Complex conjugate pomeranchuk trajectories and pomeranchuk theorem for differential cross-sections

Abstract

The Pomcranchuk theorem for the differential cross-sections, ar~ asymptotic equality of the particle-particle and particle-antiparticle elastic differential cross-sections, has been derived for nonoscillating amplitudes by several authors (~)using the Phragm~nLindelSf theorem (2). The same method was used to show that the Pomeranchuk theorem for the differential cross-sections cannot be satisfied if the dominating Regge trajectory is complex (a). In this paper we will investigate an asymptotic ratio of the particle-particle and particle-antiparticle elastic differential cross-sections for the case when the scattering is dominated by a pair of complex conjugate trajectories. This case might be of interest for the following reason. It has been suggested (4) that any Regge trajectory meets the corresponding ReggePomeranehuk cut and develops some sort of singularity near t ~ 0, Consequently,, one can expect that all Regge trajectories will have left-hand branch cuts. (4) in t from some negative t (eventually from t : -- c~) to t : t 0, where Itl << 1. Furthermore, it has been shown (5) that because of the collision of the Regge trajectory with the corresponding Regge-Pomeranehuk cut a new pole can come from the second sheet of the complex momentum plane and form a pair of complex conjugate poles with the original pole. If the pair of complex conjugate poles is on the physical sheet and if for some region of negative t the real part of the Regge trajectory is larger than the real part of the branch point trajectory (~) (cut), then in this t region the scattering ampli