Some Aspects of Complex Angular Momentum and Three-Particle States

In this paper we try to gain some understanding of the problem of how complex singularities affect the continuation of multiparticle amplitudes to complex angular momentum $J$ by examining a few simple pole contributions to a production amplitude. We find that in order to compute the effect of a three-particle state on a two-particle amplitude of definite signature the unitarity integral has to be considerably reorganized. This has the result of requiring us to deal with three different production amplitudes of definite signature, each of which provides a different continuation to complex $J$. We find also that the phase-space integrations which occur in unitarity-like equations must be performed over suitably deformed contours when $J$ is complex. For integer angular momentum, of course, the deformation has no effect. We elucidate this situation by introducing doubly projected partial-wave amplitudes in which both the total and a sub-angular momentum are projected out.