Four-Loop Nonplanar Cusp Anomalous Dimension in N=4 Supersymmetric Yang-Mills Theory

TL;DR

The paper computes the first non-planar four-loop correction to the light-like cusp anomalous dimension in ${\\cal N}=4$ SYM by a detailed Sudakov form factor analysis. It employs a novel uniformly transcendental (UT) basis of integrals, constructed via dLog representations and leading-singularity tests, enabling a precise numerical evaluation with controlled uncertainties. The result demonstrates a nonzero non-planar CAD, signaling a breakdown of quadratic Casimir scaling at four loops and reinforcing maximal transcendentality in the non-planar sector. The approach suggests broad applicability to complex higher-loop calculations in gauge theories, including potential insights for QCD. Note: subsequent works have reported Casimir-scaling violations in QCD, underscoring the generality of the finding.

Abstract

The light-like cusp anomalous dimension is a universal function that controls infrared divergences in quite general quantum field theories. In the maximally supersymmetric Yang-Mills theory this function is fixed fully by integrability to the three-loop order. At four loops a non-planar correction appears which we obtain for the first time from a numerical computation of the Sudakov form factor. Key ingredients are widely applicable methods to control the number-theoretic aspects of the appearing integrals. Our result shows explicitly that quadratic Casimir scaling breaks down at four loops.

Figures (1)

• Figure 1: Integral topologies that contribute to the non-planar form factor at four loops in a basis of UT integrals.