Bremsstrahlung function, leading Lüscher correction at weak coupling and localization

We discuss the near BPS expansion of the generalized cusp anomalous dimen- sion with L units of R-charge. Integrability provides an exact solution, obtained by solving a general TBA equation in the appropriate limit: we propose here an alternative method based on supersymmetric localization. The basic idea is to relate the computation to the vacuum expectation value of certain 1/8 BPS Wilson loops with local operator insertions along the contour. These observables localize on a two-dimensional gauge theory on S2, opening the possibility of exact calculations. As a test of our proposal, we reproduce the leading Lu ̈scher correction at weak coupling to the generalized cusp anomalous dimension. This result is also checked against a genuine Feynman diagram approach in N = 4 Super Yang-Mills theory.