Nonlinear Gluon Evolution in the Color Glass Condensate: I

TL;DR

This work develops a principled derivation of nonlinear gluon evolution at small x within the Color Glass Condensate, formulating a functional Fokker-Planck equation for the color-source weight W_τ[ρ] by matching quantum corrections to a classical MV framework. It introduces a covariant-gauge formulation of the color source, a Schwinger-Keldysh contour for the quantum MV model, and a p^− RG flow that resums leading logarithms in rapidity; in the weak-field limit, it reproduces BFKL behavior, while in the saturation regime it sets the stage for nonlinear evolution that reduces to BK at large N_c. The paper also provides explicit constructions of the background-field propagators essential for higher-order calculations and clarifies subtleties related to gauge choices, boundary conditions, and the longitudinal structure of the source. Together, these elements establish a first-principles CGC framework for universal high-energy hadronic dynamics and prepare for the quantitative evaluation of the evolution coefficients in the companion paper.

Abstract

We consider a nonlinear evolution equation recently proposed to describe the small-$x$ hadronic physics in the regime of very high gluon density. This is a functional Fokker-Planck equation in terms of a classical random color source, which represents the color charge density of the partons with large $x$. In the saturation regime of interest, the coefficients of this equation must be known to all orders in the source strength. In this first paper of a series of two, we carefully derive the evolution equation, via a matching between classical and quantum correlations, and set up the framework for the exact background source calculation of its coefficients. We address and clarify many of the subtleties and ambiguities which have plagued past attempts at an explicit construction of this equation. We also introduce the physical interpretation of the saturation regime at small $x$ as a Color Glass Condensate. In the second paper we shall evaluate the expressions derived here, and compare them to known results in various limits.

Figures (7)

• Figure 1: Tree (a) and one-loop (b) contributions to $\chi$ to quadratic order in $\rho$. Wavy lines with a blob denote insertions of the electric background field ${\cal F}^{+i}\,$; the thick internal wavy lines are propagators of the semi-fast gluons; the external dotted lines carry soft momenta, and couple to the fields $\delta A^-$.
• Figure 2: Contributions to the induced current to low orders in $\rho$. (a) A contribution to $\sigma_a\,$; the continuous line represents the classical source $\rho$. (b) A contribution to $\delta{\cal J}^\mu_a\,$; the thick internal line, with momentum $p^+$, is the propagator $\langle a^i a^i \rangle$ of the semi-fast gluons; the other, thiner, internal lines are the propagators $G^{i-}$ and $G^{-i}$ of the soft fields.
• Figure 3: Complex-time contour for the quantum MV model: $C=C_+\cup C_-$.
• Figure 4: A typical $n$-point vertex (here, $n=3$) generated by the action $S_W$. This is non-local in time, but local in the spatial coordinates. The continuous line represents the source $\rho$, while the wavy lines are $A^-$ gluon fields.
• Figure 5: 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']}).