On the rotator Hamiltonian for the SU$(N)\times\,$SU$(N)$ sigma-model in the delta-regime

TL;DR

This paper analyzes the SU(N)×SU(N) sigma-model in the delta-regime, showing the rotator approximation's agreement with chiral perturbation theory and exploring spectrum corrections for N=3.

Contribution

It demonstrates the accuracy of the rotator approximation in the delta-regime and relates spectrum deviations to quadratic Casimir invariants, supported by nonperturbative results.

Findings

Isospin susceptibility matches chiral perturbation theory up to NNLO.

Deviations involve terms vanishing as 1/ell and exponentially.

Spectrum corrections relate to quadratic Casimir invariant.

Abstract

We investigate some properties of the standard rotator approximation of the SU$(N)\times\,$SU$(N)$ sigma-model in the delta-regime. In particular we show that the isospin susceptibility calculated in this framework agrees with that computed by chiral perturbation theory up to next-to-next to leading order in the limit $\ell=L_t/L\to\infty\,.$ The difference between the results involves terms vanishing like $1/\ell\,,$ plus terms vanishing exponentially with $\ell\,$. As we have previously shown for the O($n$) model, this deviation can be described by a correction to the rotator spectrum proportional to the square of the quadratic Casimir invariant. Here we confront this expectation with analytic nonperturbative results on the spectrum in 2 dimensions for $N=3\,.$