Three-Particle Scattering—A Relativistic Theory

Abstract

We give relativistic extensions of the Faddeev equations for three-particle scattering. They are linear integral equations for six amplitudes, one of the particles being off the mass shell. They satisfy exactly three-particle unitarity. These equations are obtained by applying the techniques introduced by Blankenbecler and Sugar to multiladder diagrams. Accordingly, the two-particle scattering amplitudes, which appear in the kernel of the equations, depend on the energy of the third particle. We study the equations for these amplitudes. If one replaces these two-body amplitudes by phenomenological amplitudes satisfying unitarity, one gets phenomenological relativistic equations for the three-body problem. When the two-body amplitudes are approximated by the contributions of bound states and resonances, the equations can be reduced to a set of integral equations in one variable. As a by-product of this study, the following results have been obtained: (a) a new proof of unitarity for the Lippmann-Schwinger and Faddeev equations; (b) a proof of the analytic properties of a resonance form factor in the nonrelativistic theory; (c) a proof of the asymptotic behavior of these form factors, which uses functional-analysis techniques and is a rather general method for investigating the asymptotic behavior of solutions of Fredholm equations.