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Nonlinear Gluon Evolution in the Color Glass Condensate: II

Elena Ferreiro, Edmond Iancu, Andrei Leonidov, Larry McLerran

TL;DR

This work completes the renormalization group treatment of the Color Glass Condensate by deriving explicit one-loop real and virtual coefficients for the functional weight W_τ[α], yielding a Hamiltonian (Fokker-Planck-like) evolution in rapidity that encompasses nonlinear small-x dynamics. The authors show that the evolution reduces to the BFKL equation in the weak-field limit and reproduces the Balitsky–Kovchegov (BK) nonlinear equation in the appropriate large-N_c/dipole limit, establishing equivalence with Weigert's functional equation and Balitsky's hierarchy. Their analysis provides a gauge-consistent, all-orders-in-field description of gluon saturation and its impact on high-energy scattering, with potential applications to deep inelastic scattering and heavy-ion collisions. The results pave the way for nonperturbative explorations, including lattice studies and1/N_c corrections, to quantify saturation phenomena and the associated scale Q_s(τ).

Abstract

We complete the construction of the renormalization group equation (RGE) for the Color Glass Condenstate begun in Paper I. This is the equation which governs the evolution with rapidity of the statistical weight function for the color glass field. The coefficients in this equation --- one-loop real and virtual contributions --- are computed explicitly, to all orders in the color glass field. The resulting RGE can be interpreted as the imaginary-time evolution equation, with rapidity as the ``imaginary time'', for a quantum field theory in two spatial dimensions. In the weak field limit it reduces to the BFKL equation. In the general non-linear case, it is equivalent to an equation by Weigert which summarizes in functional form the evolution equations for Wilson line operators previously derived by Balitsky and Kovchegov.

Nonlinear Gluon Evolution in the Color Glass Condensate: II

TL;DR

This work completes the renormalization group treatment of the Color Glass Condensate by deriving explicit one-loop real and virtual coefficients for the functional weight W_τ[α], yielding a Hamiltonian (Fokker-Planck-like) evolution in rapidity that encompasses nonlinear small-x dynamics. The authors show that the evolution reduces to the BFKL equation in the weak-field limit and reproduces the Balitsky–Kovchegov (BK) nonlinear equation in the appropriate large-N_c/dipole limit, establishing equivalence with Weigert's functional equation and Balitsky's hierarchy. Their analysis provides a gauge-consistent, all-orders-in-field description of gluon saturation and its impact on high-energy scattering, with potential applications to deep inelastic scattering and heavy-ion collisions. The results pave the way for nonperturbative explorations, including lattice studies and1/N_c corrections, to quantify saturation phenomena and the associated scale Q_s(τ).

Abstract

We complete the construction of the renormalization group equation (RGE) for the Color Glass Condenstate begun in Paper I. This is the equation which governs the evolution with rapidity of the statistical weight function for the color glass field. The coefficients in this equation --- one-loop real and virtual contributions --- are computed explicitly, to all orders in the color glass field. The resulting RGE can be interpreted as the imaginary-time evolution equation, with rapidity as the ``imaginary time'', for a quantum field theory in two spatial dimensions. In the weak field limit it reduces to the BFKL equation. In the general non-linear case, it is equivalent to an equation by Weigert which summarizes in functional form the evolution equations for Wilson line operators previously derived by Balitsky and Kovchegov.

Paper Structure

This paper contains 27 sections, 196 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The effective theory for the CGC
  3. The quantum evolution of the effective theory
  4. Feynman rules for $\chi$ and $\sigma$
  5. The induced charge-charge correlator, or "real correction"
  6. Non-linear effects and their dependence upon the gauge-fixing prescription
  7. Explicit calculation of $\hat{\chi}(\vec{x}, \vec{y})$
  8. The two-dimensional correlator $\chi(x_\perp,y_\perp)$
  9. From $\chi$ to $\eta$
  10. The induced color charge, of "virtual correction"
  11. Explicit calculation of $\hat{\sigma}(\vec{x})$
  12. From $\sigma$ to $\nu$
  13. The Renormalization Group Equation
  14. General properties
  15. The RGE in Hamiltonian form
  16. ...and 12 more sections

Figures (2)

  • Figure 1: Feynman diagrams for the four contributions to $\chi$ given in eq. (\ref{['chi2']}).
  • Figure 2: Feynman diagrams for $\hat{\sigma}_1$ (a) and $\hat{\sigma}_2$ (b,c,d). The wavy line with a blob denotes the background field propagator of the semi-fast gluons; the continuous line represents the source $\rho$; the precise vertices can be read off eq. (\ref{['rho2']}).