Large N approach to Kaon decays and mixing 28 years later: ΔI = 1/2 rule, B̂_K, and ΔM_K

We review and update our results for K → ππ decays and K⁰-K̄⁰ mixing obtained by us in the 1980s within an analytic approximate approach based on the dual representation of QCD as a theory of weakly interacting mesons for large N, where N is the number of colors. In our analytic approach the Standard Model dynamics behind the enhancement of ReA₀ and suppression of ReA₂, the so-called ΔI = 1/2 rule for K → ππ decays, has a simple structure: the usual octet enhancement through the long but slow quark-gluon renormalization group evolution down to the scales O(1 GeV) is continued as a short but fast meson evolution down to zero momentum scales at which the factorization of hadronic matrix elements is at work. The inclusion of lowest-lying vector meson contributions in addition to the pseudoscalar ones and of Wilson coefficients in a momentum scheme improves significantly the matching between quark-gluon and meson evolutions. In particular, the anomalous dimension matrix governing the meson evolution exhibits the structure of the known anomalous dimension matrix in the quark-gluon evolution. While this physical picture did not yet emerge from lattice simulations, the recent results on ReA₂ and ReA₀ from the RBC-UKQCD collaboration give support for its correctness. In particular, the signs of the two main contractions found numerically by these authors follow uniquely from our analytic approach. Though the current-current operators dominate the ΔI = 1/2 rule, working with matching scales O(1 GeV) we find that the presence of QCD-penguin operator Q₆ is required to obtain satisfactory result for ReA₀. At NLO in 1/N we obtain R = ReA₀/ReA₂ = 16.0 ± 1.5 which amounts to an order of magnitude enhancement over the strict large N limit value √2. We also update our results for the parameter B̂_K, finding B̂_K = 0.73 ± 0.02. The smallness of 1/N corrections to the large N value B̂_K = 3/4 results within our approach from an approximate cancelation between pseudoscalar and vector meson one-loop contributions. We also summarize the status of ΔM_K in this approach.