Large N approach to Kaon decays and mixing 28 years later: Delta I=1/2 rule, hat{B}_K,and Delta M_K

We review and update our results for $Krightarrow pi pi $ decays and $K^0$ – $bar{K}^0$ 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 $hbox {Re}A_0$ and suppression of $hbox {Re}A_2$ , the so-called $Delta I=1/2$ rule for $Krightarrow pi pi $ decays, has a simple structure: the usual octet enhancement through the long but slow quark–gluon renormalization group evolution down to the scales $mathcal{O}(1, {hbox { GeV}})$ 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 $hbox {Re}A_2$ and $hbox {Re}A_0$ 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 $Delta I=1/2$ rule, working with matching scales $mathcal{O}(1 , {hbox { GeV}})$ we find that the presence of QCD-penguin operator $Q_6$ is required to obtain satisfactory result for $hbox {Re}A_0$ . At NLO in $1/N$ we obtain $R=hbox {Re}A_0/hbox {Re}A_2= 16.0pm 1.5$ which amounts to an order of magnitude enhancement over the strict large $N$ limit value $sqrt{2}$ . We also update our results for the parameter $hat{B}_K$ , finding $hat{B}_K=0.73pm 0.02$ . The smallness of $1/N$ corrections to the large $N$ value $hat{B}_K=3/4$ results within our approach from an approximate cancelation between pseudoscalar and vector meson one-loop contributions. We also summarize the status of $Delta M_K$ in this approach.