Soft edge of hadron scattering and minijet models for the total and inelastic pp cross sections at LHC and beyond

We show that the onset and rise of QCD minijets provide the dynamical mechanism behind the appearance of a soft edge in pp collisions around CERN intersecting storage rings (ISR) energies, and thus such a soft edge is built in our minijet model with soft gluon resummation. Here the model is optimized for the LHC at s=7, 8 TeV and predictions made for higher LHC and cosmic ray energies. Further, we provide a phenomenological picture to discuss the breakup of the total cross section into its elastic, uncorrelated, and correlated inelastic pieces in the framework of a one-channel eikonal function.