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

We present our result for the $Ktopipi$ 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 ${bf p}=(0,0,2pi/L)$ 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 $48^3times 64$ ($La=4.4,{rm fm}$) 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.