  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.
