Role of the $\sigma$ resonance in determining the convergence of chiral perturbation theory

The dimensionless parameter $\xi =M_{\pi }^2/(16{\pi }^2{F}_{\pi }^2)$, where $F_{\pi }$ is the pion decay constant and $M_{\pi }$ 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 $\sigma$-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, $\xi \lesssim 0.002$ is necessary before 1-loop chiral perturbation theory predicts the data within 1%. For $\xi >0.0035$ 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 $\sigma$-resonance in our model. Our findings may be useful for lattice QCD studies.

Figure 1

Finite-size scaling of $Y_v$, $Y_c$, and ${\chi }_{\sigma }$ at $m=0.0002$ (top left), $m=0.001$ (top right), $m=0.0035$ (bottom left), and $m=0.0065$ (bottom right). The solid lines are fits of the data to the expected finite-size scaling form from chiral perturbation theory while dashed lines are fits to a constant.

Figure 2

Rescaled and subtracted quantities $R_{F_{\pi }}$, $R_{⟨\overline{q}q⟩}$, and $R_{M_{\pi }^2}$ defined in Eqs. (8). The solid lines are plots of the fits discussed in the text. The dashed lines show the linear region for larger values of ${\xi }^{\prime }$. The knee is estimated roughly as the point where the two lines cross.