Non-planar data of $\mathcal N=4$ SYM

TL;DR

This paper computes the complete non-planar four-loop integrand of four-point functions in $ ext{N}=4$ SYM using twistor space reformulation, enabling the extraction of non-planar anomalous dimensions and structure constants for twist-two operators.

Contribution

It fully determines the non-planar four-loop integrand of four-point functions in $ ext{N}=4$ SYM using twistor space, advancing the understanding of non-planar corrections.

Findings

Computed non-planar anomalous dimensions for twist-two operators up to spin eight.

Fixed the integrand of four-point functions at four loops completely.

Confirmed some results with existing literature.

Abstract

The four-point function of length-two half-BPS operators in $\mathcal{N}=4$ SYM receives non-planar corrections starting at four loops. Previous work relied on the analysis of symmetries and logarithmic divergences to fix the integrand up to four constants. In this work, we compute those undetermined coefficients and fix the integrand completely by using the reformulation of $\mathcal{N}=4$ SYM in twistor space. The final integrand can be written as a combination of finite conformal integrals and we have used the method of asymptotic expansions to extract non-planar anomalous dimensions and structure constants for twist-two operators up to spin eight. Some of the results were already know in the literature and we have found agreement with them.