Coulomb Interference in High-Energy Scattering

The relativistic Coulomb interference problem is carefully examined in order to evaluate critically the equations used in analyzing experiments designed to test the forward $\pi -N$ dispersion relations. We show that with a suitable interpretation the nonrelativistic Bethe formula for the phase difference $\phi$ between the strong and electromagnetic contributions is valid. However, for high energy near forward scattering there are unknown contributions which can change this phase by $O\left(\alpha \right)$ rad; this is to be compared with a magnitude of $\phi$ of $\sim 2\alpha$ rad. We further show that two previous relativistic calculations of $\phi$ are incomplete. The effects of calculable radiative corrections such as soft photon emission or vacuum polarization are examined. For the former it is found that although their magnitude is relatively large, they can, in general, be neglected. The reason for this is that by folding an accurate measurement of the total cross section into the data analysis a compensation of errors is induced. No such compensation takes place for the case of vacuum polarization, and its contribution could be important. Finally, we investigate the effects of small admixtures to the strong interactions which are not of the form $e^{a+b^{2}t}$. We find that they are unlikely to be of significance here provided they can be assumed to be $\lesssim 20\%$ of the complete amplitude.