Three-point functions in ${\cal N}=4$ SYM: the hexagon proposal at three loops

TL;DR

This paper tests the hexagon proposal for three-point functions in ${ m extbf{N}=4}$ SYM by computing wrapping corrections at three loops in the $SL(2)$ sector, confirming predictions with existing four-point correlator results.

Contribution

It provides the first explicit three-loop calculation of wrapping corrections in the hexagon framework for the $SL(2)$ sector, validating the proposal's accuracy.

Findings

Wrapping corrections at three loops match known results

The hexagon proposal accurately predicts structure constants

Finite-size effects are essential for precise three-point function computations

Abstract

Basso, Komatsu and Vieira recently proposed an all-loop framework for the computation of three-point functions of single-trace operators of ${\cal N}=4$ super-Yang-Mills, the "hexagon program". This proposal results in several remarkable predictions, including the three-point function of two protected operators with an unprotected one in the $SU(2)$ and $SL(2)$ sectors. Such predictions consist of an "asymptotic" part---similar in spirit to the asymptotic Bethe Ansatz of Beisert and Staudacher for two-point functions---as well as additional finite-size "wrapping" L\"uscher-like corrections. The focus of this paper is on such wrapping corrections, which we compute at three-loops in the $SL(2)$ sector. The resulting structure constants perfectly match the ones obtained in the literature from four-point correlators of protected operators.