Finite-volume effects in the $\delta$-regime

TL;DR

This paper develops a systematic perturbative approach to analyze finite-volume effects in the $O(N)$ sigma-model within the $ ext{delta}$-regime, accurately reproducing known energy spectra up to next-to-next-to-leading order.

Contribution

It introduces a threshold expansion method to handle power-counting violations, enabling precise calculations of the finite-volume energy spectrum in the $ ext{delta}$-regime.

Findings

Reproduces known energy spectrum results up to NNLO.

Provides a systematic perturbative framework for finite-volume effects.

Addresses power-counting violations with threshold expansion.

Abstract

We derive a systematic perturbative expansion for the finite-volume energy spectrum of the non-linear $O(N)$ $\sigma$-model in the $\delta$-regime. The violation of the power-counting rules that emerges after the separation of the fast and slow modes is dealt with to all orders by use of the threshold expansion. The known result for the rest-frame energy spectrum up-to-and-including next-to-next-to-leading order is reproduced.