Finite size spectrum of SU(N) principal chiral field from discrete Hirota dynamics

Using recently proposed method of discrete Hirota dynamics for integrable (1+1)(1+1)D quantum field theories on a finite space circle of length L   we derive and test numerically a finite system of nonlinear integral equations for the exact spectrum of energies of SU(N)×SU(N)SU(N)×SU(N) principal chiral field model as functions of mL, where m   is the mass scale. We propose a determinant solution of the underlying Y-system, or Hirota equation, in terms of Wronskian determinants of N×NN×N matrices parameterized by N−1N−1 functions of the spectral parameter θ with the known analytic properties at finite L  . Although the method works in principle for any state, the explicit equations are written for states in the U(1)U(1) sector only. For N>2N>2, we encounter and clarify a few subtleties in these equations related to the presence of bound states, absent in the previously considered N=2N=2 case. As a demonstration of efficiency of our method, we solve these equations numerically for a few low-lying states at N=3N=3 in a wide range of mL.