$\Delta I = 3/2$ and $\Delta I = 1/2$ channels of $K\to\pi\pi$ decay at the physical point with periodic boundary conditions

TL;DR

This paper reports a lattice QCD calculation of $K o\pi\pi$ decay amplitudes and direct CP violation measure $\varepsilon' ext{, using periodic boundary conditions and the variational method to resolve two-pion states at the physical point.

Contribution

It introduces a new approach using the variational method with periodic boundary conditions to extract $K o\pi\pi$ decay amplitudes at the physical point.

Findings

Successful resolution of two-pion spectrum up to on-shell energy

Promising preliminary results on decay amplitudes and $\varepsilon'$

Motivation for future calculations on finer lattices

Abstract

We present a lattice calculation of the $K\to\pi\pi$ matrix elements and amplitudes with both the $\Delta I = 3/2$ and 1/2 channels and $\varepsilon'$, the measure of direct $CP$ violation. We use periodic boundary conditions (PBC), where the correct kinematics of $K\to\pi\pi$ can be achieved via an excited two-pion final state. To overcome the difficulty associated with the extraction of excited states, our previous work \cite{Bai:2015nea,RBC:2020kdj} successfully employed G-parity boundary conditions, where pions are forced to have non-zero momentum enabling the $I=0$ two-pion ground state to express the on-shell kinematics of the $K\to\pi\pi$ decay. Here instead we overcome the problem using the variational method which allows us to resolve the two-pion spectrum and matrix elements up to the relevant energy where the decay amplitude is on-shell. In this paper we report an exploratory calculation of $K\to\pi\pi$ decay amplitudes and $\varepsilon'$ using PBC on a coarser lattice size of $24^3\times64$ with inverse lattice spacing $a^{-1}=1.023$ GeV and the physical pion and kaon masses. The results are promising enough to motivate us to continue our measurements on finer lattice ensembles in order to improve the precision in the near future.