Positivity of hexagon perturbation theory

TL;DR

This paper proves that the finite-volume perturbation theory for the hexagon approach in integrable models is positive, ensuring consistency with weak-coupling expansions in computing correlation functions in $ ext{N}=4$ SYM.

Contribution

It demonstrates the positivity of the hexagon finite-volume perturbation theory, addressing potential issues with negative powers of the coupling and ensuring compatibility with weak-coupling expansions.

Findings

Finite-volume corrections are positive in the hexagon formalism.

Positivity ensures the formalism's consistency at weak coupling.

Supports the reliability of the hexagon approach for arbitrary n-point functions.

Abstract

The hexagon-form-factor program was proposed as a way to compute three- and higher-point correlation functions in $\mathcal{N}=4$ super-symmetric Yang-Mills theory and in the dual AdS$_5\times$S$^5$ superstring theory, by exploiting the integrability of the theory in the 't Hooft limit. This approach is reminiscent of the asymptotic Bethe ansatz in that it applies to a large-volume expansion. Finite-volume corrections can be incorporated through L\"uscher-like formulae, though the systematics of this expansion is largely unexplored so far. Strikingly, finite-volume corrections may feature negative powers of the 't Hooft coupling $g$ in the small-$g$ expansion, potentially leading to a breakdown of the formalism. In this work we show that the finite-volume perturbation theory for the hexagon is positive and thereby compatible with the weak-coupling expansion for arbitrary $n$-point functions.