$\gamma W$-exchange contributions in neutron $\beta$ decay in the forward-angle limit

In this work, the contributions from $\gamma W$-exchange in neutron $\beta$ decay are estimated at the amplitude level. Using a general form for the electromagnetic (EM) form factors (FFs) of the proton, the EM FFs of the neutron, and the weak FFs of the $Wnp$ interaction as inputs, we present analytical expressions for the inner part of the $\gamma W$-exchange amplitude under the forward angle limit. The differences and relations between our method and those used in the references are discussed. To compare our numerical results with those provided in the references, we consider three types of FFs as examples. The numerical results show that when the favored EM FFs are used, our result for the contribution from the Fermi part is consistent with those reported in the references, while our results for the Gamow-Teller parts are about 7\% and 13\% larger than those reported in Ref.~\cite{Hayen-2021}.