Role of theσresonance in determining the convergence of chiral perturbation theory

Abstract

The dimensionless parameter $\ensuremath{\xi}={M}_{\ensuremath{\pi}}^{2}/(16{\ensuremath{\pi}}^{2}{F}_{\ensuremath{\pi}}^{2})$, where ${F}_{\ensuremath{\pi}}$ is the pion decay constant and ${M}_{\ensuremath{\pi}}$ is the pion mass, is expected to control the convergence of chiral perturbation theory applicable to QCD. Here we demonstrate that a strongly coupled lattice gauge theory model with the same symmetries as two-flavor QCD but with a much lighter $\ensuremath{\sigma}$-resonance is different. We first confirm that the leading low-energy constants appearing in the chiral Lagrangian are the same when calculated from the $p$-regime and the $ϵ$-regime as expected. However, $\ensuremath{\xi}\ensuremath{\lesssim}0.002$ is necessary before 1-loop chiral perturbation theory predicts the data within 1%. For $\ensuremath{\xi}>0.0035$ the data begin to deviate dramatically from 1-loop chiral perturbation theory predictions. We argue that this qualitative change is due to the presence of a light $\ensuremath{\sigma}$-resonance in our model. Our findings may be useful for lattice QCD studies.