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.

We investigate the mass spectra of closed-charm mesons with $J^{PC}=1^{--}$, for hybrid charmonium, molecules, and diquark-antidiquark operators, in quenched lattice QCD with exact chiral symmetry. For two lattice volumes ${24}^3\times 48$ and ${20}^3\times 40$, each of 100 gauge configurations generated with single-plaquette action at $\beta =6.1$, we compute point-to-point quark propagators and measure the time-correlation functions of these exotic meson operators. For the molecular operator $\{(\overline{\mathbf{q}}{\gamma }_5{\gamma }_i\mathbf{c})(\overline{\mathbf{c}}{\gamma }_5\mathbf{q})-(\overline{\mathbf{c}}{\gamma }_5{\gamma }_i\mathbf{q})(\overline{\mathbf{q}}{\gamma }_5\mathbf{c})\}$, it detects a resonance with mass around $4238\pm 31\mathrm{MeV}$, which is naturally identified with $Y(4260)$. Further, for any molecular and diquark-antidiquark operator, it detects heavier exotic charmed mesons, with $(\mathbf{c}\mathbf{s}\overline{\mathbf{c}}\overline{\mathbf{s}})$ around $4450\pm 100\mathrm{MeV}$, and $(\mathbf{c}\mathbf{c}\overline{\mathbf{c}}\overline{\mathbf{c}})$ around $6400\pm 50\mathrm{MeV}$.

We test the convergence property of the chiral perturbation theory using a lattice QCD calculation of pion mass and decay constant with two dynamical quark flavors. The lattice calculation is performed using the overlap fermion formulation, which realizes exact chiral symmetry at finite lattice spacing. By comparing various expansion prescriptions, we find that the chiral expansion is well saturated at the next-to-leading order for pions lighter than $\sim 450\mathrm{MeV}$. Better convergence behavior is found, in particular, for a resummed expansion parameter $\xi$, with which the lattice data in the pion mass region 290–750 MeV can be fitted well with the next-to-next-to-leading order formulas. We obtain the results in two-flavor QCD for the low energy constants ${\overline{l}}_3$ and ${\overline{l}}_4$ as well as the pion decay constant, the chiral condensate, and the average up and down quark mass.

We discuss the salient features of the Zolotarev optimal rational approximation for the inverse square root function, in particular, for its applications in lattice QCD with the overlap Dirac quark. The theoretical error bound for the matrix-vector multiplication $H_w{(H}_w^2{)}^{-1/2}Y$ is derived. We check that the error bound is always satisfied amply, for any QCD gauge configurations we have tested. An empirical formula for the error bound is determined, together with its numerical values (by evaluating elliptic functions) listed in Table II as well as plotted in Fig. 3. Our results suggest that, with the Zolotarev approximation to ${(H}_w^2{)}^{-1/2},$ one can essentially preserve the exact chiral symmetry of the overlap Dirac operator to very high precision, for any gauge configuration on a finite lattice.