Strongly Coupled Soft Functions

Abstract

The renormalization of operators built out of Wilson lines that meet at an angle (cusp) involve what is known as the cusp anomalous dimension, a universal object appearing in many processes in QCD due to the divergences from gluonic interactions. In this paper, we consider the vacuum expectation value of a pair of cusped Wilson lines separated in the transverse direction. This configuration can be related to a simple observable in heavy quark transverse momentum-dependent (TMD) fragmentation as well as the TMD soft function. We compute the expectation values of Wilson loops in $\mathcal{N}=4$ super Yang-Mills theory at strong coupling via the AdS/CFT correspondence, an approach complementary to perturbative calculations of the cusp anomalous dimension. We first present a thorough analysis, directly in Minkowski signature, of the Nambu-Goto action and its saddle points for a Wilson line configuration with a single cusp. We find that there are two different classes of saddle points whose contributions dominate different regions of parameter space. We calculate the cusp anomalous dimension $\Gamma_{\rm cusp}[\Delta\eta, \Delta\theta]$ for the whole range of $\Delta\eta$ from this setup, focusing on the cases $\Delta \theta = 0$ and $\Delta \theta = \pi$, and compare it with previous results in literature. We then use the techniques we developed to compute the expectation value of the two-cusp Wilson loop with transverse separation, and determine the profile of the transverse coordinate on the extremal surface in the large rapidity limit. We discuss the possible generalizations of our setup and results to soft functions in other gauge theories with a holographic dual.