diff --git "a/19FQT4oBgHgl3EQfFDWe/content/tmp_files/load_file.txt" "b/19FQT4oBgHgl3EQfFDWe/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/19FQT4oBgHgl3EQfFDWe/content/tmp_files/load_file.txt" @@ -0,0 +1,1692 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf,len=1691 +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' E-mail: eyyuan@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='cn, b w@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='cn, zjhuang@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='cn, xinan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='zhou@ucas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' We focus on the simplest 1 2-BPS operators which correspond to the super gluons in the massless current multiplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' Our results reveal a number of interesting properties enriched by the color structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='13240v1 [hep-th] 30 Jan 2023 Contents 1 Introduction 2 2 Preliminaries 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='1 Four-point correlators 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='2 Projectors and color decomposition 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='3 Spectrum and conformal block decomposition 9 3 Leading logarithmic singularities 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='1 Recursion by unitarity 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='2 Hidden conformal symmetry 14 4 One-loop correlator 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='1 Ansatz 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='2 Constraints 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='3 Results at one loop 21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='4 Comparison with the Mellin space result 25 5 Two-loop correlator 26 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='1 Color structures at two loops 26 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='2 Ansatz and constraints 28 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' In the holographic limit, these observables are expanded in powers of 1/c with respect to the large central charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' However, to extract nontrivial dynamical information one needs to go to higher orders in 1/c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' Computing these subleading contributions is in general intractable from the CFT side alone as the theory is strongly coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' In fact, just at the tree level, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=', at order 1/c, the computation of general four-point functions remained an unsolved problem for almost two decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' At the moment, there is already a wealth of results at tree level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='1 By contrast, our understanding for loop level correlators is much more limited, even in the paradigmatic example of IIB supergravity on AdS5 × S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' The first one-loop correlator was computed in [15, 16] for the stress tensor multiplet in position space and later in Mellin space [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' The calculation was generalized to four-point functions with higher KK levels in [18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' However, explicit one-loop results are still case-by-case with the exception for the ⟨22pp⟩ family in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' At two loops and higher, the situation is more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' The strategy at one loop, which is based on the AdS unitarity method [21], now requires the additional input of multi-trace operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' Such information is not yet available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' In this way, the four-point two-loop correlator of stress tensor multiplets has also been bootstrapped [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 1See [14] for a recent review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 2For example, at two loops there are exchange contributions from triple-trace operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' These can be in principle extracted from tree-level five-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' However, only five-point functions of the form ⟨pp222⟩ have been computed [22, 23] while extracting the data requires all ⟨pqr22⟩ five-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' – 2 – In this paper, we continue to explore the loop-level calculation of holographic correla- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' However, instead of considering correlators of super gravitons, we will focus on super gluons of SYM in AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' More precisely, we consider a decoupling sector of certain 4d N = 2 SCFTs in the holographic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' The near horizon geometries in both cases include an AdS5 ×S3 subspace which hosts localized degrees of freedom corresponding to the gluons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' The resulting physics in 8d is the same regardless of the model we choose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' Strictly speaking, the decoupling happens only at the leading order and correlators at subleading orders include gravity contributions as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' The motivations for considering super gluon correlators in such a setup are two fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' First, as we already mentioned, holographic correlators are on-shell scattering amplitudes in AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' It is natural to wonder if various remarkable properties of flat-space amplitudes admit generalizations in curved backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' In particular, does the double copy relation [31], which famously states gravity is the “square” of YM, still holds in AdS?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' To this end, it makes sense to decouple gravity and study the amplitudes of just SYM in AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' In fact, analysis of this model at tree level already showed evidence for such a generalization at four points [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' Second, the super gluon case also provides a useful playground for acquiring deeper understandings of various results from the supergravity setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' Let us briefly outline our strategy and the key results of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' Our approach is similar to that of [15, 19, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' We first make an ansatz in position space which requires a set of building block functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' In other words, the correlators are assumed to be linear combinations of SVMPLs with rational functions of the cross ratios as coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' However, this turns out to be a bit too general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' By unitarity this symmetry extends to the leading part of the Lorentzian singularities at arbitrary loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='1a) H2-loop sugra = � ∆(8)�2 L2-loop sugra + 5 4H1-loop sugra − 1 16Htree sugra , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='1b) together with additional lower-order correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='2a) H2-loop SYM = � ∆(4)�2 L2-loop SYM + � H1-loop SYM + � Htree