The $K\to (\ddot{I}€\ddot{I}€{)}_{I=2}$ Decay Amplitude from Lattice QCD

Abstract

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 π-mesons with isospin 2. We find Re$A_2=(1.436\pm 0.063_{\text{stat}}\pm 0.258_{\text{syst}})10^{-8}\text{GeV}$ in good agreement with the experimental result and for the hitherto unknown imaginary part we find {Im}$A_2=-(6.83\pm 0.51_{\text{stat}}\pm 1.30_{\text{syst}})10^{-13}\mathrm{GeV}$. Moreover combining our result for Im\,$A_2$ with experimental values of Re\,$A_2$, Re\,$A_0$ and $\hat{I}{\mu }^{\prime }/\hat{I}\mu$, we obtain the following value for the unknown ratio Im\,$A_0$/Re\,$A_0$ within the Standard Model: $\mathrm{Im}{A}_0/\mathrm{Re}{A}_0=-1.63(19{)}_{\mathrm{stat}}(20{)}_{\mathrm{syst}}\times 10^{-4}$. One consequence of these results is that the contribution from Im\,$A_2$ to the direct CP violation parameter $\hat{I}{\mu }^{\prime }$ (the so-called Electroweak Penguin, EWP, contribution) is Re$(\hat{I}{\mu }^{\prime }/\hat{I}\mu {)}_{\mathrm{EWP}}=-(6.52\pm 0.49_{\text{stat}}\pm 1.24_{\text{syst}})\times 10^{-4}$. 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.

5 pages, 1 figure