Rise and Fall of Relativistic Trajectories

The general problem of rising and falling Regge trajectories is investigated in a fully relativistic model in which an arbitrary two-particle interaction is unitarized by summing generalized ladder diagrams. The trajectories are extracted from the infinite sum by an extension of the technique previously applied to ladder diagrams in an ordinary scalar field theory. Any basic interaction which vanishes faster than any power of the momentum transfer t generates infinitely falling trajectories. A physical explanation of falling trajectories is given in terms of the scattering of infinitely composite particles where the standard single-particle-exchange interaction is modified by exponentially damped form factors. A variety of models for rising trajectories is constructed, all of which violate either the Jaffe bound on form factors or the Froissart bound on forward scattering amplitudes. If the interaction is single-particle exchange with form factors e-γ(-t)β, the trajectory has Reα→+∞ as s→∞ if β≥1, but it is not of narrow width. Energy-dependent interactions based on Regge-pole exchange, energy-dependent coupling constants, or direct form factors yield rising trajectories, but not narrow widths. Moreover, the energy dependence must be such that the Froissart bound appears to be grossly violated.