Null octagon from Deift-Zhou steepest descent

TL;DR

This paper applies the Deift-Zhou steepest descent method to analyze a Riemann-Hilbert problem related to a special class of correlation functions in supersymmetric Yang-Mills theory, deriving an exact octagonal anomalous dimension.

Contribution

It introduces a new formulation of the octagonal anomalous dimension as a convolution involving the non-linear phase and Fermi distribution, solved via Riemann-Hilbert techniques.

Findings

Reproduces the exact octagonal anomalous dimension in 't Hooft coupling.

Provides a novel convolutional formulation in the infinite chemical potential limit.

Utilizes the Deift-Zhou steepest descent to solve the Riemann-Hilbert problem.

Abstract

A special class of four-point correlation functions in the maximally supersymmetric Yang-Mills theory is given by the square of the Fredholm determinant of a generalized Bessel kernel. In this note, we re-express its logarithmic derivatives in terms of a two-dimensional Riemann-Hilbert problem. We solve the latter in the null limit making use of the Deift-Zhou steepest descent. We reproduce the exact octagonal anomalous dimension in 't Hooft coupling and provide its novel formulation as a convolution of the non-linear quasiclassical phase with the Fermi distribution in the limit of the infinite chemical potential.