Regge-Pole Exchange and Direct-Channel Resonances in Models for High-Energy Scattering Amplitudes

The behavior of the forward and backward $\pi N$ scattering amplitudes for momenta of 1-5 BeV/c has been analyzed recently using models in which Breit-Wigner amplitudes describing direct-channel resonances are added to a background amplitude given by the Regge-pole-exchange model. Although remarkably successful in practice, the model has severe theoretical limitations, especially with regard to the treatment of the tails of the resonant terms, double counting of the background contributions, and the restriction to the Breit-Wigner approximation for sets of isolated resonances. The theory of Regge-pole-plus-resonance (RPR) models is examined in detail for both single-channel potential scattering and the many-channel relativistic case. A modified RPR model is developed in which (i) the double-counting problems are eliminated, and (ii) direct-channel resonances are described in terms of their Regge-trajectory functions. There is no difficulty with the tails of the resonant amplitudes in this formulation of the RPR model. Moreover, the contributions of the entire set of resonances on a given Regge trajectory can be included in the scattering amplitude. The relevance of these modifications of the RPR model to past analyses of $\pi N$ scattering is discussed briefly.