Two--Loop Anomalous Dimension Matrix for $ΔS=1$ Weak Non-Leptonic Decays II: ${\cal O}(α_e α_s)$

We calculate the $10\times 10$ two--loop anomalous dimension matrix to order $\ord(\alpha_e \alpha_s)$ in the dimensional regularization scheme with anticommuting $\gamma_5$ (NDR) which is necessary for the extension of the $\Delta S=1$ weak Hamiltonian involving electroweak penguins beyond the leading logarithmic approximation. We demonstrate, how a direct calculation of penguin diagrams involving $\gamma_5$ in closed fermion loops can be avoided thus allowing a consistent calculation of two--loop anomalous dimensions in the simplest renormalization scheme with anticommuting $\gamma_5$ in $D$ dimensions. We give the necessary one--loop finite terms which allow to obtain the corresponding two--loop anomalous dimension matrix in the HV scheme with non--anticommuting $\gamma_5$.