Direct CP violation and the $\mathrm{\Delta }I=1\mathrm{/}2$ rule in $K\to \pi \pi$ decay from the Standard Model

We present a lattice QCD calculation of the $\mathrm{\Delta }I=1\mathrm{/}2$, $K\to \pi \pi$ decay amplitude $A_0$ and ${\epsilon }^{\mathrm{\prime }}$, the measure of direct CP-violation in $K\to \pi \pi$ decay, improving our 2015 calculation of these quantities. Both calculations were performed with physical kinematics on a ${32}^3\times 64$ lattice with an inverse lattice spacing of $a^{-1}=1.3784(68)$ GeV. However, the current calculation includes nearly four times the statistics and numerous technical improvements allowing us to more reliably isolate the $\pi \pi$ ground-state and more accurately relate the lattice operators to those defined in the Standard Model. We find $\mathrm{R}\mathrm{e}(A_0)=2.99(0.32)(0.59)\times {10}^{-7}$ GeV and $\mathrm{I}\mathrm{m}(A_0)=-6.98(0.62)(1.44)\times {10}^{-11}$ GeV, where the errors are statistical and systematic, respectively. The former agrees well with the experimental result $\mathrm{R}\mathrm{e}(A_0)=3.3201(18)\times {10}^{-7}$ GeV. These results for $A_0$ can be combined with our earlier lattice calculation of $A_2$ to obtain $\mathrm{R}\mathrm{e}({\epsilon }^{\mathrm{\prime }}\mathrm{/}\epsilon )=21.7(2.6)(6.2)(5.0)\times {10}^{-4}$, where the third error represents omitted isospin breaking effects, and Re$(A_0)$/Re$(A_2)=19.9(2.3)(4.4)$. The first agrees well with the experimental result of ${\rm Re}(\varepsilon’/\varepsilon)=16.6(2.3)\times 10^{-4}$\mathrm{.}AcomparisonofthesecondwiththeobservedratioRe$(A_0)/$Re$(A_2) = 22.45(6)$,demonstratestheStandardModeloriginofthis\mathrm{"}$\Delta I = 1/2$ rule" enhancement.