Three-Body Problem with Charged Particles

The formal modification, by long-range potentials, of the scattering equations describing three particles interacting through short-range potentials is discussed in terms of the exact Coulomb Green's function, whose mathematical properties are known. This Green's function can be explicitly written down in several special cases; approximate forms of the modified theory are described for these instances. The application of the Coulomb-modified three-particle scattering theory to the $\alpha$-cluster model of ${\mathrm{C}}^{12}$ and to deuteron-induced nuclear reactions is described in some detail. For the three-$\alpha$ model, an heuristic generalization of the local plane-wave approximation is used to obtain a simple approximate form for the Coulomb Green's function. This new theory may have some relevance to the problem of helium burning in stars. Finally, in the Appendix a discussion is given of why the methods of multiparticle collision theory fail in the problem of calculating the general three-body Coulomb Green's function at positive, real energies. A possibly useful, mathematically well-defined method for constructing this Green's function at negative real energies is also given in the Appendix.