The pion mass in finite volume

TL;DR

This paper calculates the finite-volume shift of the pion mass using chiral perturbation theory and Lüscher's formula, providing insights into the convergence and subleading effects in the large volume limit.

Contribution

It combines two-flavor chiral perturbation theory with Lüscher's formula to accurately determine pion mass shifts in finite volume, including higher-order corrections.

Findings

Quantifies pion mass shift due to finite volume.

Assesses convergence of chiral series in finite volume.

Evaluates size of subleading corrections.

Abstract

We determine the relative pion mass shift $M_\pi(L)/M_\pi-1$ due to the finite spatial extent $L$ of the box by means of two-flavor chiral perturbation theory and the one-particle L\"uscher formula. We use as input the expression for the infinite volume $\pi\pi$ forward scattering amplitude up to next-to-next-to-leading order and can therefore control the convergence of the chiral series. A comparison to the full leading order chiral expression for the pion mass in finite volume allows us to check the size of subleading terms in the large-$L$ expansion.