Coherence in Three-Body Final States

Abstract

The amplitude for a reaction leading to three free particles is studied in the context of nonrelativistic scattering theory with particular emphasis on the distribution of information about final-state interactions of a given pair. We show that contrary to the simple notions often read into the Faddeev equations, terms ending with the interaction of a given pair do not carry the major information about interactions of that pair. When a coherent combination of all terms contributing to the process is taken, a factor clearly carrying the features of a pair's final-state interactions appears multiplying the entire amplitude. This factor carries the elastic-scattering phase of the pair. It will dominate the variation of the amplitude with respect to final-state interactions of the pair if certain other terms vary slowly, and this seems plausible. Application of this result to triangle singularities shows that the coherent combination of terms with and without final rescattering changes the singularity qualitatively. The assignment of a final-state interaction factor onto the entire amplitude for each pair suggests a product-type final-state wave function for the process. The implications of this for overlapping resonances are discussed with particular reference to static models and the absence of the Peierls mechanism. The case of two particles interacting independently with a fixed center, for which the product wave function is exact, is discussed from the Faddeev viewpoint and shows that a multiple-scattering approach can be misleading.