Watson's Theorem When There Are Three Strongly Interacting Particles in the Final State

Abstract

Amado has recently conjectured the following extension of Watson's final-state-interaction theorem: The phase of an amplitude leading to a final state with three strongly interacting particles, when considered as a function of the relative energy of one pair, in a given-pair partial wave, all other variables being held constant, is the scattering phase of that pair, $\delta$. This conjecture is investigated in two simple, complementary models. In the first, a final-state interaction is grafted on to a specific production amplitude; as expected, if the production amplitude is real, the phase of the total amplitude is $\delta$. The question of the phase turns out to be intimately related to the question of whether triangle singularities are observable or not; Amado's results on this point are corrected. In the second model, the production mechanism is ignored and attention is focused on the possibility that there may be two ("overlapping") resonances in the final state. Here the theorem does not hold, in general (however it does turn out that the corrections are either small or easily calculable). In this case too, the phase question is linked with the observability of triangle singularities, and the rather negative conclusions reached by Schmid (who also treated this second model) are criticized. A basic tool is a theorem that the on- and off-shell parts of a rescattering graph contribute equally near a singularity which is near the physical region.