Nonlinear integral equations for the sausage model

The sausage model, first proposed by Fateev, Onofri, and Zamolodchikov, is a deformation of the O(3) sigma model preserving integrability. The target space is deformed from the sphere to sausage shape by a deformation parameter v. This model is defined by a factorizable S-matrix which is obtained by deforming that of the O(3) sigma model by a parameter $\lambda$. Clues for the deformed sigma model are provided by various UV and IR information through the thermodynamic Bethe ansatz (TBA) analysis based on the S-matrix. Application of TBA to the sausage model is, however, limited to the case of $\frac{1}{\lambda}$ integer where the coupled integral equations can be truncated to a finite number. In this paper, we propose a finite set of nonlinear integral equations (NLIEs), which are applicable to generic value of $\lambda$. Our derivation is based on T-Q relations extracted from the truncated TBA equations. For a consistency check, we compute next-leading order corrections of the vacuum energy and extract the S-matrix information in the IR limit. We also solved the NLIE both analytically and numerically in the UV limit to get the effective central charge and compared with that of the zero-mode dynamics to obtain exact relation between v and $\lambda$.