +P I1I2|I3I4 +a +Ca , +(2.13) +with coefficients Ca, or equivalently +Ca = dim(Ra)−1P I1I2|I3I4 +a +CI1I2I3I4 . +(2.14) +The efficiency of these projectors comes from the fact that the set of irreducible rep- +resentations arising in adj ⊗ adj depends only on the gauge group GF but not on the +perturbative order. As a result, the color decomposition of the reduced correlator H as +well as any term H(L+1) in the expansion (2.9) can be carried out in a uniform manner. +Generically, we have +HI1I2I3I4 = +� +a∈adj⊗adj +P I1I2|I3I4 +a +Ha , +(2.15) +and H(L+1) follows similarly. Furthermore, the idempotency condition (2.12) also makes +the recursive relation between different loop levels very simple, as will be further illustrated +in the next section. +As a simple example for the use of projectors, let us quickly review the tree-level +correlator H(1), which was computed in [13]. Its takes the following form +H(1) = csH(1) +s ++ ctH(1) +t ++ cuH(1) +u , +(2.16) +with +H(1) +s += u3 +3 +� +2∂u + (1 + v)∂u∂v + u∂2 +u +� ¯D1111, +(2.17a) +H(1) +t += −u3 +3 +� +2∂v + v∂2 +v + (1 + u)∂u∂v +� ¯D1111, +(2.17b) +H(1) +u += u3 +3 +� +2∂v + v∂2 +v − 2∂u + (u − v)∂u∂v − u∂2 +u +� ¯D1111 . +(2.17c) +Here ¯D1111 is an example of the ¯D-functions which are contact Witten diagrams in AdS 6 +¯D1111(z, ¯z) = +1 +z − ¯z +� +2Li2(z) − 2Li2(¯z) + log(z¯z) log +�1 − z +1 − ¯z +�� +. +(2.18) +cs/t/u are color factors built from structure constants +(cs)I1I2I3I4 = fI1I2JfJ I3I4, +(ct)I1I2I3I4 = fI1I4JfJ I2I3, +(cu)I1I2I3I4 = fI1I3JfJ I4I2, +(2.19) +which are diagrammatically depicted in Figure 1. Note again in this decomposition the +kinematic factors H(1) +s/t/u are independent of the gauge group GF . +When decomposing using the projectors, let us assume that we are working with the +gauge group E8. In this case adj ⊗ adj includes altogether five irreducible representations +6For a review of the precise definition and general properties of ¯D-functions, see Appendix C of [14]. +– 8 – + +I1 +I4 +I3 +I2 +cs = +ct = +I1 +I4 +I3 +I2 +cu = +I1 +I4 +I3 +I2 +Figure 1: Tree color structures cs, ct and cu. +1, 3875, 27000, 248 (adj), and 30380. The former three representations are parity even +and the latter two are paritty odd. +Note that (cs)I1I2I3I4 already represents the exchange of the adjoint representation 248 +in the s-channel, it is therefore proportional to the projector P I1I2|I3I4 +248 +, and we have +(cs)a ≡ P I1I2|I3I4 +248 +(cs)I1I2I3I4 = ψ2h∨(0 +↑ +1 +, +0 +↑ +3875 +, +0 +↑ +27000 +, 1 +↑ +248 +, +0 +↑ +30380 +)T , +(2.20) +where h∨ is the dual Coxeter number, ψ2 is the length squared of the longest root. By +contrast the decomposition of ct, cu involves a mixture of different s-channel projectors +(ct)a = −ψ2h∨ +� +1, 1 +5, − 1 +30, 1 +2, 0 +�T +, +(cu)a = ψ2h∨ +� +1, 1 +5, − 1 +30, −1 +2, 0 +�T +. +(2.21a) +One easily sees that the Jacobi identity (cs)a + (ct)a + (cu)a = 0 is satisfied. Consequently +the coefficients in the projector decomposition of the whole tree-level correlator H(1) are +H(1) +1 +=ψ2h∨ � +−H(1) +t ++ H(1) +u +� +, +(2.22a) +H(1) +3875 =ψ2h∨ +5 +� +−H(1) +t ++ H(1) +u +� +, +(2.22b) +H(1) +27000 = − ψ2h∨ +30 +� +−H(1) +t ++ H(1) +u +� +, +(2.22c) +H(1) +248 =ψ2h∨ +2 +� +2H(1) +s +− H(1) +t +− H(1) +u +� +, +(2.22d) +H(1) +30380 =0. +(2.22e) +Quite remarkably, at this specific level the coefficients of projectors with equal parity are +in fact the same up to some overall constant factors, as was observed in a more general +setup in [13]. +2.3 +Spectrum and conformal block decomposition +As mentioned before our computation partly relies on the existing data of operators ob- +tained from lower loops, so it is helpful to have a quick look at the structure of OPE +and the related block expansion. Thanks to the 4d N = 2 superconformal symmetry, the +correlator GI1I2I3I4 admits a decomposition into superconformal blocks in correspondence +to the exchanges of different superconformal multiplets in the four-point function. The +– 9 – + +relevant sueprmultiplets are listed in Table 1 and a complete classification can be found in +[35]. The OPE of two 1 +2-BPS multipelts B1 contains the following supermultiplets +B1 ⊗ B1 ≃ +2 +� +p=0 +Bp ⊕ +� +ℓ≥0 +� +� +1 +� +p=0 +Cp,( ℓ +2 , ℓ +2) +� +∆ +A∆ +0,( ℓ +2 , ℓ +2) +� +� . +(2.23) +Here Bp and Cp,( ℓ +2 , ℓ +2 ) are protected multiplets and their twists τ = ∆ − ℓ are bounded +from above by the allowed R-symmetry charges. In contrast, there is no upper bound on +the twists of the long multiplets A∆ +0,( ℓ +2 , ℓ +2 ) and their dimensions are not protected. Instead, +they have a