SYM , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='2b) where ¯Htree SYM and � Htree SYM are “tree-like” correlators and � H1-loop SYM is a “one-loop-like” corre- lator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' We will be more precise about the meaning of “tree-like” and “one-loop-like”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' But for the moment it suffices to say they are characterized by the transcendental degrees of SVMPLs expected at each loop order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' Note that, unlike supergravity, super gluon correlators have different color structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' Therefore, we make such an ansatz for each independent color structure and assume the correlator to be a linear combination of all these structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' To perform the bootstrap, we impose a number of consistency conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' These are Leading logarithmic singularities Crossing symmetry together with a few other constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' Here the leading logarithmic singularities rely only on the tree-level data and can be computed at any loop order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' We review in Section 2 some preliminaries of super gluon four-point functions which include the superconformal kinematics, color structure and superconformal block decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' In Section 4 we introduce the position – 4 – space method and demonstrate it by bootstrapping the one-loop correlator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' In Section 5 we apply the method to the two-loop correlator and obtain the full answer by imposing constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' In Section 6 we outline a few future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' The paper also has several appendices where we include further technical details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' In Appendix A we give a brief review of the properties of SVMPLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' Appendix B contains the complete analytic result for the reduced correlator at one loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' The details of the flat-space two-loop amplitude are presented in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' In Appendix D we discuss the computations related to the twist-4 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 2 Preliminaries In this paper, we consider holographic correlators corresponding to super gluon scattering in AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' To be concrete, we consider SYM in AdS5 ×S3 which arises as a decoupling sector of certain 4d N = 2 SCFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' Here GF depends on the theory and is a gauge group from the bulk perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' These are the massless and massive AdS gluons and their super partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' Their conformal dimensions are fully fixed by R-symmetry and therefore are independent of the bulk theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' More precisely, the superconformal primaries of these 1 2-BPS are scalar operators Ok labelled by an integer k = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='. They have conformal dimension ∆ = k and transform in the spin-k 2 representation of the SU(2)R R-symmetry group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' Moreover, they transform in the adjoint representaiton of the flavor group GF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' We will call the fields dual to these superprimaries the super gluons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' The real spinning gluon fields are superconformal descendants in the multiplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' By contrast, the super gravitons and their super partners live in the full ten dimen- sional spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' Unlike the supergluons, their KK spectrum depends on the specific theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' However, an interesting fact of all these 4d N = 2 SCFTs is that there is a hier- archy in the couplings at large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' Therefore, for large N the lead- ing contribution to the super gluon correlators comes only from the 8d SYM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' Subleading corrections in 1/N will in general contain graviton contributions as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' – 5 – the perspective of exploring curved space generalizations of gauge theory amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' This section serves to provide some preliminary features of this correlator, which will be used in our bootstrap computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='1 Four-point correlators With all indices restored, the super gluon operator O2 has the form OI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='a1a2 2 (x) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='1) where I = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' , dim(GF ) is the flavor symmetry index and ai = 1, 2 are the SU(2)R R- symmetry indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' It is convenient to contract the R-symmetry indices with two-dimensional polarization spinors OI 2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' v) = OI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='a1a2 2 va1va2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='2) The four-point function GI1I2I3I4(xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' vi) = ⟨OI1 2 (x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' v1)OI2 2 (x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' v2)OI3 2 (x3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' v3)OI4 2 (x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' v4)⟩ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='3) is therefore a function of both the spacetime coordinates xi and the internal space spinors vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' Exploiting the bosonic symmetries, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' α) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='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) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' α) �� α=1/x = 0 , x = z or ¯z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='6) Solving these identities, we can decompose the correlator into two parts GI1I2I3I4(z, ¯z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' α) = GI1I2I3I4 0 (z, ¯z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' α) + RHI1I2I3I4(z, ¯z) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='7) where R = (1 − zα)(1 − ¯zα) z¯z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='8) Note that our definiton of R is different from that of [13] by a z¯z in the denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' The first term GI1I2I3I4 0 is protected, while dynamical information of the correlator is encoded in the reduced correlator HI1I2I3I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' – 6 – We study the expansion of the correlator with respect to the large flavor central charge CJ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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) + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='9) This expansion has a nice interpretation from the bulk point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' The first correction H(1) is the tree-level scat- tering of the super gluons, which has been obtained