Final-State Interactions among Three Particles. II. Explicit Evaluation of the First Rescattering Correction

Abstract

The lowest approximation to the three-body final-state interaction problem is a simple superposition of 3 two-body interaction terms, as given by the Watson final-state formula. The first rescattering correction corresponds to the graph of perturbation and dispersion theory; for example, the process $W\ensuremath{\rightarrow}2+(13)\ensuremath{\rightarrow}1+(23)\ensuremath{\rightarrow}1+2+3$, i.e., where first particles 1 and 3 interact strongly, then separate in such a way that 2 and 3 come together and interact strongly. For the case in which the (13) interaction is dominated by a resonance, we discuss the kinematic conditions under which such a rescattering correction will contribute appreciably to the total transition amplitude. We then evaluate this term exactly in the non-relativistic (N.R.) case; that is, we evaluate the N.R. triangle graph. An analogous expression is proposed for the relativistic case, with suitable reinterpretation of the variables. The method is generalized to include spin and angular momentum, and to incorporate (23) rescattering to all orders, for the cases in which this can be parametrized either by a scattering length or by a resonance. Specific application is made to the $\ensuremath{\pi}\ensuremath{\pi}N$ case.