Analytic properties of unitarization schemes

Saturation is now a very popular term, but it has a very wide meaning. It includes shadowing and antishadowing processes, the saturation of the FroissartMartin bound [1], gluon merging at small x and geometrical scaling, unitarization effects, etc. In the S-matrix language, saturation means that we reach the maximum possible scattering. This happens for a specific value of the impact parameter, and is directly connected with the unitary property of the scattering amplitude. It is that last meaning that we shall use in our work. The most important results on the energy dependence of diffractive hadronic scattering were obtained from first principles (analyticity, unitarity and Lorentz invariance), which lead to specific analytic forms for the scattering amplitude as a function of its kinematical parameters − s, t, and u. Analytic S-matrix theory relates the high-energy behaviour of hadronic scattering to the singularities of the scattering amplitude in the complex angular momentum plane. One of the important theorems is the Froissart-Martin bound, which states that the high-energy cross section for the scattering of hadrons is limited by