# Isospin susceptibility in the O($n$) sigma-model in the delta-regime

## TL;DR

This paper calculates the isospin susceptibility in the O(n) sigma-model within the delta-regime using chiral perturbation theory, including effects of explicit symmetry breaking and finite-volume corrections, up to third order.

## Contribution

It extends previous computations by including a small explicit symmetry breaking term and analyzing finite-volume effects with higher-order chiral perturbation theory in the delta-regime.

## Key findings

- Consistency with previous mass gap and susceptibility results in the chiral limit.
- Derived the susceptibility's behavior with explicit symmetry breaking to third order.
- Identified a correction to the rotator spectrum proportional to the quadratic Casimir invariant.

## Abstract

We compute the isospin susceptibility in an effective O($n$) scalar field theory (in $d=4$ dimensions), to third order in chiral perturbation theory ($\chi$PT) in the delta--regime using the quantum mechanical rotator picture. This is done in the presence of an additional coupling, involving a parameter $\eta$, describing the effect of a small explicit symmetry breaking term (quark mass). For the chiral limit $\eta=0$ we demonstrate consistency with our previous $\chi$PT computations of the finite-volume mass gap and isospin susceptibility. For the massive case by computing the leading mass effect in the susceptibility using $\chi$PT with dimensional regularization, we determine the $\chi$PT expansion for $\eta$ to third order. The behavior of the shape coefficients for long tube geometry obtained here might be of broader interest. The susceptibility calculated from the rotator approximation differs from the $\chi$PT result in terms vanishing like $1/\ell$ for $\ell=L_t/L_s\to\infty$. We show that this deviation can be described by a correction to the rotator spectrum proportional to the square of the quadratic Casimir invariant.