Hexagonalization of Correlation Functions

TL;DR

This paper introduces a nonperturbative hexagonalization framework for calculating correlation functions in N=4 SYM, leveraging integrability and a novel decomposition into hexagon form factors, and validates it with perturbative results.

Contribution

It develops a new hexagonalization method for correlation functions, differing from OPE, and demonstrates its effectiveness through explicit one-loop calculations.

Findings

Accurately computes four-point functions of BPS operators.

Matches perturbative data at one loop.

Proposes a new approach for conformal integrals.

Abstract

We propose a nonperturbative framework to study general correlation functions of single-trace operators in $\mathcal{N}=4$ 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.