arXiv:hep-ph/9512380v1 15 Dec 1995 M P I − P h/95 − 104 T U M − T 31 − 100/95 F ERM ILAB − P U B − 95/305 − T SLAC − P U B 7009 November 1995) to appear in Reviews of Modern Physics WEAK DECAYS BEYOND LEADING LOGARITHMS AbstractWe review the present status of QCD corrections to weak decays beyond the leading logarithmic approximation including particle-antiparticle mixing and rare and CP violating decays. After presenting the basic formalism for these calculations we discuss in detail the effective hamiltonians for all decays for which the next-to-leading corrections are known. Subsequently, we present the phenomenological implications of these calculations. In particular we update the values of various parameters and we incorporate new information on m t in view of the recent top quark discovery. One of the central issues in our review are the theoretical uncertainties related to renormalization scale ambiguities which are substantially reduced by including next-to-leading order corrections. The impact of this theoretical improvement on the determination of the Cabibbo-Kobayashi-Maskawa matrix is then illustrated in various cases. *

We use the recently calculated two-loop anomalous dimensions of current-current operators, QCD and electroweak penguin operators to construct the effective Hamiltonian for ∆S = 1 transitions beyond the leading logarithmic approximation. We solve the renormalization group equations involving α s and α up to two-loop level and we give the numerical values of Wilson coefficient functions C i (µ) beyond the leading logarithmic approximation in various renormalization schemes. Numerical results for the Wilson coefficients in ∆B = 1 and ∆C = 1 Hamiltonians are also given. We discuss several aspects of renormalization scheme dependence and demonstrate the scheme independence of physical quantities. We stress that the scheme dependence of the Wilson coefficients C i (µ) can only be cancelled by the one present in the hadronic matrix elements Q i (µ) . This requires also the calculation of O(α) corrections to Q i (µ) . We propose a new semi-phenomenological approach to hadronic matrix elements which incorporates the data for CP -conserving K → ππ amplitudes and allows to determine the matrix elements of all (V − A) ⊗ (V − A) operators in any renormalization scheme. Our renormalization group analysis of all hadronic matrix elements Q i (µ) reveals certain interesting features. We compare critically our treatment of these matrix elements with those given in the literature. When matrix elements of dominant QCD penguin (Q 6 ) and electroweak penguin (Q 8 ) operators are kept fixed the effect of next-to-leading order corrections is to lower considerably ε ′ /ε in the 't Hooft-Veltman (HV) renormalization scheme with a smaller effect in the dimensional regularization scheme with anticommuting γ 5 (NDR). Taking m t = 130 GeV, Λ MS = 300 MeV and calculating Q 6 and Q 8 in the 1/N approach with m s (1 GeV) = 175 MeV, we find in the NDR scheme ε ′ /ε = (6.7 ± 2.6) × 10 −4 in agreement with the experimental findings of E731. We point out however that the increase of Q 6 by only a factor of two gives ε ′ /ε = (20.0 ± 6.5) × 10 −4 in agreement with the result of NA31. The dependence of ε ′ /ε on Λ MS , m t and Q 6,8 is presented .A detailed anatomy of various contributions and comparison with the analyses of Rome and Dortmund groups are given.