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We report a direct lattice calculation of the $K$ to $\pi\pi$ decay matrixelements for both the $\Delta I=1/2$ and 3/2 amplitudes $A_0$ and $A_2$ on 2+1flavor, domain wall fermion, $16^3\times32\times16$ lattices. This is acomplete calculation in which all contractions for the required ten, four-quarkoperators are evaluated, including the disconnected graphs in which no quarkline connects the initial kaon and final two-pion states. These latticeoperators are non-perturbatively renormalized using the Rome-Southampton methodand the quadratic divergences are studied and removed. This is an important butnotoriously difficult calculation, requiring high statistics on a large volume.In this paper we take a major step towards the computation of the physical$K\to\pi\pi$ amplitudes by performing a complete calculation at unphysicalkinematics with pions of mass 422\,MeV at rest in the kaon rest frame. Withthis simplification we are able to resolve Re$(A_0)$ from zero for the firsttime, with a 25% statistical error and can develop and evaluate methods forcomputing the complete, complex amplitude $A_0$, a calculation central tounderstanding the $\Delta =1/2$ rule and testing the standard model of CPviolation in the kaon system.

$K$ to $\pi\pi$ Decay amplitudes from Lattice QCD