On the Probabilistic Interpretation of the Evolution Equations with Pomeron Loops in QCD

We study some structural aspects of the evolution equations with Pomeron loops recently derived in QCD at high energy and for a large number of colors, with the purpose of clarifying their probabilistic interpretation. We show that, in spite of their appealing dipolar structure and of the self–duality of the underlying Hamiltonian, these equations cannot be given a meaningful interpretation in terms of a system of dipoles which evolves through dissociation (one dipole splitting into two) and recombination (two dipoles merging into one). The problem comes from the saturation effects, which cannot be described as dipole recombination, not even effectively. We establish this by showing that a (probabilistically meaningful) dipolar evolution in either the target or the projectile wavefunction cannot reproduce the actual evolution equations in QCD. 1 Membre du Centre National de la Recherche Scientifique (CNRS), France. 2 On leave from the Fundamental Theoretical Physics group of the University of Liège. 3 Present address: ECT, Villa Tambosi, Strada delle Tabarelle 286, I-38050 Villazzano (TN), Italy Preprint submitted to Elsevier Science 2 February 2008