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 $m_{0}a=0.006\mathrm{}$. 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 $O\left(a^2\right)$ 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.