Radiative corrections to neutron $β$ decay with explicit $Δ(1232)$ degrees of freedom

Abstract

We study electromagnetic radiative corrections to neutron $β$ decay within chiral perturbation theory extended to include the $Δ(1232)$ resonance as an explicit degree of freedom. Building on a previous analysis performed in the $Δ$-less theory, we identify and compute additional loop contributions with intermediate $Δ$ 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 $Δ$-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 $Δ$-induced contribution to the weak magnetism term, which is an order of magnitude smaller than the pion-loop result and negligible for upcoming experiments.