# $K$ to $ππ$ Decay amplitudes from Lattice QCD

T. Blum, P. A. Boyle, N. H. Christ, N. Garron, E. Goode, T. Izubuchi, C. Lehner, Q. Liu, R. D. Mawhinney, C. T. Sachrajda, A. Soni, C. Sturm, H. Yin, R. Zhou

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## Abstract

We report a direct lattice calculation of the $K$ to $\pi\pi$ decay matrix elements for both the $\Delta I=1/2$ and 3/2 amplitudes $A_0$ and $A_2$ on 2+1 flavor, domain wall fermion, $16^3\times32\times16$ lattices. 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 non-perturbatively 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 paper we take a major step towards the computation of the physical $K\to\pi\pi$ amplitudes by performing a complete calculation at unphysical kinematics with pions of mass 422\,MeV at rest in the kaon rest frame. With this simplification we are able to resolve Re$(A_0)$ from zero for the first time, with a 25% statistical error and can develop and evaluate methods for computing the complete, complex amplitude $A_0$, a calculation central to understanding the $\Delta =1/2$ rule and testing the standard model of CP violation in the kaon system.
Original language English 114503 40 Physical Review D 84 https://doi.org/10.1103/PhysRevD.84.114503 Published - 14 Jun 2011

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