Effective field theory for radiative corrections to charged-current processes I: Vector coupling

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.