Master integrals for the four-loop Sudakov form factor

TL;DR

This work tackles the challenge of the four-loop non-planar cusp anomalous dimension in $\mathcal{N}=4$ SYM by computing the four-loop Sudakov form factor and performing an explicit IBP reduction using Reduze, followed by an independent algebraic-geometry cross-check with Mint. It demonstrates that the form factor integrand is independent of the free parameter after IBP cancellations and identifies two topologies that vanish, while assembling a concrete master-integral basis shared by generic form factors, including QCD. The study also highlights rare mismatches between Mint and Reduze and discusses substantial hurdles in numerically evaluating the non-planar master integrals, outlining practical strategies (sector decomposition, symmetry exploitation) and future directions (MB representations, differential equations) toward completing the four-loop non-planar cusp computation. Overall, the paper provides a robust, parameter-free integrand structure and a comprehensive master-integral basis that will underpin future progress in precision multi-loop IR analyses in gauge theories. The results have broad relevance for IR structure in quantum field theories and for pushing non-planar computations toward a full four-loop prediction in both ${\cal N}=4$ SYM and QCD.

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.