Fermionic correlators from integrability

Komatsu

2017

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150

338

We propose a nonperturbative framework to study general correlation functions of single-trace operators in N = 4 supersymmetric Yang-Mills theory at large N . The basic strategy is to decompose them into fundamental building blocks called the hexagon form factors, which were introduced earlier to study structure constants using integrability. The decomposition is akin to a triangulation of a Riemann surface, and we thus call it hexagonalization. We propose a set of rules to glue the hexagons together based on symmetry, which naturally incorporate the dependence on the conformal and the R-symmetry cross ratios. Our method is conceptually different from the conventional operator product expansion and automatically takes into account multi-trace operators exchanged in OPE channels. To illustrate the idea in simple set-ups, we compute four-point functions of BPS operators of arbitrary lengths and correlation functions of one Konishi operator and three short BPS operators, all at one loop. In all cases, the results are in perfect agreement with the perturbative data. We also suggest that our method can be a useful tool to study conformal integrals, and show it explicitly for the case of ladder integrals.

Section: Review Of the Hexagon Formalismmentioning

confidence: 99%

Hexagonalization of correlation functions

Komatsu

2017

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150

338

“…The S-matrix acts diagonally in the state above as it gives a product of the elements D 12 , see [25,30]. In our conventions this element has the value -1, thus we have S · |0, 0 = (−1) (ab) |0, 0 .…”

Section: B31 Case Imentioning

confidence: 99%

“…• There is a factor of (−1) F 1 (−1) F 2 where F i is the fermion number of each state. These signs appear because in the string frame the one-particle hexagon form factor differs by a factor of −i for bosonic and fermionic indices [30].…”

Section: D1 Explicit Form Of the Integrandmentioning

confidence: 99%

Hexagonalization of correlation functions II: two-particle contributions

Komatsu

2018

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133

In this work, we compute one-loop planar five-point functions in N = 4 superYang-Mills using integrability. As in the previous work, we decompose the correlation functions into hexagon form factors and glue them using the weight factors which depend on the cross-ratios. The main new ingredient in the computation, as compared to the four-point functions studied in the previous paper, is the two-particle mirror contribution. We develop techniques to evaluate it and find agreement with the perturbative results in all the cases we analyzed. In addition, we consider next-to-extremal four-point functions, which are known to be protected, and show that the sum of one-particle and two-particle contributions at one loop adds up to zero as expected. The tools developed in this work would be useful for computing higher-particle contributions which would be relevant for more complicated quantities such as higher-loop corrections and non-planar correlators.

“…(2.19) can be obtained by implementing mirror moves, or crossing transformations [70], on the magnons in (2.16), following the rules spelled out in the appendices of Refs. [24] and [38]. Performing these manipulations gives the form factor (2.19) as a S matrix element with arguments u, w −2γ , v −4γ ; see middle panel in 4.…”

Section: )mentioning

confidence: 99%

Hexagons and correlators in the fishnet theory

Basso

Caetano

2019

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We investigate the hexagon formalism in the planar 4d conformal fishnet theory. This theory arises from N = 4 SYM by a deformation that preserves both conformal symmetry and integrability. Based on this relation, we obtain the hexagon form factors for a large class of states, including the BMN vacuum, some excited states, and the Lagrangian density. We apply these form factors to the computation of several correlators and match the results with direct Feynman diagrammatic calculations. We also study the renormalisation of the hexagon form factor expansion for a family of diagonal structure constants and test the procedure at higher orders through comparison with a known universal formula for the Lagrangian insertion. arXiv:1812.09794v1 [hep-th]