The BFKL Pomeron Calculus in the dipole approach

Abstract

In this paper we continue to pursue a goal of finding an effective theory for high energy interaction in QCD based on the colour dipole approach, for which the BFKL Pomeron Calculus gives a low energy limit. The key problem, that we try to solve in this paper is the probabilistic interpretation of the BFKL Pomeron Calculus in terms of the colourless dipoles and their interactions. We demonstrate that the BFKL Pomeron Calculus has two equivalent descriptions : (i) one is the generating functional which gives a clear probabilistic interpretation of the processes of high energy scattering and also provides a Hamiltonian-like description of the system of interacting dipoles; (ii) the second is the Langevin equation with a specific noise term which is rather complicated. We found that at high energies this Langevin equation can be reduced to the Langevin equation for directed percolation in the momentum space if the impact parameter is large, namely, $b>> 1/k$, where $k$ is the transverse momentum of a dipole. Unfortunately, this simplified form of Langevin equation is not applicable for summation of Pomeron loops, where one integrates over all possible values of impact parameter. We show that the BFKL Pomeron calculus with two vertices (splitting $P\to P+P$ and merging $P+P\to P$ of Pomerons) can be interpreted as a system of colourless dipoles with two processes: the decay of one dipole into two and the merging of two dipoles into one dipole. However, a number of assumptions we have to make on the way to simplify the noise term in the Langevin equation and/or to apply the probabilistic interpretation, therefore, we can consider both of these approaches in the present form only as the QCD motivated models.