The Wilson renormalization group for low x physics: Gluon evolution at finite parton density

TL;DR

The paper derives a complete Wilson renormalization group equation for low-x gluon evolution at arbitrary parton density by combining an MV-like effective action with a Wilsonian RG that integrates out fast modes. It computes the critical coefficient functions σ[ρ] and χ[ρ] from fluctuations in a nontrivial background, yielding a functional RG for the color-charge density weight F[ρ] that governs the x-dependence of gluonic observables. In the weak-field limit, the formalism reproduces BFKL dynamics, while in the high-density regime it sets the stage for nonlinear saturation through a functional evolution. The work lays groundwork for understanding saturation, fixed points, and potential extensions to hadron-hadron collisions within a two-dimensional effective theory framework for low-x QCD.

Abstract

We derive the complete Wilson renormalization group equation which governs the evolution of the gluon distribution and other gluonic observables at low $x$ and arbitrary density.

Figures (9)

• Figure 1: Diagram contributing to the gluon distribution at lowest order in $\alpha_s$ (but to all orders in $\rho$) (a) and a typical order $\alpha_s$ correction (b). The horizontal line represents a propagator in the presence of the full, $\rho$-induced background field
• Figure 2: The small $\rho$ limit of the diagrams shown in Fig. \ref{['fig:distcorr']}: The lowest order terms shown in (a) correspond to those in Eq.(\ref{['relation']}). (b) is the small $\rho$ limit of Fig.\ref{['fig:distcorr']}b with all gluon propagators perturbative.
• Figure 3: Original vertex (a) and vertex modified by the emission of an additional fast gluon (b). "$+$" components of momenta are ordered from top to bottom
• Figure 4: Diagrammatic representation of $\delta J_1$ in terms of classical and fluctuation fields. The coupling to a $\delta A_\mu$ field has been indicated by a curly line whereas slow modes $a_\mu$ have been symbolized by dashed ones.
• Figure 5: Diagrammatic representation of $\delta J_2$. Symbols as in Fig.\ref{['fig:deltaJ1']}