Analytic Solution of Bremsstrahlung TBA

TL;DR

This paper analytically solves the near BPS limit of the quark-anti-quark potential in planar N=4 SYM using TBA equations, revealing new results for arbitrary R-charge L and connecting to matrix models and string theory.

Contribution

It introduces an exact analytical solution for the vacuum potential in the near BPS limit, generalizing previous results and establishing links to matrix models and classical string algebraic curves.

Findings

Analytic expression for the potential at arbitrary L

Matching with localization results at L=0

Classical limit described by a matrix model and string theory

Abstract

We consider the quark--anti-quark potential on the three sphere or the generalized cusp anomalous dimension in planar N=4 SYM. We concentrate on the vacuum potential in the near BPS limit with $L$ units of R-charge. Equivalently, we study the anomalous dimension of a super-Wilson loop with L local fields inserted at a cusp. The system is described by a recently proposed infinite set of non-linear integral equations of the Thermodynamic Bethe Ansatz (TBA) type. That system of TBA equations is very similar to the one of the spectral problem but simplifies a bit in the near BPS limit. Using techniques based on the Y-system of functional equations we first reduced the infinite system of TBA equations to a Finite set of Nonlinear Integral Equations (FiNLIE). Then we solve the FiNLIE system analytically, obtaining a simple analytic result for the potential! Surprisingly, we find that the system has equivalent descriptions in terms of an effective Baxter equation and in terms of a matrix model. At L=0, our result matches the one obtained before using localization techniques. At all other L's, the result is new. Having a new parameter, L, allows us to take the large L classical limit. We use the matrix model description to solve the classical limit and match the result with a string theory computation. Moreover, we find that the classical string algebraic curve matches the algebraic curve arising from the matrix model.