Master integrals for the four-loop Sudakov form factor

TL;DR

This paper computes and simplifies the master integrals needed for the four-loop non-planar Sudakov form factor in $ ext{N}=4$ SYM, advancing understanding of infrared divergences and providing tools applicable to QCD.

Contribution

It introduces a reduction of four-loop integrals using IBP identities with a modified Reduze, and cross-checks master integrals with algebraic-geometry techniques, aiding future computations in gauge theories.

Findings

Master integrals are reduced and simplified using IBP and algebraic-geometry methods.

The form factor is shown to be independent of a free parameter after IBP reduction.

Two integral topologies vanish after reduction.

Abstract

The light-like cusp anomalous dimension is a universal function in the analysis of infrared divergences. In maximally ($\mathcal{N}=4$) supersymmetric Yang-Mills theory (SYM) in the planar limit, it is known, in principle, to all loop orders. The non-planar corrections are not known in any theory, with the first appearing at the four-loop order. The simplest quantity which contains this correction is the four-loop two-point form factor of the stress tensor multiplet. This form factor was largely obtained in integrand form in a previous work for $\mathcal{N}=4$ SYM, up to a free parameter. In this work, a reduction of the appearing integrals obtained by solving integration-by-parts (IBP) identities using a modified version of Reduze is reported. The form factor is shown to be independent of the remaining parameter at integrand level due to an intricate pattern of cancellations after IBP reduction. Moreover, two of the integral topologies vanish after reduction. The appearing master integrals are cross-checked using independent algebraic-geometry techniques explored in the Mint package. The latter results provide the basis of master integrals applicable to generic form factors, including those in Quantum Chromodynamics. Discrepancies between explicitly solving the IBP relations and the MINT approach are highlighted. Remaining bottlenecks to completing the computation of the four-loop non-planar cusp anomalous dimension in $\mathcal{N}=4$ SYM and beyond are identified.