Integrated correlators in $\mathcal{N}=4$ super Yang-Mills and periods

TL;DR

This paper investigates integrated four-point correlators in $ =4$ super Yang-Mills theory, revealing their relation to conformal Feynman graph periods, enabling high-loop computations and uncovering new relations among these periods.

Contribution

It demonstrates that integrated correlators are expressible as linear combinations of conformal Feynman graph periods, allowing explicit high-loop calculations and linking to supersymmetric localization results.

Findings

Computed four-loop planar integrated correlator.

Matched localization results with perturbative calculations.

Predicted a six-loop integral period from five-loop localization data.

Abstract

We study perturbative aspects of recently proposed integrated four-point correlators in $\mathcal{N}=4$ supersymmetric Yang-Mills with all classical gauge groups using standard Feynman diagram computations. We argue that perturbative contributions of the integrated correlators are given by linear combinations of periods of certain conformal Feynman graphs, which were originally introduced for the construction of perturbative loop integrands of the un-integrated correlator. This observation allows us to evaluate the integrated correlators to high loop orders. We explicitly compute one of the integrated correlators up to four loops in the planar limit, and up to three loops for the other integrated correlator, and find agreement with the results obtained from supersymmetric localisation. The identification between the integrated correlators and certain periods also implies non-trivial relations among these periods, given that one may compute the integrated correlators using localisation. We illustrate this idea by considering one of the integrated correlators at five loops in the planar limit, where the localisation result leads to a prediction for the period of a certain six-loop integral.