An impact parameter representation for soft gluon radiation is applied to
obtain both the initial decrease of the total cross-section (σtot)
for proton-proton collisions as well as the later rise of σtot with
energy for both pp and ppˉ. The non-perturbative soft part of the
eikonal includes only limited low energy gluon emission and leads to the
initial decrease in the proton-proton cross- section. On the other hand, the
rapid rise in the hard, perturbative jet part of the eikonal is tamed into the
experimentally observed mild increase by soft gluon radiation whose maximum
energy rises slowly with energy.Comment: 30 pages, 6 figures. Version accepted for publication in Physical
Review D. Additional section with explanatory material added making the paper
more self contained and two figures changed to have a complete summary of the
available accelerator dat