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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.

The determination of α s from hadronic τ decays is revisited, with a special emphasis on the question of higher-order perturbative corrections and different possibilities of resumming the perturbative series with the renormalisation group: fixed-order (FOPT) vs. contour-improved perturbation theory (CIPT). The difference between these approaches has evolved into a systematic effect that does not go away as higher orders in the perturbative expansion are added. We attempt to clarify under which circumstances one or the other approach provides a better approximation to the true result. To this end, we propose to describe the Adler function series by a model that includes the exactly known coefficients and theoretical constraints on the large-order behaviour originating from the operator product expansion and the renormalisation group. Within this framework we find that while CIPT is unable to account for the fully resummed series, FOPT smoothly approaches the Borel sum, before the expected divergent behaviour sets in at even higher orders. Employing FOPT up to the fifth order to determine α s in the MS scheme, we obtain α s (M τ ) = 0.320 +0.012 −0.007 , corresponding to α s (M Z ) = 0.1185 +0.0014 −0.0009 . Improving this result by including yet higher orders from our model yields α s (M τ ) = 0.316 ± 0.006, which after evolution leads to α s (M Z ) = 0.1180 ± 0.0008. Our results are lower than previous values obtained from τ decays.

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