Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Langevin equation for high energy evolution with pomeron loops

E. Iancu, D. N. Triantafyllopoulos

TL;DR

This work identifies a fundamental gap in the Balitsky–JIMWLK framework: the omission of gluon-number fluctuations prevents a correct description of both dilute and saturation regimes. By merging Mueller's dipole picture at low density with CGC/JIMWLK at high density, the authors derive a fluctuating hierarchy that generates pomeron loops and, after impact-parameter coarse-graining, a Langevin equation in the sFKPP universality class. The stochastic BK-like evolution reveals that fluctuations seed high-momentum, low-density regions and ultimately drive non-linear effects earlier than predicted by BFKL, signaling a breakdown of the mean-field approximation. These results establish a concrete link between high-energy QCD evolution and stochastic reaction-diffusion systems, offering a framework to study fluctuations, saturation, and their interplay in a unified setting.

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.

A Langevin equation for high energy evolution with pomeron loops

TL;DR

This work identifies a fundamental gap in the Balitsky–JIMWLK framework: the omission of gluon-number fluctuations prevents a correct description of both dilute and saturation regimes. By merging Mueller's dipole picture at low density with CGC/JIMWLK at high density, the authors derive a fluctuating hierarchy that generates pomeron loops and, after impact-parameter coarse-graining, a Langevin equation in the sFKPP universality class. The stochastic BK-like evolution reveals that fluctuations seed high-momentum, low-density regions and ultimately drive non-linear effects earlier than predicted by BFKL, signaling a breakdown of the mean-field approximation. These results establish a concrete link between high-energy QCD evolution and stochastic reaction-diffusion systems, offering a framework to study fluctuations, saturation, and their interplay in a unified setting.

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.

Paper Structure

This paper contains 14 sections, 99 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Physical motivation
  3. The Balitsky--JIMWLK equations and beyond
  4. A toy model borrowed from statistical physics
  5. Fluctuations and evolution in the dipole picture
  6. Fluctuations and saturation in dipole--CGC scattering
  7. From dipole densities to scattering amplitudes
  8. The equations for the scattering amplitudes
  9. The Langevin equation
  10. Physical discussion
  11. Relation with the sFKPP equation
  12. Some results from sFKPP equation and their consequences for QCD
  13. Front diffusion and the breakdown of the BFKL approximation
  14. A one--dimensional reaction-diffusion model

Figures (10)

  • Figure 1: The geometry of dipole splitting
  • Figure 2: Diagrams for single dipole scattering: (a) the tree--level contribution; (b,c,d) one step evolution of the projectile; (e,f) one step evolution of the target.
  • Figure 3: A typical gluon cascade which contributes to a classical color field configuration in the color glass condensate
  • Figure 4: Diagrams for dipole pair scattering: (a) tree--level contribution; (b, c) one step projectile evolution; (d, e) one step target evolution; (f) a diagram suppressed at large $N_c$; (g) the missing diagram, which is of leading order in both $\alpha_s$ and $N_c$.
  • Figure 5: The one--step evolution of the average dipole number density as described by Eq. (\ref{['evolnumber']}).
  • ...and 5 more figures