We report the results of a calculation of the $\overrightarrow{K}\pi \pi$ matrix elements relevant for the $\Delta I=1/2\mathrm{}$ rule and ${\epsilon }^{\prime }/\epsilon$ in quenched lattice QCD using domain wall fermions at a fixed lattice spacing $a^{-1}\sim 2\mathrm{GeV}.$ 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 $\overrightarrow{K}\pi$ and $\overrightarrow{K}|0〉$ matrix elements of dimension six, four-fermion operators. Through lowest order chiral perturbation theory these yield $\overrightarrow{K}\pi \pi$ matrix elements, which we then normalize to continuum values through a nonperturbative renormalization technique. For the ratio of isospin amplitudes $|A_0|/|A_2|$ 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 ${\epsilon }^{\prime }/\epsilon ,$ 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 $(17.2\mathrm{}\pm 1.8)\times {10}^{-4}.$ Because we find a large cancellation between the $I=0\mathrm{}$ and $I=2\mathrm{}$ contributions to ${\epsilon }^{\prime }/\epsilon ,$ 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 also calculate the kaon B parameter $B_K$ and find $B_{K,\overline{\mathrm{MS}}}\left(2\mathrm{}\mathrm{GeV}\right)=0.532\mathrm{}\left(11\right).$ 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.

• Received 19 July 2002

DOI:https://doi.org/10.1103/PhysRevD.68.114506