  The CKM unitarity precision test of the Standard Model requires a systematic
treatment of electromagnetic and strong corrections for semi-leptonic decays.
Electromagnetic corrections require the renormalization of a semileptonic
four-fermion operator. In this work we calculate the
$\mathcal{O}(\alpha\alpha_s)$ perturbative scheme conversion between the
$\bar{\rm MS}$ 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 QCD corrections to the conversion factor that do not
vanish for $\alpha=0$ and which generate an artificial dependence on the
lattice matching scale that would only disappear after summing all orders of
perturbation theory. This can be traced to the violation of a Ward identity
that holds in tha $\alpha =0$ limit. We show how to remedy this by judicious
choices of projector, and prove that the Wilson coefficients in those schemes
are free from pure QCD contributions. The resulting Wilson coefficients (and
operator matrix elements) have greatly reduced scale dependence. Our choice of
the $\bar{\rm MS}$ scheme over the traditional $W$-mass scheme is motivated by
the fact that, besides being more tractable at higher orders, unlike the latter
it allows for a transparent separation of scales. We exploit this to obtain
renormalization-group-improved leading-log and next-to-leading-log strong
corrections to the electromagnetic contributions and study the (QED-induced)
dependence on the lattice matching scale.
