Four-Loop Nonplanar Cusp Anomalous Dimension in $\mathcal{N}=4$ Supersymmetric Yang-Mills Theory

The lightlike cusp anomalous dimension is a universal function that controls infrared divergences in quite general gauge theories. In the maximally supersymmetric Yang-Mills theory this function is fixed fully by integrability to the three-loop order. At four loops a nonplanar 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.

Figure 1

Integral topologies that contribute to the nonplanar form factor at four loops in a basis of UT integrals.