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

Understand

We report a direct lattice calculation of the $K$ to $\pi\pi$ decay matrix elements for both $\Delta I=1/2$ and $3/2$ channels on 2+1 flavor, domain wall fermion, $16^3\times32$ 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 $A_2$.