Regge Poles and Unequal-Mass Scattering Processes

It is not clear from the Regge representation that the asympotic form s^α(u) holds in the backward scattering of unequal-mass particles, because the cosine of the u-channel scattering angle remains small as s increases. In this paper we use a representation for the scattering amplitude first suggested by Khuri to show that the form s^α(u) is valid throughout the backward region. However, in order to ensure the analyticity of the amplitude defined by the Khuri representation at $u=0$, it is necessary that Regge trajectories occur in families whose zero-energy intercepts are spaced by integers. Denoting the leading or parent trajectory by α₀(u), we find that daughter trajectories αₖ(u) must exist, of signature (-1)ᵏ relative to the parent, satisfying αₖ(0)=α₀(0)-k. We then study Bethe-Salpeter models and find that this daughter-trajectory hypothesis is satisfied for any Bethe-Salpeter amplitude which Reggeizes in the first place. This fact follows elegantly from the four-dimensional symmetry of Bethe-Salpeter equations at zero total energy. Some phenomenological implications of the daughter-trajectory hypothesis are discussed. We have also characterized the behavior of partial-wave amplitudes in unequal-mass scattering at $u=0$ and find the hitherto unsuspected result a(u,l)~u^-α(0), where α(u) is the leading u-channel Regge trajectory.