Calculation of $K \to \pi\pi$ decay amplitudes with an improved Wilson fermion action in a nonzero momentum frame in lattice QCD

TL;DR

This paper computes $K o\pi\pi$ decay amplitudes using improved Wilson fermions in a nonzero momentum frame on the lattice, providing results relevant for understanding CP violation and decay processes.

Contribution

It introduces a method to calculate physical $K o\pi\pi$ decay amplitudes with nonzero pion momenta on the lattice using improved Wilson fermions, advancing previous techniques.

Findings

Calculated ${ m Re}A_2$, ${ m Re}A_0$, and $rac{ m ext{ extit{epsilon}}'}{ ext{ extit{epsilon}}}$ with statistical errors.

Results show weak dependence on the matching scale $q^*$.

Systematic errors from renormalization factors are estimated at 1.3% for ${ m Re}A_2$ and 11% for ${ m Re}A_0$.

Abstract

We present our result for the $K\to\pi\pi$ decay amplitudes for both the $\Delta I=1/2$ and $3/2$ processes with the improved Wilson fermion action. In order to realize the physical kinematics, where the pions in the final state have finite momenta, we consider the decay process $K({\bf p}) \to \pi({\bf p}) + \pi({\bf 0})$ in the nonzero momentum frame with momentum ${\bf p}=(0,0,2\pi/L)$ on the lattice. Our calculations are carried out with $N_f=2+1$ gauge configurations generated with the Iwasaki gauge action and nonperturbatively $O(a)$-improved Wilson fermion action at $a=0.091\,{\rm fm}$ ($1/a=2.176\,{\rm GeV}$), $m_\pi=260\,{\rm MeV}$, and $m_K=570\,{\rm MeV}$ on a $48^3\times 64$ ($La=4.4\,{\rm fm}$) lattice. For these parameters the energy of the $K$ meson is set at that of two-pion in the final state. We obtain ${\rm Re}A_2 = 2.431(19) \times10^{ -8}\,{\rm GeV}$, ${\rm Re}A_0 = 51(28) \times10^{ -8}\,{\rm GeV}$, and $\epsilon'/\epsilon = 1.9(5.7) \times10^{-3}$ for a matching scale $q^* =1/a$ where the errors are statistical. The dependence on the matching scale $q^*$ of these values is weak. The systematic error arising from the renormalization factors is expected to be around $1.3\%$ for ${\rm Re}A_2$ and $11 \%$ for ${\rm Re}A_0$. Prospects toward calculations with the physical quark mass are discussed.