Radiative corrections to neutron $\beta$ decay with explicit $\Delta (1232)$ degrees of freedom
We study electromagnetic radiative corrections to neutron $\beta$ decay within chiral perturbation theory extended to include the $\Delta(1232)$ resonance as an explicit degree of freedom. Building on a previous analysis performed in the $\Delta$-less theory, we identify and compute additional loop contributions with intermediate $\Delta$ states at leading and next-to-leading order in the small-scale expansion, and match the results to the pionless effective field theory. We find that the electromagnetic shift in the axial coupling is reduced by approximately a factor of two compared to the $\Delta$-less result. The correction remains at the percent level and is phenomenologically significant for the comparison between experimental extractions and lattice-QCD determinations of the nucleon axial charge. We also obtain a new $\Delta$-induced contribution to the weak magnetism term, which is an order of magnitude smaller than the pion-loop result and negligible for upcoming experiments.