Direct CP violation and the $\hat{I}”I=1/2$ rule in $K\to \ddot{I}€\ddot{I}€$ decay from the Standard Model

Abstract

We present a lattice QCD calculation of the $\hat{I}”I=1/2$, $K\to \ddot{I}€\ddot{I}€$ decay amplitude $A_0$ and ${\epsilon }^{\prime }$, the measure of direct CP-violation in $K\to \ddot{I}€\ddot{I}€$ 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 $\ddot{I}€\ddot{I}€$ ground-state and more accurately relate the lattice operators to those defined in the Standard Model. We find $\mathrm{Re}(A_0)=2.99(0.32)(0.59)\times 10^{-7}$ GeV and $\mathrm{Im}(A_0)=-6.98(0.62)(1.44)\times 10^{-11}$ GeV, where the errors are statistical and systematic, respectively. The former agrees well with the experimental result $\mathrm{Re}(A_0)=3.3201(18)\times 10^{-7}$ GeV. These results for $A_0$ can be combined with our earlier lattice calculation of $A_2$ to obtain $\mathrm{Re}({\epsilon }^{\prime }/\epsilon )=21.7(2.6)(6.2)(5.0)\times 10^{-4}$, 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 $\mathrm{Re}({\epsilon }^{\prime }/\epsilon )=16.6(2.3)\times 10^{-4}$. A comparison of the second with the observed ratio Re$(A_0)/$Re$(A_2)=22.45(6)$, demonstrates the Standard Model origin of this "$\hat{I}”I=1/2$ rule" enhancement.

Updated to published version. 95 pages, 12 figures