Calculation of $K to pipi$ decay amplitudes with improved Wilson fermion in 2+1 flavor lattice QCD

We present results for the $Ktopipi$ decay amplitudes for both the $Delta I=1/2$ and $3/2$ channels. This calculation is carried out on 480 gauge configurations in $N_f=2+1$ QCD generated over 12,000 trajectories with the Iwasaki gauge action and non-perturbatively $O(a)$-improved Wilson fermion action at $a=0.091,{rm fm}$, $m_pi=280,{rm MeV}$ and $m_K=580,{rm MeV}$ on a $32^3times 64$ ($La=2.9,{rm fm}$) lattice. For the quark loops in the Penguin and disconnected contributions in the $I=0$ channel, the combined hopping parameter expansion and truncated solver techniques work very well for variance reduction. We obtain, for the first time with a Wilson-type fermion action, that ${rm Re}A_0 = 60(36) times10^{ -8},{rm GeV}$ and ${rm Im}A_0 =-67(56) times10^{-12},{rm GeV}$ for a matching scale $q^* =1/a$. The dependence on the matching scale is weak.