Pion scattering at finite volume within the Inverse Amplitude Method

TL;DR

This paper develops a comprehensive finite-volume analysis of pion-pion scattering using Chiral Perturbation Theory and the Inverse Amplitude Method, accounting for all relevant loop contributions and group symmetries to improve lattice QCD data interpretation.

Contribution

It extends the IAM to finite volume, including discretization effects in all channels and a novel projection method, providing more accurate predictions of two-pion energy levels in lattice simulations.

Findings

Sizable finite-volume corrections for $m_\pi L \lesssim 2$

Results align with lattice data, improving previous analyses

Method enhances phase-shift extraction accuracy

Abstract

We study the effect of a finite volume for pion-pion scattering within Chiral Perturbation Theory (ChPT) and the Inverse Amplitude Method (IAM) in a $L^3$ box (rest frame). Our full ChPT calculation takes into account the discretization not only in the $s$-channel loops but also in the $t,u$- channels and tadpole contributions. Hence, not only the unitarity right-hand cut but also the left-hand one continuum contributions are calculated in the finite volume. A proper extension of the standard Veltman-Pasarino identities is needed, as well as a suitable projection on the internal space spanned by the irreducible representations (irreps) of the octahedral group, based on either a finite set of cubic harmonics or the matrices which represent the irreps properly. From the ChPT we construct the IAM in the internal space, which provides the full volume dependence of the interacting energy levels of two-pions scattering in the finite volume. Our results for various low-energy constants sets show sizable corrections with respect to previous analyses in the literature for $ m_\pi L \lesssim 2$, being compatible with energy levels lattice data. We expect that our analysis and results will help to optimize the process of determination of energy levels and phase-shifts with higher accuracy.