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

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

1992-11-25

arxiv.org/abs/hep-ph/9211304v1

Nucl.Phys. B400 (1993) 37-74

Physics

High Energy Physics

High Energy Physics - Phenomenology

33 pages

Scientific paper

10.1016/0550-3213(93)90397-8

We calculate the two-loop $10 \times 10$ anomalous dimension matrix ${\cal O}(\alpha_s^{2})$ 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 ${\cal O}(\alpha_e\alpha_s)$, required for a consistent inclusion of electroweak penguin operators, is presented in a subsequent publication.

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