Evidence for a Constant `Edge' in Proton-Proton Scattering at Very High   Energies
  Accurate fits to $pp$ and $\bar pp$ cross section data up to Tevatron energies, incorporating the constraints imposed by analyticity and unitarity, successfully predict the results of recent LHC and cosmic ray measurements, and suggest that the cross sections approach a black disc limit asymptotically. The approach to the limit is, however, very slow. We present a simple geometric picture which explains these features in a natural way. A black disc of logarithmically growing radius is supplemented by a soft `edge' whose properties are invariant with energy. The constancy of the edge results in the prediction that the quantity $(\sigma^{TOT}-2\sigma^{El})/\surd\sigma^{TOT}$ approaches a constant at high energy. Using the existing fits, this prediction appears to be verified. The value of the limiting constant allows an estimate of the thickness of the edge, which turns out to be on the order of $1\,{\rm fm}$. One thus arrives at a picture where the proton-proton scattering at lower energies is dominated by what becomes the edge, while at higher energies it is dominated by the disc. The crossover between the two regimes is only at $\surd s\geq $ 10 TeV, accounting for the slow approach to asymptotic behavior. Some questions as to the nature of the edge are discussed. 
Martin M. Block, Loyal Durand, Francis Halzen, Leo Stodolsky, and   Thomas J. Weiler
hep-ph