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

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 $K(\mathbf{p})\to \pi (\mathbf{p})+\pi (0)$ in the nonzero momentum frame with momentum $\mathbf{p}=(0,0,2\pi /L)$ 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 $a=0.091\mathrm{fm}$ ($1/a=2.176\mathrm{GeV}$), $m_{\pi }=260\mathrm{MeV}$, and $m_K=570\mathrm{MeV}$ on a $48^3\times 64$ ($La=4.4\mathrm{fm}$) lattice. For these parameters the energy of the $K$ meson is set at that of two-pion in the final state. We obtain $\mathrm{Re}{A}_2=2.431(19)\times 10^{-8}\mathrm{GeV}$, $\mathrm{Re}{A}_0=51(28)\times 10^{-8}\mathrm{GeV}$, and ${ϵ}^{\prime }/ϵ=1.9(5.7)\times 10^{-3}$ for a matching scale $q^{\ast }=1/a$ where the errors are statistical. The dependence on the matching scale $q^{\ast }$ of these values is weak. The systematic error arising from the renormalization factors is expected to be around $1.3\%$ for $\mathrm{Re}{A}_2$ and $11\%$ for $\mathrm{Re}{A}_0$. Prospects toward calculations with the physical quark mass are discussed.

Comments: LaTeX2e, 19 pages, 5 eps figures, uses revtex4 and graphicx. arXiv admin note: substantial text overlap with arXiv:1505.05289. Published in PRD