Effective

We compute the effective Hamiltonian for non-leptonic image decays in the standard model including next-to-next-to-leading order QCD corrections. In particular, we present the complete three-loop anomalous dimension matrix describing the mixing of current–current and QCD penguin operators. The calculation is performed in an operator basis which allows to consistently use fully anticommuting image in dimensional regularization at an arbitrary number of loops. The renormalization scheme dependences and their cancellation in physical quantities is discussed in detail. Furthermore, we demonstrate how our results are transformed to a different basis of effective operators which is frequently adopted in phenomenological applications. We give all necessary two-loop constant terms which allow to obtain the three-loop anomalous dimensions and the corresponding initial conditions of the two-loop Wilson coefficients in the latter scheme. Finally, we solve the renormalization group equation and give the analytic expressions for the low-energy Wilson coefficients relevant for non-leptonic B meson decays beyond next-to-leading order in both renormalization schemes.