Quenched lattice QCD with domain wall fermions and the chiral limit

Quenched QCD simulations on three volumes $8^3\times ,$ ${12}^3\times$ and ${16}^3\times 32$ and three couplings $\beta =5.7,$ 5.85 and 6.0 using domain wall fermions provide a consistent picture of quenched QCD. We demonstrate that the small induced effects of chiral symmetry breaking inherent in this formulation can be described by a residual mass ${(m}_{\mathrm{res}})$ whose size decreases as the separation between the domain walls ${(L}_s)$ is increased. However, at stronger couplings much larger values of $L_s$ are required to achieve a given physical value of $m_{\mathrm{res}}.$ For $\beta =6.0\mathrm{}$ and $L_s=16,$ we find $m_{\mathrm{res}}{/m}_s=0.033\mathrm{}\left(3\right),$ while for $\beta =5.7,$ and $L_s=48,$ $m_{\mathrm{res}}{/m}_s=0.074\mathrm{}\left(5\right),$ where $m_s$ is the strange quark mass. These values are significantly smaller than those obtained from a more naive determination in our earlier studies. Important effects of topological near zero modes which should afflict an accurate quenched calculation are easily visible in both the chiral condensate and the pion propagator. These effects can be controlled by working at an appropriately large volume. A non-linear behavior of $m_{\pi }^2$ in the limit of small quark mass suggests the presence of additional infrared subtlety in the quenched approximation. Good scaling is seen both in masses and in $f_{\pi }$ over our entire range, with inverse lattice spacing varying between 1 and 2 GeV.