We present new results for the amplitude $A_2$ for a kaon to decay into two pions with isospin $I=2$: $\mathrm{Re}{A}_2=1.50(4{)}_{\text{stat}}(14{)}_{\text{syst}}\times {10}^{-8}\mathrm{GeV}$; $\mathrm{Im}{A}_2=-6.99(20{)}_{\text{stat}}(84{)}_{\text{syst}}\times {10}^{-13}\mathrm{GeV}$. These results were obtained from two ensembles generated at physical quark masses (in the isospin limit) with inverse lattice spacings $a^{-1}=1.728(4)\mathrm{GeV}$ 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 [1,2], that due to lattice artifacts. The only previous lattice computation of $K\to \pi \pi$ decays at physical kinematics was performed using an ensemble at a single, rather coarse, value of the lattice spacing [$a^{-1}\simeq 1.37(1)\mathrm{GeV}$]. We confirm the observation reported in [3] that there is a significant cancellation between the two dominant contributions to $\mathrm{Re}{A}_2$ which we suggest is an important ingredient in understanding the $\mathrm{\Delta }I=1/2$ rule, $\mathrm{Re}{A}_0/\mathrm{Re}{A}_2\simeq 22.5$, 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 ${\epsilon }^{\prime }/\epsilon$ is $\mathrm{Re}({\epsilon }^{\prime }/\epsilon {)}_{\mathrm{EWP}}=-(6.6\pm 1.0)\times {10}^{-4}$.

• Received 3 February 2015

DOI:https://doi.org/10.1103/PhysRevD.91.074502