Role of the σ resonance in determining the convergence of chiral perturbation theory

The dimensionless parameter ξ=M_π²/(16π²F_π²), where F_π is the pion decay constant and M_π 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 σ-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, ξ≲0.002 is necessary before 1-loop chiral perturbation theory predicts the data within 1%. For ξ>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 σ-resonance in our model. Our findings may be useful for lattice QCD studies.