New lattice approaches to the ΔI = 12 rule

Abstract Lattice QCD should allow a derivation of the ΔI = 1 2 rule from first principles, but numerical calculations to date have been plagued by a variety of problems. After a brief review of these problems, we present several new methods for calculating K → ππ amplitudes. These are designed for Wilson fermions, though they can be used also with staggered fermions. They all involve a nonperturbative determination of matching coefficients. We show how problems of operator mixing can be greatly reduced by using point-split hadronic currents, and how CP-violating parts of the K → ππ amplitudes can be calculated by introducing a fake top quark. Many of the methods can also be applied to the calculation of two body non-leptonic B-meson decays.