Reformulating the TBA equations for the quark anti-quark potential and their two loop expansion

Recently boundary thermodynamic Bethe Ansatz (BTBA) equations were inroduced to describe the cusp anomalous dimension in the N=4 super Yang-Mills theory.. The equations contain imaginary chemical potentials and singular boundary fugacities, which make its systematic expansion problematic. We propose an alternative formulation based on real chemical potentials and additional source terms. We expand our equations to double wrapping order and find complete agreement with the direct two-loop gauge theory computation of the cusp anomalous dimension.