Regge Poles and Unequal-Mass Scattering Processes

Abstract

It is not clear from the Regge representation that the asympotic form ${s}^{\ensuremath{\alpha}(u)}$ 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 ${s}^{\ensuremath{\alpha}(u)}$ 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 ${\ensuremath{\alpha}}_{0}(u)$, we find that daughter trajectories ${\ensuremath{\alpha}}_{k}(u)$ must exist, of signature ${(\ensuremath{-}1)}^{k}$ relative to the parent, satisfying ${\ensuremath{\alpha}}_{k}(0)={\ensuremath{\alpha}}_{0}(0)\ensuremath{-}k$. 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 $a(u,l)\ensuremath{\sim}{u}^{\ensuremath{-}\ensuremath{\alpha}(0)}$, where $\ensuremath{\alpha}(u)$ is the leading $u$-channel Regge trajectory.