A Langevin equation for high-energy evolution with pomeron loops

Abstract

We show that the Balitsky-JIMWLK equations proposed to describe non-linear evolution in QCD at high energy fail to include the effects of fluctuations in the gluon number, and thus to correctly describe both the low density regime and the approach towards saturation. On the other hand, these fluctuations are correctly encoded (in the limit where the number of colors is large) in Mueller's color dipole picture, which however neglects saturation. By combining the dipole picture at low density with the JIMWLK evolution at high density, we construct a generalization of the Balitsky hierarchy which includes the particle number fluctuations, and thus the pomeron loops. After an additional coarse-graining in impact parameter space, this hierarchy is shown to reduce to a Langevin equation in the universality class of the stochastic Fisher-Kolmogorov-Petrovsky-Piscounov (sFKPP) equation. This equation implies that the non-linear effects in the evolution become important already in the high momentum regime where the average density is small, which signals the breakdown of the BFKL approximation.

On the probabilistic interpretation of the evolution equations with pomeron loops in QCD

“…In this section, we shall briefly recall the evolution equations for dipole scattering amplitudes in QCD at high energy and large N c , together with their physical interpretation in terms of splitting and merging processes in either the target or the projectile [4,9,10].…”

“…However, as pointed out in Refs. [2][3][4], the fluctuation terms are in fact leading-order effects whenever the target is so dilute (or, equivalently, the external dipole (x, y) is so small) that T (x, y) < ∼ α 2 s . Note also that T = 1 is a fixed point of the evolution described by these equations 6 , which physically corresponds to the 'black disk' limit at high energy.…”

“…Over the last year, an intense activity in the field of high-energy QCD has been triggered by the observations that (i) the gluon number fluctuations at low energy play an important role in the evolution towards gluon saturation with increasing energy [1][2][3], and (ii) the relevant fluctuations are in fact missed [4] by the existing approaches to nonlinear evolution in QCD at high energy, namely, the Balitsky-JIMWLK equations [5][6][7][8].…”

“…Specifically, the nonlinear terms responsible for unitarity corrections can be seen as dipole splitting in the wavefunction of the projectile [5,27,28,35], whereas the fluctuation terms relevant at low energy represent dipole splitting in the wavefunction of the target [3,4,22] (see Sect. 2 below for details).…”

“…More generally, this would establish an explicit correspondence between high-energy evolution in QCD and some Markovian "reactiondiffusion processes" which are intensively studied in the context of statistical physics (see the textbook [36], and the review papers [37] for more recent developments), from where one could borrow some methods to QCD. Such a correspondence has been already proven to be useful, both at the level of the mean field approximation (the BK equation) [38], where it shed a new light on the 'geometric scaling' property of the solution [39][40][41][42], and in analyzing the role of the particle number fluctuations under asymptotic conditions (very high energy and arbitrarily small coupling constant) [3,4]. But, clearly, it would be extremely useful to extend this correspondence to a more interesting, non-asymptotic, regime, and rely on it in order to translate to QCD the huge experience accumulated with this problem in the context of statistical physics.…”