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

We present results for the $K\to \pi \pi$ 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 $a=0.091\mathrm{f}\mathrm{m}$, $m_{\pi }=280\mathrm{M}\mathrm{e}\mathrm{V}$ and $m_K=580\mathrm{M}\mathrm{e}\mathrm{V}$ on a ${32}^3\times 64$ ($La=2.9\mathrm{f}\mathrm{m}$) 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 $\mathrm{R}\mathrm{e}{A}_0=60(36)\times {10}^{-8}\mathrm{G}\mathrm{e}\mathrm{V}$ and ${\rm Im}A_0 =-67(56) \times10^{-12},{\rm GeV}$foramatchingscale$q^* =1/a$. The dependence on the matching scale is weak.