Kaon Matrix Elements and CP-violation from Quenched Lattice QCD: (I) the 3-flavor case

We report the results of a calculation of the $K \to \pi \pi$ matrix elements relevant for the $\DIhalf$ rule and $\epe$ in quenched lattice QCD using domain wall fermions. Working in the three-quark effective theory, where only the $u$, $d$ and $s$ quarks enter and which is known perturbatively to next-to-leading order, we calculate the lattice $K \to \pi$ and $K \to |0>$ matrix elements of dimension six, four-fermion operators. Through lowest order chiral perturbation theory these yield $K \to \pi \pi$ matrix elements, which we then normalize to continuum values through a non-perturbative renormalization technique. For the $\DIhalf$ rule we find a value of $25.3 \pm 1.8$ (statistical error only) compared to the experimental value of 22.2, with individual isospin amplitudes 10-20% below the experimental values. For $\epe$, using known central values for standard model parameters, we calculate $(-4.0 \pm 2.3) \times 10^{-4}$ (statistical error only) compared to the current experimental average of $(17.2 \pm 1.8) \times 10^{-4}$. Because we find a large cancellation between the $I = 0$ and $I = 2$ contributions to $\epe$, the result may be very sensitive to the approximations employed. Among these are the use of: quenched QCD, lowest order chiral perturbation theory and continuum perturbation theory below 1.3 GeV. We have also calculated the kaon $B$ parameter, $B_K$ and find $B_K(2 {\rm GeV}) = 0.513(11)$. Although currently unable to give a reliable systematic error, we have control over statistical errors and more simulations will yield information about the effects of the approximations on this first-principles determination of these important quantities.