One–dimensional model for QCD at high energy

We propose a stochastic particle model in (1+1)–dimensions, with one dimension corresponding to rapidity and the other one to the transverse size of a dipole in QCD, which mimics high–energy evolution and scattering in QCD in the presence of both saturation and particle–number fluctuations, and hence of Pomeron loops. The model evolves via non–linear particle splitting, with a non–local splitting rate which is constrained by boost–invariance and multiple scattering. The splitting rate saturates at high density, so like the gluon emission rate in the JIMWLK evolution. In the mean field approximation obtained by ignoring fluctuations, the model exhibits the hallmarks of the BK equation, namely a BFKL–like evolution at low density, the formation of a traveling wave, and geometric scaling. In the full evolution including fluctuations, the geometric scaling is washed out at high energy and replaced by diffusive scaling. It is likely that the model belongs to the universality class of the reaction–diffusion process. The analysis of the model sheds new light on the Pomeron loops equations in QCD and their possible improvements. 1 E-mail addresses: [email protected] (E. Iancu), [email protected] (J.T. de Santana Amaral), [email protected] (G. Soyez), [email protected] (D.N. Triantafyllopoulos). 2 Membre du Centre National de la Recherche Scientifique (CNRS), France. 3 On leave from the Fundamental Theoretical Physics group of the University of Liège. Preprint submitted to Elsevier 2 February 2008