Non-planar integrated correlator in $\mathcal{N}=4$ SYM

TL;DR

This paper extends the analysis of integrated correlators in $ ext{N}=4$ SYM to the non-planar four-loop sector, revealing the role of multiple zeta values and confirming predictions from supersymmetric localization.

Contribution

It introduces a method to express non-planar four-loop integrated correlators as linear combinations of periods of $f$-graphs, including non-trivial multiple zeta values.

Findings

Multiple zeta values cancel in the sum, leaving a single zeta value.

Results agree with supersymmetric localization predictions.

Extension from planar to non-planar sectors at four loops.

Abstract

Integrated correlator of four superconformal stress-tensor primaries in $SU(N)$ $\mathcal{N}=4$ super Yang-Mills (SYM) theory in the perturbative limit takes a remarkably simple form, where the $L$-loop coefficient is given by a rational multiple of $\zeta(2L+1)$. In this letter, we extend the previous analysis of expressing the perturbative integrated correlator as a linear combination of periods of $f$-graphs, graphical representations for loop integrands, to the non-planar sector at four loops. At this loop order, multiple zeta values make their first appearance when evaluating periods of non-planar $f$-graphs, but cancel non-trivially in the weighted sum. The remaining single zeta value, along with the rational number prefactor, makes a perfect agreement with the prediction from supersymmmetric localisation.