Resummation of projectile-target multiple scatterings and parton saturation

In the framework of a toy model which possesses the main features of QCD in the high-energy limit, we conduct a numerical study of scattering amplitudes constructed from parton splittings and projectile-target multiple interactions, in a way that unitarizes the amplitudes without however explicit saturation in the wave function of the incoming states. This calculation is performed in two different ways. One of these formulations, the closest to field theory, involves the numerical resummation of a factorially divergent series, for which we develop appropriate numerical tools. We accurately compare the properties of the resulting amplitudes with what would be expected if saturation were explicitly included in the evolution of the states. We observe that the amplitudes have similar properties in a small but finite range of rapidity in the beginning of the evolution, as expected. Some of the features of reaction-diffusion processes are already present in that range, even when wave function saturation is left out of the model.