Preliminary results of $\mathrm{\Delta }I=1\mathrm{/}2$ and $3\mathrm{/}2$, $K$ to $\pi \pi$ Decay Amplitudes from Lattice QCD

We report a direct lattice calculation of the $K$ to $\pi \pi$ decay matrix elements for both $\mathrm{\Delta }I=1\mathrm{/}2$ and $3\mathrm{/}2$ channels on 2+1 flavor, domain wall fermion, ${16}^3\times 32$ lattices with zero $\pi \pi$ relative momentum and $m_{\pi }=420$ MeV. All $K^0$ to $\pi \pi$ contractions are carefully listed and calculated. The decay into the isospin zero $\pi \pi$ final state, which receives contributions from the disconnected graphs, is very difficult to calculate, but a clear signal in the similar disconnected $\pi \pi$ 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 $\mathrm{\Delta }I=1\mathrm{/}2$ rule in the $420$ MeV pion system.