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Preliminary results of $ΔI=1/2$ and $3/2$, $K$ to $ππ$ Decay Amplitudes from Lattice QCD

We report a direct lattice calculation of the $K$ to $ππ$ decay matrix elements for both $ΔI=1/2$ and $3/2$ channels on 2+1 flavor, domain wall fermion, $16^3\times32$ lattices with zero $ππ$ relative momentum and $m_π=420$ MeV. All $K^0$ to $ππ$ contractions are carefully listed and calculated. The decay into the isospin zero $ππ$ final state, which receives contributions from the disconnected graphs, is very difficult to calculate, but a clear signal in the similar disconnected $ππ$ correlator can be seen. Preliminary results, some with large errors, will be presented for the various contributions to the renormalized weak matrix elements $A_0$ and $A_2$. We obtain Re$(A_0)$ with $25%$ error in the case of zero momentum on shell decay, and find a factor of 6 enhancement for the $ΔI=1/2$ rule in the $420$ MeV pion system.