Multiple-Scattering Expansions for Nonrelativistic Three-Body Collision Problems. III. Comments on the CoulombTMatrix and the Application of the Faddeev-Watson Expansion to Rearrangement Collision

An application of the Faddeev-Watson multiple-scattering expansion for rearrangement collision to an arbitrary three-body Coulomb system is presented. The importance of the Coulomb cuts on the energy shell is emphasized. A demonstration of how the second-order Born term and the truncated second-order multiple-scattering term considered by Carpenter and Tuan get canceled out in the second-order multiple-scattering term is given. It is then clear that considerable problems remain in the determination of the high-energy behavior of the cross section for rearrangement collisions. A number of errors which have recently appeared in the literature concerning the treatment of the branch-point singularities and the intermediate two-body bound- and antibound-state poles in the off-shell two-body Coulomb $T$ matrix are rectified. Detailed expressions for the three-body rearrangement-collision amplitude in the first-order multiple-scattering approximation are derived and expressed as a sum of three contributions coming from (a) the bound-state poles in the initial and final wave functions, (b) the intermediate two-body antibound-state poles, and (c) the branch-point singularities in the off-shell two-body Coulomb $T$ matrix.