The dimensionless parameter $xi = M^2/(16 pi^2 F^2)$, where $F$ is the pion decay constant in the chiral limit and $M$ is the pion mass at leading order in the quark 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 $sigma$-resonance is different. Our model allows us to study efficiently the convergence of chiral perturbation theory as a function of $xi$. We first confirm that the leading low energy constants appearing in the chiral Lagrangian are the same when calculated from the $epsilon$-regime and the $p$-regime. However, $xi lesssim 0.002$ is necessary before 1-loop chiral perturbation theory predicts the data within 1%. However, for $xi > 0.0035$ the data begin to deviate qualitatively from 1-loop chiral perturbation theory predictions. We argue that this qualitative change is due to the presence of a light $sigma$-resonance in our model. Our findings may be useful for lattice QCD studies.
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