Simple Rules for Evanescent Operators in One-Loop Basis Transformations

Basis transformations often involve Fierz and other relations which are only valid in $D=4$ dimensions. In general $D$ space-time dimensions however, evanescent operators have to be introduced, in order to preserve such identities. Such evanescent operators contribute to one-loop basis transformations as well as to two-loop renormalization group running. We present a simple procedure on how to systematically change basis at the one-loop level by obtaining shifts due to evanescent operators. As an example we apply this method to derive the one-loop basis transformation from the BMU (Buras, Misiak and Urban) basis useful for NLO QCD calculations, to the JMS (Jenkins, Manohar and Stoffer) basis used in the matching to the SMEFT.