Exact g-function without strings

We propose a new approach to compute exact $g$-function for integrable quantum field theories with non-diagonal scattering S-matrices. The approach is based on an integrable lattice regularization of the quantum field theory. The exact $g$-function is encoded in the overlap of the integrable boundary state and the ground state on the lattice, which can be computed exactly by Bethe ansatz. In the continuum limit, after subtracting the contribution proportional to the volume of the closed channel, we obtain the exact $g$-function, given in terms of the counting function which is the solution of a nonlinear integral equation. The resulting $g$-function contains two parts, the scalar part, which depends on the boundary parameters and the ratio of Fredholm determinants, which is universal. This approach bypasses the difficulties of dealing with magnetic excitations for non-diagonal scattering theories in the framework of thermodynamic Bethe ansatz. We obtain numerical and analytical results of the exact $g$-function for the prototypical sine-Gordon theory with various integrable boundary conditions.