Casimir squared correction to the standard rotator Hamiltonian for the O($n$) sigma-model in the delta-regime

TL;DR

This paper investigates the correction to the standard rotator Hamiltonian in the O(n) sigma-model, focusing on the Casimir squared correction, and compares theoretical predictions with nonperturbative spectral results in 2 and 3 dimensions.

Contribution

It provides a detailed analysis of the Casimir squared correction to the rotator Hamiltonian and compares it with nonperturbative spectral data for the O(n) sigma-model.

Findings

Confirmation of the Casimir squared correction in 2D spectra

Analytic nonperturbative results support the correction hypothesis

Extension of analysis to 3D case

Abstract

In a previous paper we found that the isospin susceptibility of the O($n$) sigma-model calculated in the standard rotator approximation differs from the next-to-next to leading order chiral perturbation theory result in terms vanishing like $1/\ell\,,$ for $\ell=L_t/L\to\infty$ and further showed that this deviation could 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, by Balog and Heged\"us for $n=3,4$ and by Gromov, Kazakov and Vieira for $n=4$. We also consider the case of 3 dimensions.