Kaon to two pions decays from lattice QCD

We report a direct lattice calculation of the K to ππ decay matrix elements for both the ΔI = 1/2 and 3/2 amplitudes A0 and A2 on a 2+1 flavor, domain wall fermion, 163 × 32 × 16 lattice ensemble and a 243 × 64 × 16 lattice ensemble. This is a complete calculation in which all contractions for the required ten, four-quark operators are evaluated, including the disconnected graphs in which no quark line connects the initial kaon and final two-pion states. These lattice operators are nonperturbatively renormalized using the Rome-Southampton method and the quadratic divergences are studied and removed. This is an important but notoriously difficult calculation, requiring high statistics on a large volume. In this work we take a major step towards the computation of the physical K → ππ amplitudes by performing a complete calculation at unphysical kinematics with pions of mass 422MeV and 329MeV at rest in the kaon rest frame. With this simplification we are able to resolve Re(A0) from zero for the first time, with a 25% statistical error on the 163 lattice and 15% on the 243 lattice. The complex amplitude A2 is calculated with small statistical errors. We obtain the ΔI = 1/2 rule with a