Hidden Relation between Re ection Amplitudesand Thermodynamic Bethe

In this paper we compute the scaling functions of the eeective central charges for various quantum integrable models in a deep ultraviolet region R ! 0 using two independent methods. One is based on the \reeection amplitudes" of the (super-)Liouville eld 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 reeection amplitudes. The other method is to solve the standard ther-modynamic Bethe ansatz (TBA) equations for the integrable models in a perturbative series of 1=(const: ? ln R). The constant factor which is not xed in the lowest order computations can be identiied 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 rst method. Our results show that these two methods are complementary to each other. While the reeection amplitudes are connrmed by the numerical TBA analysis, the analytic structures of the TBA equations become clear only when the reeection amplitudes are introduced.