Study of Branch Points in the Angular-Momentum Plane

A study is made of the Amati-Fubini-Stanghellini (AFS) type of approximation to the amplitudes associated with the exchange of a single Regge pole and an elementary spinless particle, and with the exchange of two Regge poles. The location, motion, and nature of the singularities in the complex-angular-momentum plane of the $s$ reaction which appear in these approximations, and their cancellation in the full diagram, are considered in detail; the singularities are found to be of two general types: branch points whose positions are independent of, and dependent on, particle masses. Only the former singularities determine the asymptotic behavior of the AFS amplitudes in the physical scattering region, while the latter appear only on the physical sheet via the mass-independent branch points at unphysical momentum transfers. The same method used in the study of the AFS approximation to the diagrams which do not have the AFS-type singularities is applied to the analysis of the Mandelstam diagrams for which the above-mentioned cancellation of the cuts does not occur; the analysis, although less rigorous, suggests that the location and nature of the singularities in the $j$ plane are the same as those found for the AFS type of approximations to the simpler versions of these diagrams. With a number of approximations which, although plausible, are hard to justify rigorously, an estimate is made of the contribution to the amplitude coming from the angular-momentum cut.