in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' The decomposition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='10) is not the most convenient for practical computations as the color factors CI1I2I3I4 are highly redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' So instead one often seeks for other types of color decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' In this paper we call them dynamical factors to remind the readers that they contains the dynamical information of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' – 7 – In particular from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='12) we also have P I1I2|I3I4 a P I1I2|I3I4 b = δabdim(Ra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' Therefore ev- ery color factor appearing in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='10) receives a unique decomposition onto the s-channel projectors CI1I2I3I4 = � a∈adj⊗adj P I1I2|I3I4 a Ca , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='13) with coefficients Ca, or equivalently Ca = dim(Ra)−1P I1I2|I3I4 a CI1I2I3I4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' As a result, the color decomposition of the reduced correlator H as well as any term H(L+1) in the expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='9) can be carried out in a uniform manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' Generically, we have HI1I2I3I4 = � a∈adj⊗adj P I1I2|I3I4 a Ha , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='15) and H(L+1) follows similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' Furthermore, the idempotency condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='12) also makes the recursive relation between different loop levels very simple, as will be further illustrated in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' As a simple example for the use of projectors, let us quickly review the tree-level correlator H(1), which was computed in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' Its takes the following form H(1) = csH(1) s + ctH(1) t + cuH(1) u , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='16) with H(1) s = u3 3 � 2∂u + (1 + v)∂u∂v + u∂2 u � ¯D1111, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='17a) H(1) t = −u3 3 � 2∂v + v∂2 v + (1 + u)∂u∂v � ¯D1111, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='17b) H(1) u = u3 3 � 2∂v + v∂2 v − 2∂u + (u − v)∂u∂v − u∂2 u � ¯D1111 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='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 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='18) cs/t/u are color factors built from structure constants (cs)I1I2I3I4 = fI1I2JfJ I3I4, (ct)I1I2I3I4 = fI1I4JfJ I2I3, (cu)I1I2I3I4 = fI1I3JfJ I4I2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='19) which are diagrammatically depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' Note again in this decomposition the kinematic factors H(1) s/t/u are independent of the gauge group GF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' When decomposing using the projectors, let us assume that we are working with the gauge group E8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' – 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 1, 3875, 27000, 248 (adj), and 30380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' The former three representations are parity even and the latter two are paritty odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='20) where h∨ is the dual Coxeter number, ψ2 is the length squared of the longest root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='21a) One easily sees that the Jacobi identity (cs)a + (ct)a + (cu)a = 0 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='22a) H(1) 3875 =ψ2h∨ 5 � −H(1) t + H(1) u � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='22b) H(1) 27000 = − ψ2h∨ 30 � −H(1) t + H(1) u � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='22c) H(1) 248 =ψ2h∨ 2 � 2H(1) s − H(1) t − H(1) u � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='22d) H(1) 30380 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' The – 9 – relevant sueprmultiplets are listed in Table 1 and a complete classification can be found in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' The long multiplets play a key role in the paper as the loop corrections correspond to precisely the contribution of these multipelts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' We will focus on the reduced correlator H which has already taken superconformal symmetry into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' In this way the superconformal block decomposition simply reduces to just the ordinary conformal block decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' As superconformal symmetry and gauge symmetry commute, this directly passes through the color projector decomposition, and in terms of each component in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='15) this reads [35] Ha(z, ¯z) = � τa,ℓ aagτa+2,ℓ(z, ¯z) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='24) where τa and ℓ sum over the spectrum of the supermultiplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' Note that the shift in τ by 2 in the ordinary conformal block gτ+2,ℓ is a consequence of the superconformal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='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 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='26a) aa(τ, ℓ) =a(0) a + aF a(1) a + a2 F a(2) a + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='26b) 7This bound is stronger than the unitarity bound in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' – 10 – Substituting the above expansion into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='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 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='27) The first term H(0) a receives contributions only from long operators whose a(0) a are non- vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' From large N factorization, H(0) a is given by the disconnected correlator and these contributing operators can only be double-trace operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' However, these operators are degenerate at the classical level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' For instance, among the double-trace operators : O2□n−2∂ℓO2 : , : O3□n−3∂ℓO3 : , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' , : On∂ℓOn : (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='28) all have classical twist τ (0) = 2n and spin ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' Consequently, each term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='27) should not be literally understood as the contribution from a single operator, but rather in an averaged sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' Therefore, in a precise description it is necessary to use an extra label i to distinguish different operators in the degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' For generic CFTs such relations were formulated in [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' This means that the defining data for a correlator is necessarily encoded in its singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' When viewed in the perturbative expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='27) these singularities are sourced at small u by the log(u) factors arising from the derivatives acting on the conformal block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' Recall in the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content='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!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQfFDWe/content/2301.13240v1.pdf'} +page_content=' logp(u) � τ0,ℓ ⟨a(0) a (γ(1) a )p⟩ gτ0,ℓ(z, ¯z) + � terms with logk