$\mathrm{\Delta }$ resonance contributions to QED radiative corrections in neutron and inverse beta decay

We incorporate the $\mathrm{\Delta }(1232)$ resonance into pion-induced QED radiative corrections to neutron decay and inverse beta decay (IBD). Within the framework of heavy-baryon chiral perturbation theory with explicit $\mathrm{\Delta }$ degrees of freedom, we compute additional contributions and study their impact on IBD cross sections and on the renormalization of the nucleon isovector vector and axial-vector charges. $\mathrm{\Delta }$ resonance does not renormalize the vector charge. For the axial-vector charge, including the $\mathrm{\Delta }$ resonance restores power counting and significantly reduces the QED radiative correction to the experiment-over-lattice-QCD ratio $g_A\mathrm{/}\left(g_A^{\mathrm{Q}\mathrm{C}\mathrm{D}}{g}_V\right)$, bringing it to zero within radiative-correction uncertainties. $\mathrm{\Delta }$ resonance increases the pion-induced QED radiative corrections to IBD by a factor $1.2$-$1.3$.