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

Abstract

We review and update our results for K → ππ decays and K 0 -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 ReA 0 and suppression of ReA 2 , the socalled I = 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 quarkgluon 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 Re A 2 and ReA 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 I = 1/2 rule, working with matching scales O(1 GeV) we find that the presence of QCD-penguin operator Q 6 is required to obtain satisfactory result for Re A 0 . At NLO in 1/N we obtain R = ReA 0 /Re A 2 = 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 parameterB K , finding a e-mail: aburas@ph.tum.deB K = 0.73 ± 0.02. The smallness of 1/N corrections to the large N valueB 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 M K in this approach.

New physics patterns in ε′/ε and ε K with implications for rare kaon decays and ∆M K

“…While, the improvement on the estimate of isospin corrections, final state interactions (FSI) [10][11][12][13][14][15] and the inclusion of NNLO QCD corrections could in principle increase ε /ε with respect to the one in (1.1), it is rather unlikely that values of ε /ε violating the upper bound in (1.3) will be found within the SM. After all, until now, lattice QCD confirmed most of earlier results on K meson flavour physics obtained in the large N approach (see [2,16]). …”

“…• ∆M K is uniquely suppressed with the suppression increasing with increasing κ ε and M Z : Large N approach [16] indicates that long distance contributions enhance this ratio by roughly 20%. First lattice calculations [75] are still subject to large uncertainties and also the large error in (5.14) precludes any definite conclusions at present whether NP should enhance or suppress this ratio.…”

π0−η−η′mixing in a generalized multiquark interaction scheme

“…It affects the value of the K 0 → π 0 π 0 transition through the dominant QCD Q 6 penguin operator, which is one of the sources of uncertainties in the determination of ð ϵ 0 ϵ Þ CP [18], for a recent review see [19]. In chiral perturbation theory (ChPT) [2,20,21] the π 0 − η mixing angle occurs already at order p 2 and was first evaluated to order p 4 in the context of K l3 form factors in [21].…”

Study of kaon decay to two pions

“…It has therefore been studied with many different theoretical methods [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. In recent years these efforts have been extended to include lattice QCD studies, with recent results reported in Ref.…”

“…Next-to-leading order contributions were considered within the large N c approach in Refs. [9][10][11]. The potentially important role of the trace anomaly in weak K decays, especially in regard to the ΔI ¼ 1=2 rule, was discussed in Ref.…”