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We present results for the $K\to\pi\pi$ decay amplitudes for both the $\DeltaI=1/2$ and $3/2$ channels. This calculation is carried out on 480 gaugeconfigurations in $N_f=2+1$ QCD generated over 12,000 trajectories with theIwasaki gauge action and non-perturbatively $O(a)$-improved Wilson fermionaction at $a=0.091\,{\rm fm}$, $m_\pi=280\,{\rm MeV}$ and $m_K=580\,{\rm MeV}$on a $32^3\times 64$ ($La=2.9\,{\rm fm}$) lattice. For the quark loops in thePenguin and disconnected contributions in the $I=0$ channel, the combinedhopping parameter expansion and truncated solver techniques work very well forvariance reduction. We obtain, for the first time with a Wilson-type fermionaction, 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 $q^* =1/a$. Thedependence on the matching scale is weak.

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