Proton-proton cross-sections: the interplay between density and radius
  We argue that there are two mechanisms responsable for the growth of high energy cross-sections - \sigma_{tot} and \sigma_{el}, say - in pp collisions. One is by the increase of matter \emph{density}, resulting in the black disk saturation. The other is by exclusively radial expansion affecting the periphery of the overlap area. Within this simple model we can reproduce all available data in pp from ISR to LHC. In order to achieve a fast growth in the very high energy cosmic ray energies, we propose a fast black disk saturation followed by the dominance of geometric scaling (GS). 
R. Concei\c{c}\~ao, J. Dias de Deus, M. Pimenta
hep-ph