The anatomy of ε′/ε beyond leading logarithms with improved hadronic matrix elements

Abstract

We use the recently calculated two-loop anomalous dimensions of current-current operators, QCD and electroweak penguin operators to construct the effective hamiltonian for ΔS = 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 ΔB = 1 and ΔC = 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.