Addendum: Nonlinear integral equations for the sausage model (2017 J. Phys. A: Math. Theor. 50 314005)

In [1], here below referred as I, we have written the set of non-linear integral equations (NLIEs) governing the finite size effects of the vacuum as well as the thermodynamics for the integrable deformation of O(3) non-linear sigma model (NLSM), getting it from a manipulation, inspired by those introduced years ago by Suzuki [3, 4], of the larger set of Thermodynamic Bethe Ansatz (TBA) equations of the model, known since the original paper by Fateev, Onofri and Zamolodchikov [2]. However, one can realize that (I3.24)5, a crucial relation in our derivation of the sausage model NLIE, is not well-defined because neither Q nor Q¯ are analytic on the real axis. Hence Q˜ and Q˜¯ cannot be interpreted as Fourier transforms along the real line. In this addendum we examine this problem carefully and show that the derivation of the sausage model NLIE remains valid in spite of this potential difficulty.