Hidden relation between reflection amplitudes and thermodynamic Bethe ansatz

In this paper we compute the scaling functions of the effective central charges for various quantum integrable models in a deep ultraviolet region R → 0 using two independent methods. One is based on the “reflection amplitudes” of the (super-)Liouville field theory where the scaling functions are given by the conjugate momentum to the zero-modes. The conjugate momentum is quantized for the sinh-Gordon, the Bullough-Dodd, and the super sinh-Gordon models where the quantization conditions depend on the size R of the system and the reflection amplitudes. The other method is to solve the standard thermodynamic Bethe ansatz (TBA) equations for the integrable models in a perturbative series of 1/(const. - In R). The constant factor which is not fixed in the lowest order computations can be identified only when we compare the higher order corrections with the quantization conditions. Numerical TBA analysis shows a perfect match for the scaling functions obtained by the first method. Our results show that these two methods are complementary to each other. While the reflection amplitudes are confirmed by the numerical TBA analysis, the analytic structures of the TBA equations become clear only when the reflection amplitudes are introduced.