QCD Phase Shifts and Rising Total Cross-Sections

An attempt is made in QCD to explain the growth of total cross-sections with energy, without violating the Froissart bound. This is achieved by computing the phase shifts of elastic scatterings of partons rather than their amplitudes. To render that possible a general formalism of phase-shifts in QCD is developed. Computed to two-loop order, agreements with hadronic and virtual-photon total cross-sections are quite satisfactory. Predications for the slower rate of growth at higher energies are also presented.