A zero-dimensional model for high-energy scattering in QCD

We investigate a zero-dimensional toy model originally introduced by Mueller and Salam which mimics high-energy scattering in QCD in the presence of both gluon saturation and gluon number fluctuations, and hence of Pomeron loops. Unlike other toy models of the reaction-diffusion type, the model studied in this paper is consistent with boost invariance and, related to that, it exhibits a mechanism for particle saturation close to that of the JIMWLK equation in QCD, namely the saturation of the emission rate due to high-density effects. Within this model, we establish the dominant high-energy behaviour of the S-matrix element <S^n> for the scattering between a target obtained by evolving one particle and a projectile made with exactly n particles. Remarkably, we find that all such matrix elements approach the black disk limit S=0 at high rapidity Y, with the same exponential law: <S^n> ~ exp(-Y) for all values of n. This is so because the S-matrix is dominated by rare target configurations which involve only few particles. We also find that the bulk distribution for a saturated system is of the Poisson type.