Renormalization of four-quark operators, effective theory, and the role of evanescent operators

We present, in the context of dimensional regularization, a prescription to renormalize Feynman diagrams with an arbitrary number of external fermions. This prescription, which is based on the original t'Hooft-Veltman proposal to keep external particles in four dimensions, is particularly useful to define the 'renormalization' (in the context of effective Lagrangian) of physical four-quark operators without introducing any evanescent operator. The results obtained for $b\rightarrow s$ processes agree with those from the so-called naive prescription, but disagree with the ones with the introduction of evanescent operators in a renormalization group analysis. We also present an explicit two loop calculation of the mixing of the evanescent operators with the physical dimension five operators for the same processes. Particular attention is paid to the unboundedness nature of such mixing and how a formal finite transformation is effected to decouple. The inevitable mass dependence of one of these schemes in the literature is pointed out as the cause for the difference mentioned.