The ∆i = 1/2 Rule and Ε ′ /ε

The ∆I = 1/2 rule and direct CP violation ε ′ /ε in kaon decays are studied within the framework of the effective Hamiltonian approach in conjunction with generalized factorization for hadronic matrix elements. Conventionally the K → ππ matrix elements are evaluated under the factorization assumption so that O(µ) is factorized into the product of two matrix elements of single currents, governed by decay constants and form factors. However, the information of the scale and γ 5-scheme dependence of O(µ) is lost in the factorization approximation. To implement the scale dependence, it has been advocated that a physical cutoff Λ c , which is introduced to regularize the quadratic (and logarithmic) divergence of the long-distance chiral loop corrections to K → ππ amplitudes , can be identified with the renormaliza-tion scale µ of the Wilson coefficients 1. However , this chiral approach faces several difficulties: (i) The long-distance evolution of me-son loop contributions can only be extended to the scale of order 600 MeV, whereas the perturbative evaluation of Wilson coefficients cannot be reliably evolved down to the scale below 1 GeV. The conventional practice of matching chiral loop corrections to hadronic matrix elements with Wilson coefficient functions at the scale µ = (0.6−1.0) GeV requires chiral perturbation theory and/or perturba-tive QCD be pushed into the regions beyond their applicability. (ii) It is quite unnatural to match the quadratic scale dependence of chiral corrections with logarithmic µ dependence of Wilson coefficients. This means that it is necessary to apply the same renormaliza-tion scheme to regularize short-distance c(µ) and long-distance chiral corrections. (iii) It is not clear how to address the issue of γ 5-scheme dependence in the chiral approach. (iv) While the inclusion of chiral loops will make a large enhancement for A 0 , the predicted A 2 is still too large compared to experiment. This implies that nonfactorized effects other than chiral loops are needed to explain A 2. Therefore, not all the long-distance nonfactorized contributions to hadronic matrix elements are fully accounted for by chi-ral loops. (v) Finally, this approach based on chiral perturbation theory is not applicable to heavy meson decays. Therefore, it is strongly desirable to describe the nonleptonic decays of kaons and heavy mesons within the same framework. The scale and scheme problems with naive factorization will not occur in the full amplitude since Q(µ) involves vertex-type and penguin-type corrections to the hadronic …