Two-loop analysis of the pion-mass dependence of the $\rho$ meson

TL;DR

This paper develops a two-loop unitarization method for pion-mass dependence in $ ho$ meson studies, enabling more accurate analysis of lattice QCD results and assessing chiral convergence.

Contribution

It provides the first stable two-loop IAM fits for lattice QCD data and offers compact analytic expressions for partial-wave amplitudes at this order.

Findings

Two-loop IAM fits are feasible with a new fit strategy.

Chiral convergence may break down near the $ ho$ mass.

Systematic uncertainties in lattice data are crucial for reliable fits.

Abstract

Analyzing the pion-mass dependence of $\pi\pi$ scattering phase shifts beyond the low-energy region requires the unitarization of the amplitudes from chiral perturbation theory. In the two-flavor theory, unitarization via the inverse-amplitude method (IAM) can be justified from dispersion relations, which is therefore expected to provide reliable predictions for the pion-mass dependence of results from lattice QCD calculations. In this work, we provide compact analytic expression for the two-loop partial-wave amplitudes for $J=0,1,2$ required for the IAM at subleading order. To analyze the pion-mass dependence of recent lattice QCD results for the $P$-wave, we develop a fit strategy that for the first time allows us to perform stable two-loop IAM fits and assess the chiral convergence of the IAM approach. While the comparison of subsequent orders suggests a breakdown scale not much below the $\rho$ mass, a detailed understanding of the systematic uncertainties of lattice QCD data is critical to obtain acceptable fits, especially at larger pion masses.