  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 $K\to\pi\pi$ decays at physical kinematics was performed using an ensemble
at a single, rather coarse, value of the lattice spacing ($a^{-1}\simeq
1.37(1)$ 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 $\Delta I=1/2$ 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 $\epsilon^\prime/\epsilon$ is
Re($\epsilon^\prime/\epsilon)_\textrm{EWP}=-(6.6\pm 1.0)\times 10^{-4}$.

High Energy Physics - Lattice;High Energy Physics - Phenomenology