QCD Phase Shifts and Rising Total Cross-Sections

Energy dependence of $\gamma^*p$ total cross section is considered. It is known from the HERA data that the cross section grows with energy, and the rate of growth is an increasing function of the virtuality $Q^2$ of the photon $\gamma^*$. This dependence can be explained in a simple and generic way by the Froissart bound. To implement this mechanism quantitatively a theory satisfying $s$-channel unitarity is required. This is achieved by computing the total cross section from the optical theorem, and the forward elastic amplitude from the phase shifts in the impact-parameter representation. A recipe to do so in perturbative QCD is discussed, together with an expose of the advantage to calculate the elastic amplitude via phase shift, rather than direct sum of all Feynman diagrams. A two-loop computation of phase shift and total cross section is presented, and compared with the HERA data with good agreements.