Proton-Proton On Shell Optical Potential at High Energies and the   Hollownes Effect
  We analyze the usefulness of the optical potential as suggested by the double spectral Mandelstam representation at very high energies, such as in the proton-proton scattering at ISR and the LHC. Its particular meaning regarding the interpretation of the scattering data up to the maximum available measured energies is discussed. Our analysis reconstructs 3D dynamics from the effective transverse 2D impact parameter representation and suggests that besides the onset of gray nucleons at the LHC there appears an inelasticity depletion (hollowness) which precludes convolution models at the attometer scale. 
Enrique Ruiz Arriola, Wojciech Broniowski
hep-ph nucl-th