Three-particle partial wave amplitudes and unitarity conditions at complex values of angular momentum
Abstract
We have studied an analytic continuation of three-particle production partial wave amplitudes ƒ jm in complex values of total angular momentum j. In contrast to the two-particle case, there is no continuation of ƒ jm which decreases in the whole right half-plane of j. Nevertheless, at fixed values of pair energies of three particles it is possible to construct a unique continuation increasing slowly enough in the right half j-plane. If the amplitudes ƒ jm are studied in the whole three-particle physical region, it is necessary to consider at least six different analytic functions of j. This fact is a generalization of signature for the case of amplitudes ƒ jm. Each of the six continuations satisfies a simple unitarity condition in one of the pair energies at complex j. The integration domain in these unitarity conditions includes, generally speaking, the integration over nonphysical values of angles. These properties were obtained while investigating the simplest Feynman diagrams. However, as it follows from the derivation they are to be of a general character. The three-particle contribution to the unitarity condition for elastic scattering amplitude at complex j is also considered. This enables us to study the origin of Mandelstam branch points. However, we can write down the exact expression for the three-particle contribution into the unitarity condition only in the case of some simplest diagrams.
- Publication:
-
Annals of Physics
- Pub Date:
- April 1966
- DOI:
- 10.1016/0003-4916(66)90041-8
- Bibcode:
- 1966AnPhy..37..227A