Direct CP violation and the Δi=1 /2 rule in K →ππ decay from the standard model

We present a lattice QCD calculation of the ΔI=1/2, K→ππ decay amplitude A0 and µ′, the measure of direct CP violation in K→ππ decay, improving our 2015 calculation [1] of these quantities. Both calculations were performed with physical kinematics on a 323×64 lattice with an inverse lattice spacing of a-1=1.3784(68) GeV. However, the current calculation includes nearly 4 times the statistics and numerous technical improvements allowing us to more reliably isolate the ππ ground state and more accurately relate the lattice operators to those defined in the standard model. We find Re(A0)=2.99(0.32)(0.59)×10-7 GeV and Im(A0)=-6.98(0.62)(1.44)×10-11 GeV, where the errors are statistical and systematic, respectively. The former agrees well with the experimental result Re(A0)=3.3201(18)×10-7 GeV. These results for A0 can be combined with our earlier lattice calculation of A2 [2] to obtain Re( µ′/ µ)=21.7(2.6)(6.2)(5.0)×10-4, where the third error represents omitted isospin breaking effects, and Re(A0)/Re(A2)=19.9(2.3)(4.4). The first agrees well with the experimental result of Re( µ′/ µ)=16.6(2.3)×10-4. A comparison of the second with the observed ratio Re(A0)/Re(A2)=22.45(6), demonstrates the standard model origin of this "ΔI=1/2 rule"enhancement.