The rise of the proton-(anti)proton total cross section at Tevatron energies and beyond

A dispersion relation analysis of the UA4 result on the real part of the p anti-p elastic scattering amplitude (= 0.24 + or - 0.04) is presented. The interpretation is twofold. Assuming that the pp and p anti-p cross sections are asymptotically identical, a steep rise of the total cross section in the 1 to 4 TeV domain is deduced. If the cross sections are asymptotically different, there is a crossing of the total cross section of pp and p anti-p between ISR and Sp anti-pS energies followed by a steep rise of the difference of the pp and p anti-p total cross sections. It is shown that in both cases this rise can be accounted for by adding an additional term with an energy cut-off to the usual Amaldi parameterization of the total cross section: ln sq (s/scut) in the first case, or ln(s/scuto) in the second case, where sq rt scut lies around 500 GeV and sq rt scuto around 63 GeV. Both quantities can be interpreted as a threshold of a new process. For the first case, a continuous parameterization without a threshold is also proposed with an extra term of the form ln sq (1 + s/s1), where sq rt s1 equals 700 GeV.