$K\to\pi\pi$ Decays with Domain Wall Fermions: Lattice Matrix Elements

We present a lattice calculation of the $K\to\pi$ and $K\to 0$ matrix elements of the $\Delta S=1$ effective weak Hamiltonian which can be used to determine $\epsilon^\prime/\epsilon$ and the $\Delta I=1/2$ rule for $K$ decays in the Standard Model. The matrix elements for $K\to\pi\pi$ decays are related to $K\to\pi$ and $K\to 0$ using lowest order chiral perturbation theory. We also present results for the kaon $B$ parameter, $B_K$. Our quenched domain wall fermion simulation was done at $\beta=6.0$ ($a^{-1}\approx 2$ GeV), lattice size $16^3\times 32\times 16$, and domain wall height $M_5=1.8$.