Strip approximation with Regge poles and bootstrap equations for pion-pion scattering

Abstract

In an earlier paper concerned with potential scattering a modified Regge representation had been developed that is distinguished from other Regge representations by the property that its analytic structure is of just the type that is required by Mandelstam’s double dispersion relation. Here we discuss how this representation may be utilized within the framework of a relativisticS-matrix theory for pion-pion scattering. An approximation to the exact scattering amplitude, termed the strip approximation, is defined as the sum of all possible contributions to the scattering amplitude from Regge poles in the direct or one of the crossed channels. This strip approximation combines the following features: analyticity properties as described by the Mandelstam representation, exact crossing symmetry, « Regge behaviour » at high energies, description of bound states and resonances at low energies, and of forces due to the exchange of reggeized particles. Partial-wave amplitudes are discussed and the unitarity condition for complex angular momentum is introduced. A set of equations is then obtained that seems to be sufficient in principle for a self-consistent determination of the Regge trajectories, as far as they are located in the right half-plane, and of the background term of the scattering amplitude. Due to the complexity of the equations a numerical treatment is not yet in sight. Our work is related to other recent work on bootstrapping of Regge trajectories by Chew and Jones and by Frautschi, Kaus and Zachariasen.