Semileptonic weak Hamiltonian to in momentum-space subtraction schemes

The CKM unitarity precision test of the Standard Model requires a systematic treatment of electromagnetic corrections for semi-leptonic decays. Electromagnetic corrections renormalize a semi-leptonic four-fermion operator and we calculate the O(ααs ) perturbative scheme conversion between the MS¯ scheme and several momentum-space subtraction schemes, which can also be implemented on the lattice. We consider schemes defined by MOM and SMOM kinematics and emphasize the importance of the choice of projector for each scheme. The conventional projector, that has been used in the literature for MOM kinematics, generates an artificial QCD contribution to the conversion factor, which reflects in an unphysical dependence on the lattice matching scale. This can be traced to the violation of a Ward identity that holds in the case QED is neglected. We show how to remedy this by judicious choices of projector. The resulting Wilson coefficients (and operator matrix elements) have greatly reduced scale dependence.