Regge Poles and Unequal-Mass Scattering Processes

It is not clear from the Regge representation that the asympotic form $s^{\alpha \mathrm{}\left(u\right)\mathrm{}}$ 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 $s^{\alpha \mathrm{}\left(u\right)\mathrm{}}$ is valid throughout the backward region. However, in order to ensure the analyticity of the amplitude defined by the Khuri representation at $u=0\mathrm{}$, it is necessary that Regge trajectories occur in families whose zero-energy intercepts are spaced by integers. Denoting the leading or parent trajectory by ${\alpha }_0\mathrm{}\left(u\right)\mathrm{}$, we find that daughter trajectories ${\alpha }_k\mathrm{}\left(u\right)\mathrm{}$ must exist, of signature ${\mathrm{}(-1\mathrm{})\mathrm{}}^k$ relative to the parent, satisfying ${\alpha }_k\mathrm{}\left(0\right)={\alpha }_0\mathrm{}\left(0\right)-k$. 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 $u=0\mathrm{}$ and find the hitherto unsuspected result $a(u,l)\mathrm{}\sim u^{-\alpha \mathrm{}\left(0\right)\mathrm{}}$, where $\alpha \mathrm{}\left(u\right)\mathrm{}$ is the leading $u$-channel Regge trajectory.