Pion form factors from two-flavor lattice QCD with exact chiral symmetry

We calculate pion vector and scalar form factors in two-flavor lattice QCD and study the chiral behavior of the vector and scalar radii $⟨r^2{⟩}_{V,S}$. Numerical simulations are carried out on a ${16}^3\times 32$ lattice at a lattice spacing of 0.12 fm with quark masses down to $\sim m_s/6$, where $m_s$ is the physical strange quark mass. Chiral symmetry, which is essential for a direct comparison with chiral perturbation theory (ChPT), is exactly preserved in our calculation at finite lattice spacing by employing the overlap quark action. We utilize the so-called all-to-all quark propagator in order to calculate the scalar form factor including the contributions of disconnected diagrams and to improve statistical accuracy of the form factors. A detailed comparison with ChPT reveals that the next-to-next-to-leading-order contributions to the radii are essential to describe their chiral behavior in the region of quark mass from $m_s/6$ to $m_s/2$. Chiral extrapolation based on two-loop ChPT yields $⟨r^2{⟩}_V=0.409(23)(37){\mathrm{fm}}^2$ and $⟨r^2{⟩}_S=0.617(79)(66){\mathrm{fm}}^2$, which are consistent with phenomenological analysis. We also present our estimates of relevant low-energy constants.

Figure 1

Connected (leftmost diagram) and disconnected three-point functions (middle diagram). Note that $F_S(0)$ receives a contribution from the rightmost diagram due to the nonzero vacuum expectation value (VEV) of the scalar operator $S$.

Figure 2

Statistical fluctuation of connected three-point functions $C_{\pi V_4\pi ,{\varphi }_s{\varphi }_s}^{(\mathrm{conn})}(\Delta t,\Delta t^{\prime };\mathbf{p},{\mathbf{p}}^{\prime })$ (left panel) and $C_{\pi 1\pi ,{\varphi }_s{\varphi }_s}^{(\mathrm{conn})}(\Delta t,\Delta t^{\prime };\mathbf{p},{\mathbf{p}}^{\prime })$ (right panel) with $\Delta t=\Delta t^{\prime }=7$, $|\mathbf{p}|=\sqrt{2}$ and $|{\mathbf{p}}^{\prime }|=0$. Open triangles are jackknife data with a single choice of the source location $(\mathbf{x},t)$ and momentum configuration $(\mathbf{p},{\mathbf{p}}^{\prime })$, whereas the filled squares are averaged over $(\mathbf{x},t)$ and $(\mathbf{p},{\mathbf{p}}^{\prime })$ corresponding to the same value of $q^2$. Each data is normalized by its statistical average.

Figure 3

Effective value of pion energy $E_{\pi }(|\mathbf{p}|)$ at $(Q,m)=(0,0.050)$. Circles, triangles and squares show results from $C_{\pi \pi ,{\varphi }_l{\varphi }_l}(\Delta t;\mathbf{p})$, $C_{\pi \pi ,{\varphi }_l{\varphi }_s}(\Delta t;\mathbf{p})$ and $C_{\pi \pi ,{\varphi }_s{\varphi }_s}(\Delta t;\mathbf{p})$, respectively.

Figure 4

Effective value of pion energy $E_{\pi }(|\mathbf{p}|)$ at $(Q,m)=(0,0.015)$.

Figure 5

Effective value of vector form factor $F_V(\Delta t,\Delta t^{\prime };q^2)$ at $(Q,m)=(0,0.050)$. We only plot data with $\Delta t+\Delta t^{\prime }<N_t/2$. Momenta written in the legends are spatial and in units of $2\pi a/L$.

Figure 6

Effective value of $F_V(\Delta t,\Delta t^{\prime };q^2)$ at $(Q,m)=(0,0.015)$.

Figure 7

Effective value of normalized scalar form factor $F_S(\Delta t,\Delta t^{\prime };q^2)/F_S(\Delta t,\Delta t^{\prime };q_{\mathrm{ref}}^2)$ at $(Q,m)=(0,0.050)$.

Figure 8

Effective value of $F_S(\Delta t,\Delta t^{\prime };q^2)/F_S(\Delta t,\Delta t^{\prime };q_{\mathrm{ref}}^2)$ at $(Q,m)=(0,0.015)$.

Figure 9

Normalized scalar form factor $F_S(q^2)/F_S(q_{\mathrm{ref}}^2)$ with and without the contributions of disconnected diagram to $F_S(q^2)$. We use the normalization value $F_S(q_{\mathrm{ref}}^2)$ with the disconnected contributions in both data.

Figure 10

Three-point functions with vector current (left panels) and scalar operator (right panels) calculated in different topological sectors for $m=0.050$. Top and bottom panels show data at the smallest and largest nonzero $|q^2|$.

Figure 11

Vector (left panels) and scalar form factors (right panels) calculated in different topological sectors for $m=0.050$.

Figure 12

Vector form factor $F_V(q^2)$ as a function of $q^2$. Solid and dotted lines show the fit curve (35) and its error. The $q^2$ dependence from the vector meson dominance model is shown by the dashed line.

Figure 13

Contributions in the $q^2$ expansion of $F_V(q^2)-1$ at $m=0.025$. Thin solid, dashed and dot-dashed lines show $O(q^2)$, $O(q^4)$ and higher contributions, whereas the thick solid line is their total.

Figure 14

Normalized scalar form factor $F_S(q^2)/F_S(q_{\mathrm{ref}}^2)$ as a function of $q^2$. Solid and dotted lines show the cubic fit and its error.

Figure 15

Chiral fit of $⟨r^2{⟩}_V$ (left panel) and $⟨r^2{⟩}_S$ (right panel) using one-loop ChPT formulas. Filled squares are the lattice data and the value extrapolated to the physical point. In the left panel, we also plot the experimental value $⟨r^2{⟩}_V=0.437(16){\mathrm{fm}}^2$ from an analysis based on $N_f=2$ ChPT [16] (open circle) and $0.452(11){\mathrm{fm}}^2$ quoted by Particle Data Group [49] (star). The star symbol in the right panel represents $⟨r^2{⟩}_S=0.61(4){\mathrm{fm}}^2$ obtained from an indirect determination through $\pi \pi$ scattering [51].

Figure 16

Chiral behavior of radii based on two-loop ChPT with phenomenological estimates of LECs listed in Table 13. The dashed and dot-dashed lines represent contributions at NLO and NNLO, whereas the solid line is their total. We note that $r_{\{V,S\},\{r,c\}}^r$ from the resonance saturation are taken as the renormalized LECs at the resonance mass scale in this plot.

Figure 17

Simultaneous chiral fit to $⟨r^2{⟩}_V$ and $c_V$ based on two-loop formulas Eqs. (41) and (43). The experimental value for $c_V=3.85(0.60)$ is taken from Ref. [16].

Figure 18

Simultaneous chiral fit for radii $⟨r^2{⟩}_{V,S}$ and curvature $c_V$.