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Nonlinear evolution in high-density QCD

I. I. Balitsky, A. V. Belitsky

TL;DR

The paper develops a Wilson-line-based framework to study Deeply Inelastic Scattering at small x in the saturation regime, formulating the dipole evolution in a shock-wave background. It derives a generalized nonlinear evolution equation for dipole densities in the large-Nc limit, with kernels K1 (BFKL), K2 (quadratic saturation), and K3 (cubic nonlinearities); the cubic kernel K3 is computed by a meticulous two-loop diagrammatic analysis across self-energy, vertex, and box topologies, including color-algebra reduction and subtraction of multi-Regge kinematics. The results demonstrate unitarity constraints by showing K3 vanishes in the y→x limit and indicate cancellations of log-free parts, underscoring the consistency of the saturation framework beyond the standard BK equation. The work lays groundwork for numerical studies of saturation with cubic nonlinearities and foreshadows extensions to running coupling, 1/Nc corrections, and derivations from the dipole picture.

Abstract

We consider deeply inelastic scattering at very high energies in the saturation regime. The emerging picture corresponds to the propagation of a dipole, the quark-antiquark pair, in a shock wave color field of the target. We use the fomalism of Wilson lines to study the evolution of dipole densities in energy logarithms. Our analysis results into an equation in multicolor limit which sums leading logs but keeps the nonlinearities up to cubic order in densities.

Nonlinear evolution in high-density QCD

TL;DR

The paper develops a Wilson-line-based framework to study Deeply Inelastic Scattering at small x in the saturation regime, formulating the dipole evolution in a shock-wave background. It derives a generalized nonlinear evolution equation for dipole densities in the large-Nc limit, with kernels K1 (BFKL), K2 (quadratic saturation), and K3 (cubic nonlinearities); the cubic kernel K3 is computed by a meticulous two-loop diagrammatic analysis across self-energy, vertex, and box topologies, including color-algebra reduction and subtraction of multi-Regge kinematics. The results demonstrate unitarity constraints by showing K3 vanishes in the y→x limit and indicate cancellations of log-free parts, underscoring the consistency of the saturation framework beyond the standard BK equation. The work lays groundwork for numerical studies of saturation with cubic nonlinearities and foreshadows extensions to running coupling, 1/Nc corrections, and derivations from the dipole picture.

Abstract

We consider deeply inelastic scattering at very high energies in the saturation regime. The emerging picture corresponds to the propagation of a dipole, the quark-antiquark pair, in a shock wave color field of the target. We use the fomalism of Wilson lines to study the evolution of dipole densities in energy logarithms. Our analysis results into an equation in multicolor limit which sums leading logs but keeps the nonlinearities up to cubic order in densities.

Paper Structure

This paper contains 16 sections, 110 equations, 3 figures.

Table of Contents

  1. QCD of dense parton systems.
  2. High-energy limit and shock-wave.
  3. Propagators in the shock-wave background.
  4. Nonlinear evolution equation.
  5. Leading nonlinearities.
  6. Next-to-leading nonlinearities.
  7. Preliminaries.
  8. Self-energy topology.
  9. Vertex topology.
  10. Box topology.
  11. Color algebra.
  12. Subtraction of multi-Regge kinematics.
  13. Evolution kernel ${\cal K}_3$.
  14. Conclusions.
  15. Calculation of diagrams with nonabelian vertices.
  16. ...and 1 more sections

Figures (3)

  • Figure 1: Scattering of the photon probe off the hadron at high energies. The virtual quantum fluctuates into a bunch of dipoles which interact with the target.
  • Figure 2: Leading order diagrams which result into BFKL equation and the first nonlinear correction to the generalized nonlinear equation.
  • Figure 3: Samples of diagrams of different topologies inducing the next-to-leading nonlinearity, cubic in dipole densities, in the generalized nonlinear equation.