The quark anti-quark potential and the cusp anomalous dimension from a TBA equation

TL;DR

This paper derives exact integral equations for the quark-antiquark potential and cusp anomalous dimension using TBA methods, solving them perturbatively and matching previous results, advancing understanding of Wilson loops in gauge theories.

Contribution

It introduces a novel TBA-based integral equation framework for the cusp anomalous dimension, incorporating boundary reflection matrices and symmetry considerations.

Findings

Derived exact TBA equations for the cusp anomalous dimension.

Solved the equations perturbatively up to three loops.

Matched results with previous direct computational approaches.

Abstract

We derive a set of integral equations of the TBA type for the generalized cusp anomalous dimension, or the quark antiquark potential on the three sphere, as a function of the angles. We do this by considering a family of local operators on a Wilson loop with charge L. In the large L limit the problem can be solved in terms of a certain boundary reflection matrix. We determine this reflection matrix by using the symmetries and the boundary crossing equation. The cusp is introduced through a relative rotation between the two boundaries. Then the TBA trick of exchanging space and time leads to an exact equation for all values of L. The L=0 case corresponds to the cusped Wilson loop with no operators inserted. We then derive a slightly simplified integral equation which describes the small angle limit. We solve this equation up to three loops in perturbation theory and match the results that were obtained with more direct approaches.