Boundary massive sine-Gordon model at the free Fermi limit and RG flow of Casimir energy

RG flow of central charge $c_{\rm eff}$ is investigated for the two boundary sine-Gordon model at the free Fermi limit. Thermodynamic Bethe ansatz approach is used to check the non-monotonic decreasing properties of $c_{\rm eff}$, its resonance, and the modification of $c_{\rm eff}$ due to the mismatch of the periodicity of a Lagrangian parameter and that of the boundary scattering parameter. The detailed analysis uses the singularity structure of the system on the complex rapidity plane.