Effective hamiltonians for ΔS = 1 and ΔB = 1 non-leptonic decays beyond the leading logarithmic approximation

Abstract

We calculate the effective hamiltonians for ΔB = 1 and ΔS = 1 decays including next-to-leading QCD corrections. In particular we present the two-loop 6 × 6 anomalous dimension matrices describing the mixing of current-current and QCD penguin operators calculated in the dimensional regularization scheme with anticommuting γ 5 and in the 't Hooft-Veltman scheme. The renormalization scheme dependences and their cancellation for physical quantities are discussed in detail. We point out that beyond the leading order, matching between coefficients and hadronic matrix elements of local operators both in the renormalization scale μ and in the renormalization scheme for the operators has to be made. The next-to-leading corrections enhance substantially the Wilson coefficients of QCD penguin operators and these effects are stronger than the ones found for current-current operators. consequently, the contributions of QCD penguins to B-decays, the ΔI = 1 2 rule and ϵ′ / ϵ are enlarged. For μ ≈ 1 GeV and Λ MS ≈ 300 MeV , these effects amount to 20–30%. In particular we emphasize that for μ < 1 GeV the next-to-leading corrections to the coefficients of penguin operators are large, reaching 40% for μ ≈ 0.8 GeV. For such low values of μ the renormalization group improved perturbation theory cannot be trusted any longer. Moreover the results are very sensitive functions of Λ MS .