Lattice determination of theK→(ππ)I=2decay amplitudeA2

We describe the computation of the amplitude ${A}_{2}$ for a kaon to decay into two pions with isospin $I=2$. The results presented in [T. Blum et al., Phys. Rev. Lett. 108, 141601 (2012)] from an analysis of 63 gluon configurations are updated to 146 configurations giving $\mathrm{Re}{A}_{2}=1.381(46{)}_{\mathrm{stat}}(258{)}_{\mathrm{syst}}{10}^{\ensuremath{-}8}\text{ }\text{ }\mathrm{GeV}$ and $\mathrm{Im}{A}_{2}=\ensuremath{-}6.54(46{)}_{\mathrm{stat}}(120{)}_{\mathrm{syst}}{10}^{\ensuremath{-}13}\text{ }\text{ }\mathrm{GeV}$. $\mathrm{Re}{A}_{2}$ is in good agreement with the experimental result, whereas the value of $\mathrm{Im}{A}_{2}$ was hitherto unknown. We are also working toward a direct computation of the $K\ensuremath{\rightarrow}(\ensuremath{\pi}\ensuremath{\pi}{)}_{I=0}$ amplitude ${A}_{0}$ but, within the Standard Model, our result for $\mathrm{Im}{A}_{2}$ can be combined with the experimental results for $\mathrm{Re}{A}_{0}$, $\mathrm{Re}{A}_{2}$ and ${\ensuremath{\epsilon}}^{\ensuremath{'}}/\ensuremath{\epsilon}$ to give $\mathrm{Im}{A}_{0}/\mathrm{Re}{A}_{0}=\ensuremath{-}1.61(28)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$. Our result for $\mathrm{Im}{A}_{2}$ implies that the electroweak penguin (EWP) contribution to ${\ensuremath{\epsilon}}^{\ensuremath{'}}/\ensuremath{\epsilon}$ is $\mathrm{Re}({\ensuremath{\epsilon}}^{\ensuremath{'}}/\ensuremath{\epsilon}{)}_{\mathrm{EWP}}=\ensuremath{-}(6.25\ifmmode\pm\else\textpm\fi{}{0.44}_{\mathrm{stat}}\ifmmode\pm\else\textpm\fi{}{1.19}_{\mathrm{syst}})\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$.