Chiral Symmetry, Quark Mass, and Scaling of the Overlap Fermions

The chiral symmetry relation and scaling of the overlap fermions are studied numerically on the quenched lattices at 3 couplings with about the same physical volume. We find that the generalized Gell-Mann-Oakes-Renner relation is satisfied to better than 1% down to the smallest quark mass at $m_0a = 0.006$. We also obtain the quark mass from the PCAC relation and the pseudoscalar masses. The renormalization group invariant quark mass is shown to be fairly independent of scale. The $\pi$ and $\rho$ masses at a fixed $m_{\pi}/m_{\rho}$ ratio indicate small $O(a^2)$ corrections. It is found that the critical slowing down sets in abruptly at a very small quark mass close to those of the physical u and d quarks.