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 → π π decay amplitude A 0 and ϵ ′, the measure of direct C P violation in K → π π 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 π π 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(ϵ′/ϵ) = 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(ϵ′/ϵ) = 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.