We report on the first realistic \emph{ ab initio} calculation of a hadronic weak decay, that of the amplitude $A_2$ for a kaon to decay into two \pi-mesons with isospin 2. We find Re$A_2=(1.436\pm 0.063_{ \textrm{ stat}}\pm 0.258_{ \textrm{ syst}})\,10^{ -8}\,\textrm{ GeV}$ in good agreement with the experimental result and for the hitherto unknown imaginary part we find { Im}$\,A_2=-(6.83 \pm 0.51_{ \textrm{ stat}} \pm 1.30_{ \textrm{ syst}})\,10^{ -13}\,{ \rm GeV}$. Moreover combining our result for Im\,$A_2$ with experimental values of Re\,$A_2$, Re\,$A_0$ and $\epsilon^\prime/\epsilon$, we obtain the following value for the unknown ratio Im\,$A_0$/Re\,$A_0$ within the Standard Model: $\mathrm{ Im}\,A_0/\mathrm{ Re}\,A_0=-1.63(19)_{ \mathrm{ stat}}(20)_{ \mathrm{ syst}}\times10^{ -4}$. One consequence of these results is that the contribution from Im\,$A_2$ to the direct CP violation parameter $\epsilon^{ \prime}$ (the so-called Electroweak Penguin, EWP, contribution) is Re$(\epsilon^\prime/\epsilon)_{ \mathrm{ EWP}} = -(6.52 \pm 0.49_{ \textrm{ stat}} \pm 1.24_{ \textrm{ syst}}) \times 10^{ -4}$. We explain why this calculation of $A_2$ represents a major milestone for lattice QCD and discuss the exciting prospects for a full quantitative understanding of CP-violation in kaon decays.
Categories: hep-lat hep-ph