A note on momentum subtraction schemes for quark bilinears and semileptonic operators

In this work we examine a family of regularization invariant (RI) symmetric momentum (SMOM) schemes for semi-leptonic operators. By working with chirally symmetric and massless QCD, we relate the semi-leptonic operators with their corresponding flavor-changing vector currents, whose renormalization in pure QCD is protected by the Ward identity. For the latter, we extend the original RI/SMOM scheme [Sturm et al. 2009] to a family projectors suitable to be promoted to the semi-leptonic case and demonstrate their equivalence to Ref. [Gorbahn et al. 2023], relevant in particular for the perturbative calculation of the corresponding Wilson coefficients.