Chiral Properties of Pseudoscalar Mesons on a Quenched $20^4$ Lattice with Overlap Fermions

The chiral properties of the pseudoscalar mesons are studied numerically on a quenched $20^4$ lattice with the overlap fermion. We elucidate the role of the zero modes in the meson propagators, particularly that of the pseudoscalar meson. The non-perturbative renormalization constant $Z_A$ is determined from the axial Ward identity and the pion decay constants $f_{\pi}$ is calculated from which we determine the lattice spacing to be 0.157 fm.. We look for quenched chiral log in the pseudoscalar decay constants and the pseudocalar masses for both equal and unequal quark masses. We find that, given limited statistics, it is difficult to see quenched logs in pseudocalar masses with equal or unequal quark masses. Whereas, it is relatively clear to see them in the pseudoscalar matrix element and the ratio of the pseudoscalar matrix element to $f_{\pi}$. The chiral log parameter $\delta$ turns out to be consistent with that predicted from quenched chiral perturbative theory.