Abstract
We demonstrate that in the large N approach developed by the authors in collaboration with Bardeen, the parameters B6(1/2) and B8(3/2) parametrizing the K → ππ matrix elements 〈Q6〉0 and 〈Q8〉2 of the dominant QCD and electroweak operators receive both negative O 1 / N $$ \mathcal{O}\left(1/N\right) $$ corrections such that B6(1/2) ≤ B8(3/2) < 1 in agreement with the recent lattice results of the RBC-UKQCD collaboration. We also point out that the pattern of the size of the hadronic matrix elements of all QCD and electroweak penguin operators Qi contributing to the K → ππ amplitudes A0 and A2, obtained by this lattice collaboration, provides further support to our large N approach. In particular, the lattice result for the matrix element 〈Q8〉0 implies for the corresponding parameter B8(1/2) = 1.0 ± 0.2 to be compared with large N value B8(1/2) = 1.1 ± 0.1. We discuss briefly the implications of these findings for the ratio ε′/ε. In fact, with the precise value for B8(3/2) from RBC-UKQCD collaboration, our upper bound on B6(1/2) implies ε′/ε in the SM roughly by a factor of two below its experimental value (16.6± 2.3) × 10−4. We also briefly comment on the parameter B ^ K $$ {\widehat{B}}_K $$ and the ΔI = 1/2 rule.