Proton-proton hollowness at the LHC from inverse scattering

Parameterizations of the pp scattering data at the LHC collision energies indicate a hollow in the inelasticity profile of the pp interaction, with less absorption for head-on collisions than at a non-zero impact parameter. We show that some qualitatively unnoticed features may be unveiled by a judicious application of the inverse scattering problem in the eikonal approximation and interpre- tation within an optical potential model. The hollowness effect is magnifed in a 3D picture of the optical potential, and will presumably be enhanced at yet higher energies. Moreover, in 3D it sets in at much smaller energies than at the LHC. We argue that hollowness in the impact parameter is a quantum effect, relying on the build-up of the real part of the eikonal scattering phase and its possible passage through $\pi/2$. We also show that it precludes models of inelastic collisions where inelasticity is obtained by naive folding of partonic densities.