| We report the results of a calculation of the K → ππ matrix elements relevant for the ΔI = 1/2 rule and ϵ ′/ϵ in quenched lattice QCD using domain wall fermions at a fixed lattice spacing of a−1 ∼ 2 GeV. 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 → π and K → |0⟩ matrix elements of dimension six, four-fermion operators. Through lowest order chiral perturbation theory these yield K → ππ matrix elements, which we then normalize to continuum values through a non-perturbative renormalization technique. For the Δ I = 1/2 rule we find a value of 25.3 ± 1.8 (statistical error only) compared to the experimental value of 22.2, with individual isospin amplitudes 10–20% below the experimental values. For ϵ ′/ϵ; using known central values for standard model parameters, we calculate (−4.0 ± 2.3) × 10−4 (statistical error only) compared to the current experimental average of (17.2 ± 1.8) × 10−4. Because we find a large cancellation between the I = 0 and I = 2 contributions to ϵ′/ϵ, 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, BK and find BK(2 GeV) = 0.532(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. |
