Regge Poles and High-Energy Limits in Field Theory

Abstract

It is shown that the Bethe-Salpeter scattering amplitude in the ladder approximation is meromorphic in the complex angular momentum half-plane, $\mathrm{Re}l>\ensuremath{-}\frac{3}{2}$. There is always at least one Regge pole in this region.The $T$ matrix element for complex $l$ can be written in the form, ${N}_{l}{{D}_{l}}^{\ensuremath{-}1}$, where ${N}_{l}$ and ${D}_{l}$ have convergent perturbation expansions. ${D}_{l}$ has only a right-hand cut in the squared energy variable, with a branch point at the elastic scattering threshold and at each production threshold. ${N}_{l}$ has a left-hand cut, and in addition a right-hand cut beginning from the first three-particle threshold. The Regge poles are zeros of ${D}_{l}$. Much of the information about the trajectories of Regge poles is contained in the lowest-order expression for ${D}_{l}$. The general properties of the trajectories are the same as for the case of scattering from a Yukawa potential. For sufficiently small coupling constant a Regge trajectory $\ensuremath{\alpha}(s)$ may apparently be expanded in a perturbation series, valid except near thresholds in $s$.The connection between the Regge poles of the ladder graphs and the high-energy behavior of the "strip" graphs is discussed. In the $\ensuremath{\lambda}{\ensuremath{\phi}}^{3}$ theory it is shown that the second-order expression for the leading Regge trajectory, for the sum of the ladder graphs, determines the leading term in the high-energy limit of the $n\mathrm{th}$ order strip graph. This relationship has been checked in fourth-order perturbation theory, and is evidence for the consistency of a perturbation approach to the calculation of Regge trajectories.