Higher-rank sectors in the hexagon formalism and marginal deformations

TL;DR

This paper extends the hexagon formalism to higher-rank sectors in $ ext{AdS}_5/ ext{CFT}_4$ integrability, constructing entangled states for $psu(1,1|2)$ sectors and testing them against free theory and marginal deformations.

Contribution

It introduces a method to construct entangled states in higher-rank sectors within the hexagon formalism, expanding its applicability beyond rank-one cases.

Findings

Successfully constructed entangled states for $psu(1,1|2)$ sectors.

Validated the approach against free field theory for correlators.

Observed efficiency of undeformed amplitudes in marginal deformation scenarios.

Abstract

The hexagon approach provides an integrability framework for the computation of structure constants in $\mathcal{N}=4$ super Yang--Mills theory in four dimensions. Three-point functions are cut into two hexagonal patches, on which the excitations of the long-range Bethe ansatz of the spectrum problem scatter. To this end, the Bethe states representing the operators also need to be cut into two parts. In rank-one sectors such entangled states are fairly straightforward to construct so that most applications of the method have so far been restricted to this simplest case. In this article we construct entangled states for operators in $psu(1,1|2)$ sectors, importing a minimum of information from the nested Bethe ansatz. The idea is successfully tested against free field theory for a sample set of correlators with up to three higher-rank operators. Further, we take a look at the same correlators in the presence of marginal deformations of the theory. While a systematic modification of the hexagon procedure remains out of reach for now, in practical applications the undeformed amplitudes are surprisingly efficient, especially for a certain one-parameter deformation.