lower bound in the holographic limit as they are double-trace (and more +generally multi-trace) operators formed by single-trace operators.7 Let us also note that +the superprimaries of the long multiplets are only allowed to be R-symmetry singlets in +order for the representations of the entire multiplet to fit into the four-point function. The +long multiplets play a key role in the paper as the loop corrections correspond to precisely +the contribution of these multipelts. +Multiplet +Label +SU(2)R +Dimension ∆ and spin ℓ +Half-BPS +BR +R/2 +∆ = 2R, ℓ = 0 +Semi-short +CR,(ℓ/2,ℓ/2) +R/2 +∆ = 2 + 2R + ℓ +Long +A∆ +R,(ℓ/2,ℓ/2) +R/2 +∆ ≥ 2 + 2R + ℓ +Table 1: Supermultiplets that appear in the fusion rules of two B’s for N = 2 SCFTs. +We will focus on the reduced correlator H which has already taken superconformal +symmetry into account. In this way the superconformal block decomposition simply reduces +to just the ordinary conformal block decomposition. As superconformal symmetry and +gauge symmetry commute, this directly passes through the color projector decomposition, +and in terms of each component in (2.15) this reads [35] +Ha(z, ¯z) = +� +τa,ℓ +aagτa+2,ℓ(z, ¯z) , +(2.24) +where τa and ℓ sum over the spectrum of the supermultiplets. Note that the shift in τ by +2 in the ordinary conformal block gτ+2,ℓ is a consequence of the superconformal symmetry. +The detailed expression of these blocks is [36] +gτ,ℓ = +z¯z +¯z − z +� +k τ−2 +2 (z)k τ+2ℓ +2 (¯z) − k τ−2 +2 (¯z)k τ+2ℓ +2 (z) +� +, +kh(z) = zh 2F1(h, h, 2h, z). +(2.25) +Since the long multiplets are not protected, in the limit of N → ∞ their twists as well +as OPE coefficients receive perturbative corrections with respect to small aF +τa =τ0 + aF γ(1) +a ++ a2 +F γ(2) +a ++ . . . , +(2.26a) +aa(τ, ℓ) =a(0) +a ++ aF a(1) +a ++ a2 +F a(2) +a ++ . . . . +(2.26b) +7This bound is stronger than the unitarity bound in Table 1. +– 10 – + +Substituting the above expansion into (2.24) gives the following series expansion for Ha +Ha = +� +τ0,ℓ +a(0) +a gτ0+2,ℓ(z, ¯z) +� +�� +� +H(0) +a ++aF +� +τ0,ℓ +� +a(0) +a γ(1) +a ∂τ0 + a(1) +a +� +gτ0+2,ℓ(z, ¯z) +� +�� +� +H(1) +a ++ a2 +F +� +τ0,ℓ +�1 +2a(0) +a (γ(1) +a )2∂2 +τ0 + (a(1) +a γ(1) +a ++ a(0) +a γ(2) +a )∂τ0 + a(2) +a +� +gτ0+2,ℓ(z, ¯z) +� +�� +� +H(2) +a ++ . . . . +(2.27) +The first term H(0) +a +receives contributions only from long operators whose a(0) +a +are non- +vanishing. From large N factorization, H(0) +a +is given by the disconnected correlator and +these contributing operators can only be double-trace operators. However, these operators +are degenerate at the classical level. For instance, among the double-trace operators +: O2□n−2∂ℓO2 : , : O3□n−3∂ℓO3 : , . . . , : On∂ℓOn : +(2.28) +all have classical twist τ (0) = 2n and spin ℓ. Consequently, each term in (2.27) should not +be literally understood as the contribution from a single operator, but rather in an averaged +sense. Moreover, at higher orders in aF there are also higher-trace operators appearing in +the OPE 8, which can have the same twist as the double-trace operators and will enter +the mixing as well. +Therefore, in a precise description it is necessary to use an extra +label i to distinguish different operators in the degeneracy. Then the coefficient a(0) +a γ(1) +a +should in fact be understood as ⟨a(0) +a γ(1) +a ⟩ ≡ � +i a(0) +i,aγ(1) +i,a , and a(0) +a +� +γ(1) +a +�2 +as ⟨a(0) +a +� +γ(1) +a +�2 +⟩ ≡ +� +i a(0) +i,a +� +γ(1) +i,a +�2 +, and so on. +3 +Leading logarithmic singularities +As an analytic function of the kinematic variables z and ¯z, a conformal correlator can in +principle be constructed out of its singularities by dispersion-type relations, in a similar +way as the dispersion relation that generates a four-point scattering amplitude from its +physical channel discontinuities. For generic CFTs such relations were formulated in [37]. +This means that the defining data for a correlator is necessarily encoded in its singularities. +While our computation does not rely on the dispersion relations, these data still provide a +vital input in determining the loop-level corrections to the reduced correlator H. +When viewed in the perturbative expansion (2.27) these singularities are sourced at +small u by the log(u) factors arising from the derivatives acting on the conformal block. +Recall in the definition (2.27) that gτ,ℓ(z, ¯z) ∝ uτ/2, so at each order ap +F the reduced +correlator can be organized in terms of powers of log(u) +H(p) +a (z, ¯z) = +1 +2pp! logp(u) +� +τ0,ℓ +⟨a(0) +a (γ(1) +a )p⟩ gτ0,ℓ(z, ¯z) + +� +terms with logk