Pion scattering at finite volume within the Inverse Amplitude Method

TL;DR

This work develops a comprehensive framework for finite-volume pion-pion scattering by combining full Chiral Perturbation Theory at ${\mathcal{O}}(p^4)$ with the Inverse Amplitude Method (IAM) in a rest-frame $L^3$ box. It introduces a complete finite-volume treatment that includes $s$-, $t$-, and $u$-channel loops plus tadpoles, and proves that the Lorentz-covariant VP relations must be generalized to the cubic geometry, requiring projections onto the octahedral group via irreps and cubic harmonics. The finite-volume IAM is constructed as a matrix problem in the cubic-symmetry basis, yielding a quantization condition that correctly incorporates both left- and right-hand cuts and left-hand-cut effects, and it is shown to reproduce lattice energy levels and phase shifts with sizable corrections only at $m_{\pi}L\lesssim 2$. Numerical comparisons with lattice data and Bethe-Salpeter approaches highlight the importance of the full finite-volume amplitude for small volumes, while the approach remains consistent with standard Lüscher physics near threshold. The methodology provides improved energy-level predictions and a robust path to extracting phase shifts and resonance properties from lattice QCD, with potential extensions to moving frames and multi-channel processes.

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_πL \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.

Figures (15)

• Figure 1: Generic diagram for elastic pion-pion scattering amplitude, where $V$ and $Z$ account respectively for the generic interaction vertex and the renormalization of the external legs.
• Figure 2: Diagrams contributing to pion scattering in ChPT up to ${\mathcal{O}}(p^4)$
• Figure 3: Finite-volume corrections to the Lüscher approach for the $I=0,2$ energy levels near threshold, using the LECs of Ref. Hanhart:2008mx and the definition of $P$ given in Eq. \ref{['Pdeftext']}. The uncertainty bands correspond to the variation of the renormalized couplings $l_{1-4}^r$.
• Figure 4: $A_{1}^{+}$ energy levels for $I = 0$ as a function of the box size, using the LECs given in Hanhart:2008mx.
• Figure 5: Mass dependence of the ratio $m_{\pi}/f_{\pi}$. The data points correspond to lattice results provided by GWU collaboration Guo:2016zosGuo:2018zsscross:prd. The LECs are given in Tables \ref{['tab:iam']} and \ref{['tab:bs']}.