  We study radiative corrections to low-energy charged-current processes
involving nucleons, such as neutron beta decay and (anti)neutrino-nucleon
scattering within a top-down effective-field-theory approach. We first match
the Standard Model to the low-energy effective theory valid below the weak
scale and, using renormalization group equations with anomalous dimensions of
$\mathcal{O}(\alpha, \alpha \alpha_s, \alpha^2)$, evolve the resulting
effective coupling down to the hadronic scale. Here, we first match to
heavy-baryon chiral perturbation theory and subsequently, below the pion-mass
scale, to a pionless effective theory, evolving the effective vector coupling
with anomalous dimensions of $\mathcal{O}(\alpha, \alpha^2)$ all the way down
to the scale of the electron mass, relevant for beta decays. We thus provide a
new evaluation of the ``inner" radiative corrections to the vector coupling
constant and to the neutron decay rate, discussing differences with the
previous literature. Using our new result for the radiative corrections, we
update the extraction of the Cabibbo-Kobayashi-Maskawa matrix element $V_{ud}$
from the neutron decay.
