Finite Size Spectrum of SU(N) Principal Chiral Field from Discrete Hirota Dynamics

TL;DR

This paper develops a numerical method based on discrete Hirota dynamics to compute the exact finite-size spectrum of the SU(N) principal chiral field, revealing new insights into bound states for N>2.

Contribution

It introduces a determinant solution for the Y-system of the SU(N) principal chiral model and applies it to compute spectra for N=3, addressing subtleties related to bound states.

Findings

Derived nonlinear integral equations for the spectrum

Numerically computed low-lying states for N=3

Clarified bound state effects for N>2

Abstract

Using recently proposed method of discrete Hirota dynamics for integrable (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)xSU(N) principal chiral field model as functions of m L, where m is the mass scale. We propose a determinant solution of the underlying Y-system, or Hirota equation, in terms of determinants (Wronskians) of NxN matrices parameterized by N-1 functions of the spectral parameter, with the known analytical properties at finite L. Although the method works in principle for any state, the explicit equations are written for states in the U(1) sector only. For N>2, we encounter and clarify a few subtleties in these equations related to the presence of bound states, absent in the previously considered N=2 case. In particular, we solve these equations numerically for a few low-lying states for N=3 in a wide range of m L.