The Two-Loop Anomalous Dimension Matrix for $ΔS=1$ Weak Non-Leptonic Decays I: ${\cal O}(α_s^2)$

Abstract

We calculate the two-loop $10 \times 10$ anomalous dimension matrix ${\cal O}(α_s^{2})$ involving current-current operators, QCD penguin operators, and electroweak penguin operators especially relevant for $ΔS=1$ weak non-leptonic decays, but also important for $ΔB=1$ decays. The calculation is performed in two schemes for $γ_{5}$: the dimensional regularization scheme with anticommuting $γ_{5}$ (NDR), and in the 't Hooft-Veltman scheme. We demonstrate how a direct calculation of diagrams involving $γ_{5}$ in closed fermion loops can be avoided thus allowing a consistent calculation in the NDR scheme. The compatibility of the results obtained in the two schemes considered is verified and the properties of the resulting matrices are discussed. The two-loop corrections are found to be substantial. The two-loop anomalous dimension matrix ${\cal O}(α_eα_s)$, required for a consistent inclusion of electroweak penguin operators, is presented in a subsequent publication.