Ktoππdecay amplitudes from lattice QCD

Abstract

We report a direct lattice calculation of the $K$ to $\ensuremath{\pi}\ensuremath{\pi}$ decay matrix elements for both the $\ensuremath{\Delta}I=1/2$ and $3/2$ amplitudes ${A}_{0}$ and ${A}_{2}$ on $2+1$ flavor, domain wall fermion, ${16}^{3}\ifmmode\times\else\texttimes\fi{}32\ifmmode\times\else\texttimes\fi{}16$ 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 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 paper, we take a major step toward the computation of the physical $K\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ 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 $\mathrm{Re}({A}_{0})$ 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 $\ensuremath{\Delta}=1/2$ rule and testing the standard model of $CP$ violation in the kaon system.