$K\to \ddot{I}€\ddot{I}€$ $\hat{I}”I=3/2$ decay amplitude in the continuum limit

Abstract

We present new results for the amplitude $A_2$ for a kaon to decay into two pions with isospin $I=2$: Re$A_2=1.50(4{)}_{\mathrm{stat}}(14{)}_{\mathrm{syst}}\times 10^{-8}$ GeV; Im$A_2=-6.99(20{)}_{\mathrm{stat}}(84{)}_{\mathrm{syst}}\times 10^{-13}$ GeV. These results were obtained from two ensembles generated at physical quark masses (in the isospin limit) with inverse lattice spacings $a^{-1}=1.728(4)$ GeV and $2.358(7)$ GeV. We are therefore able to perform a continuum extrapolation and hence largely to remove the dominant systematic uncertainty from our earlier results, that due to lattice artefacts. The only previous lattice computation of $K\to \ddot{I}€\ddot{I}€$ decays at physical kinematics was performed using an ensemble at a single, rather coarse, value of the lattice spacing ($a^{-1}\simeq 1.37(1)$ GeV). We confirm the observation that there is a significant cancellation between the two dominant contributions to Re$A_2$ which we suggest is an important ingredient in understanding the $\hat{I}”I=1/2$ rule, Re$A_0$/Re$A_2\simeq 22.5$, where the subscript denotes the total isospin of the two-pion final state. Our result for $A_2$ implies that the electroweak penguin contribution to $\hat{I}{\mu }^{\prime }/\hat{I}\mu$ is Re($\hat{I}{\mu }^{\prime }/\hat{I}\mu {)}_{\text{EWP}}=-(6.6\pm 1.0)\times 10^{-4}$.

46 pages, 12 figures