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

We present our result for the $K\to \pi \pi$ decay amplitudes for both the $\mathrm{\Delta }I=1\mathrm{/}2$ and $3\mathrm{/}2$ 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})$inthenonzeromomentumframewithmomentum${\bf p}=(0,0,2\pi/L)$onthelattice\mathrm{.}Ourcalculationsarecarriedoutwith$N_f=2+1$ gauge configurations generated with the Iwasaki gauge action and nonperturbatively $O(a)$-improved Wilson fermion action at $a=0.091\mathrm{f}\mathrm{m}$ ($1\mathrm{/}a=2.176\mathrm{G}\mathrm{e}\mathrm{V}$), $m_{\pi }=260\mathrm{M}\mathrm{e}\mathrm{V}$, and $m_K=570\mathrm{M}\mathrm{e}\mathrm{V}$ on a ${48}^3\times 64$ ($La=4.4\mathrm{f}\mathrm{m}$) 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 $q^{\ast }=1\mathrm{/}a$ 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 $1.3\mathrm{\%}$ for ${\rm Re}A_2$and$11 %$for${\rm Re}A_0$. Prospects toward calculations with the physical quark mass are discussed.