On the Thermodynamic Bethe Ansatz Equation in Sinh-Gordon Model

Two implicit periodic structures in the solution of sinh-Gordon thermodynamic Bethe ansatz equation are considered. The analytic structure of the solution as a function of complex $\theta$ is studied to some extent both analytically and numerically. The results make a hint how the CFT integrable structures can be relevant in the sinh-Gordon and staircase models. More motivations are figured out for subsequent studies of the massless sinh-Gordon (i.e. Liouville) TBA equation.