Soft-photon expansion and soft-photon theorem

The question of the existence of the soft-photon expansion and its connection with the derivation of the soft-photon theorem is studied. We find that the soft-photon expansion does not necessarily exist for all possible bremsstrahlung cross sections, and if such an expansion does not exist for some types of cross section, then the soft-photon theorem cannot be derived for them. We define two different classes of bremsstrahlung cross section for our study, which we call the $H$-type cross section and the $R$-type cross section. We find that the $R$-type cross section can be expanded in powers of $k$ (photon momentum) with all coefficients of the expansion which are independent of $k$, while the $H$-type cross section can only have an expansion whose coefficients are all functions of $k$. Moreover, we have shown that those coefficients which are functions of $k$ must necessarily contain off-shell parameters. Therefore, we conclude that the soft-photon expansion exists and hence the soft-photon theorem can be derived only for the $R$-type cross section. This strongly implies that the $R$-type cross section is to be preferred for the study of bremsstrahlung processes.