Lattice determination of the $K\to (\ddot{I}€\ddot{I}€{)}_{I=2}$ Decay Amplitude $A_2$

Abstract

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 the letter Phys.Rev.Lett. 108 (2012) 141601 from an analysis of 63 gluon configurations are updated to 146 configurations giving Re$A_2=1.381(46{)}_{\text{stat}}(258{)}_{\text{syst}}10^{-8}$ GeV and Im$A_2=-6.54(46{)}_{\text{stat}}(120{)}_{\text{syst}}10^{-13}$ GeV. Re$A_2$ is in good agreement with the experimental result, whereas the value of Im$A_2$ was hitherto unknown. We are also working towards a direct computation of the $K\to (\ddot{I}€\ddot{I}€{)}_{I=0}$ amplitude $A_0$ but, within the standard model, our result for Im$A_2$ can be combined with the experimental results for Re$A_0$, Re$A_2$ and $\hat{I}{\mu }^{\prime }/\hat{I}\mu$ to give Im$A_0/$Re$A_0=-1.61(28)\times 10^{-4}$ . Our result for Im\,$A_2$ implies that the electroweak penguin (EWP) contribution to $\hat{I}{\mu }^{\prime }/\hat{I}\mu$ is Re$(\hat{I}{\mu }^{\prime }/\hat{I}\mu {)}_{\mathrm{EWP}}=-(6.25\pm 0.44_{\text{stat}}\pm 1.19_{\text{syst}})\times 10^{-4}$.

59 pages, 11 figures