Domain Wall Quarks and Kaon Weak Matrix Elements

We present lattice calculations of kaon matrix elements with domain wall fermions. Using lattices with $6/g^2=5.85\mathrm{}$, 6.0, and 6.3, we estimate $B_K\left(\mu \approx 2\mathrm{GeV}\right)=0.628\mathrm{}\left(47\right)$ in quenched QCD, which is consistent with previous calculations. At $6/g^2=6.0\mathrm{}$ and 5.85 we find the ratio $f_K/m_{\rho }$, in agreement with the experimental value, within errors. These results support expectations that $O\left(a\right)$ errors are exponentially suppressed in low energy $(E\ll a^{-1\mathrm{}})$ observables, and indicate that domain wall fermions have good scaling behavior at relatively strong couplings. We also demonstrate that the axial current numerically satisfies the lattice analog of the usual continuum axial Ward identity.