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

We investigate a zero-dimensional toy model originally introduced by Mueller and Salam [A.H. Mueller, G.P. Salam, Nucl. Phys. B 475 (1996) 293] 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 behavior of the S-matrix element 〈Sn〉 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: 〈Sn〉∼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.