Two-loop anomalous dimension matrix for ΔS = 1 weak non-leptonic decays (I). O( α2s)

Abstract

We calculate the two-loop 10×10 anomalous dimension matrix O( α 2 s) 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 O( αα s), required for a consistent inclusion of electroweak penguin operators, is presented in a subsequent publication.