The Two-Loop Anomalous Dimension Matrix for $\Delta S=1$ Weak Non-Leptonic Decays I: $\mathcal{O}({\alpha }_s^2)$

Abstract

We calculate the two-loop $10\times 10$ anomalous dimension matrix $\mathcal{O}({\alpha }_s^2)$ involving current-current operators, QCD penguin operators, and electroweak penguin operators especially relevant for $\Delta S=1$ weak non-leptonic decays, but also important for $\Delta B=1$ decays. The calculation is performed in two schemes for ${\gamma }_5$: the dimensional regularization scheme with anticommuting ${\gamma }_5$ (NDR), and in the 't Hooft-Veltman scheme. We demonstrate how a direct calculation of diagrams involving ${\gamma }_5$ 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 $\mathcal{O}({\alpha }_e{\alpha }_s)$, required for a consistent inclusion of electroweak penguin operators, is presented in a subsequent publication.

Comments: 33 pages