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We investigate the sub-leading contributions to the free energy of BetheAnsatz solvable (continuum) models with different boundary conditions. We showthat the Thermodynamic Bethe Ansatz approach is capable of providing the O(1)pieces if both the density of states in rapidity space and the quadraticfluctuations around the saddle point solution to the TBA are properly takeninto account. In relativistic boundary QFT the O(1) contributions are directlyrelated to the exact g-function. In this paper we provide an all-orders proofof the previous results of P. Dorey et al. on the g-function in both massiveand massless models. In addition, we derive a new result for the g-functionwhich applies to massless theories with arbitrary diagonal scattering in thebulk.

On O(1) contributions to the free energy in Bethe Ansatz systems: the exact g-function