The Cross Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory

The cross or soft anomalous dimension matrix describes the renormalization of Wilson loops with a self-intersection and is an important object in the study of infrared divergences of scattering amplitudes. In this paper it is studied for the Maldacena-Wilson loop in $$ \mathcal{N}=4 $$ supersymmetric Yang-Mills theory and Euclidean kinematics. We consider both the strong-coupling description in terms of minimal surfaces in AdS5 as well as the weak-coupling side up to the two-loop level. In either case, the coefficients of the cross anomalous dimension matrix can be expressed in terms of the cusp anomalous dimension. The strong-coupling description displays a Gross-Ooguri phase transition and we argue that the cross anomalous dimension is an interesting object to study in an integrability-based approach.