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

TL;DR

This paper presents preliminary lattice QCD calculations of $K$ to $22$ decay amplitudes for both isospin channels, revealing a significant enhancement in the $22$ decay amplitude consistent with the $22$ rule.

Contribution

First direct lattice calculation of $K$ to $22$ decay amplitudes including disconnected diagrams at physical pion mass.

Findings

Re($A_0$) with 25% error for zero momentum decay

Observed 6-fold enhancement in $22$ decay amplitude

Identified challenges in calculating disconnected diagrams

Abstract

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 $A_2$. 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 $\Delta I=1/2$ rule in the $420$ MeV pion system.