High-energy asymptotic behavior of the Bourrely-Soffer-Wu model for   elastic scattering
  Some time ago, an accurate phenomenological approach, the BSW model, was developed for proton-proton and antiproton-proton elastic scattering cross sections at center-of-mass energies above 10 GeV. This model has been used to give successful theoretical predictions for these processes, at successive collider energies. The BSW model involves a combination of integrals that, while computable numerically at fairly high energies, require some mathematical analysis to reveal the high-energy asymptotic behavior. In this paper we present a high-energy asymptotic representation of the scattering amplitude at moderate momentum transfer, for the leading order in an expansion parameter closely related to the logarithm of the center-of-mass energy. The fact that the expansion parameter goes as the logarithm of the energy means that the asymptotic behavior is accurate only for energies greatly beyond any foreseeable experiment. However, we compare the asymptotic representation against the numerically calculated model for energies in a less extreme region of energy. The asymptotic representation is given by a simple formula which, in particular, exhibits the oscillations of the differential cross section with momentum transfer. We also compare the BSW asymptotic behavior with the Singh-Roy unitarity upper bound for the diffraction peak. 
Claude Bourrely, John M. Myers, Jacques Soffer, Tai Tsun Wu
hep-ph