The three-loop cusp anomalous dimension in QCD and its supersymmetric extensions

TL;DR

This paper analytically computes the three-loop cusp anomalous dimension in QCD and supersymmetric theories, revealing universality properties and predicting four-loop corrections, with detailed calculations of Wilson line integrals.

Contribution

It provides the first three-loop result for the cusp anomalous dimension in supersymmetric theories and uncovers a universal behavior across different models.

Findings

Three-loop cusp anomalous dimension in supersymmetric theories is new.

Universality of the cusp anomalous dimension across QCD and supersymmetric models.

Prediction of nonplanar four-loop corrections based on three-loop results.

Abstract

We present the details of the analytic calculation of the three-loop angle-dependent cusp anomalous dimension in QCD and its supersymmetric extensions, including the maximally supersymmetric $\mathcal{N}=4$ super Yang-Mills theory. The three-loop result in the latter theory is new and confirms a conjecture made in our previous paper. We study various physical limits of the cusp anomalous dimension and discuss its relation to the quark-antiquark potential including the effects of broken conformal symmetry in QCD. We find that the cusp anomalous dimension viewed as a function of the cusp angle and the new effective coupling given by light-like cusp anomalous dimension reveals a remarkable universality property -- it takes the same form in QCD and its supersymmetric extensions, to three loops at least. We exploit this universality property and make use of the known result for the three-loop quark-antiquark potential to predict the special class of nonplanar corrections to the cusp anomalous dimensions at four loops. Finally, we also discuss in detail the computation of all necessary Wilson line integrals up to three loops using the method of leading singularities and differential equations.