Computation of the electromagnetic pion form factor from lattice QCD in the $\epsilon$ regime

We calculate the electromagnetic pion form factor in lattice QCD with $2+1$ flavors of the dynamical overlap quarks. Up and down quark masses are set below their physical values so that the system is in the so-called $\epsilon$ regime with the small size of our lattice $\sim 1.8\mathrm{fm}$. The finite volume corrections are generally expected to be $\sim 100\%$ in the $\epsilon$ regime. We, however, find a way to automatically cancel the dominant part of them. Inserting nonzero momenta and taking appropriate ratios of the two- and three-point functions, we can eliminate the contribution from the zero-momentum pion mode. Then the remaining finite volume effect is a small perturbation from the nonzero modes. Our lattice data agree with this theoretical prediction and the extracted pion charge radius is consistent with the experiment.

Figure 1

Ratio, $C_{PP}^{2\mathrm{pt}}(t,\mathbf{p})/{\mathrm{\Delta }}_t{C}_{PP}^{2\mathrm{pt}}(t,\mathbf{0})$, of two-point correlators at different momenta where we set $t_{\text{ref}}=16$. Lattice data are plotted together with the expectation of ChPT at the leading order without mass (ignoring $Z$) (solid curve) and with mass $(m_{\pi }^2+\mathrm{\Delta }{m}_{\pi }^2{)}^{1/2}=100\mathrm{MeV}$ (dashed curve). Different symbols represent $\mathbf{p}=(1,0,0)$, (1, 1, 0), (1, 1, 1), and (2, 0, 0) in units of $2\pi /L$. Here, the rotationally symmetric correlators are averaged in the data.

Figure 2

$F_V^1(t,t^{\prime },q^2=-0.40[{\mathrm{GeV}}^2])$ (top panel) and $F_V^2(t,t^{\prime },q^2=-0.11[{\mathrm{GeV}}^2])$ (bottom panel) are plotted as a function of $t^{\prime }$. The combination of initial and final momenta are shown in the plots in units of $2\pi /L$. Data for different $t$’s are plotted with different symbols. A constant fit and their error are shown by bands.

Figure 3

Lattice results for $F_V(q^2)$ as a function of $q^2$. The lattice data in the $\epsilon$ regime ($m_{ud}a=0.002$, filled symbols) and in the $p$ regime ($m_{ud}a=0.015$, open symbols) are plotted. The $\epsilon$ regime data are obtained either from the ratio $F_V^1$ (circles) or $F_V^2$ (squares).

Figure 4

Pion charge radius as a function of $m_{\pi }^2$. A result from this work obtained in the $\epsilon$ regime (circle) is plotted together with the data at heavier up and down quarks (square, from [43]). The experimental result is shown at $m_{\pi }=135\mathrm{MeV}$. The fit curves are those of next-to-leading order ChPT as well as a simple polynomial (quadratic).