The four-point function of determinant operators in $\mathcal{N}=4$ SYM

TL;DR

This paper computes the four-point function of determinant operators in $ ext{N}=4$ SYM at weak coupling, using perturbative and matrix integral methods, and discusses implications for phase transitions in integrability.

Contribution

It introduces a semi-classical large-$N$ approach to evaluate four-point functions of determinants, extending previous methods to include one-loop corrections.

Findings

Perturbative four-point function computed at finite $N$

Matrix integral approach generalized for one-loop corrections

Results suggest a phase transition in an integrability framework

Abstract

We calculate the four-point function of $1/2$-BPS determinant operators in $\mathcal{N}=4$ SYM at next-to-leading order at weak coupling. We use two complementary methods recently developed for a class of determinant three-point functions: one is based on Feynman diagrams and it extracts perturbative data at finite $N$, while the other one expresses a generic correlator of determinants as the zero-dimensional integral over an auxiliary matrix field. We generalise the latter approach to calculate one-loop corrections and we solve the four-point function in a semi-classical approach at large $N$. The results allow to comment on the order of the phase transition that the four-point function is expected to exhibit in an exact integrability-based description.