Unitarization of Gluon Distribution in the Doubly Logarithmic Regime at High Density

TL;DR

The paper tackles unitarization of perturbative gluon growth at high energies by deriving nonlinear evolution equations for multi-gluon correlators within a Wilson renormalization-group framework and reformulating them in terms of chromoelectric-field correlators. In the doubly logarithmic regime, the evolution becomes local in transverse momentum and reveals a dynamically generated mass that slows evolution, yielding logarithmic growth of the gluon density at small impact parameter and more rapid growth peripherally. A Gaussian truncation is introduced to obtain a closed, local equation for the gluon density xg(x,Q,b_perp), providing a tractable bridge between weak- and high-density regimes and connecting to other saturation formalisms. The results illuminate a perturbative pathway to unitarization and offer a framework to study the crossover between DGLAP-like evolution and high-density nonlinear dynamics, with potential relevance for DIS and high-energy hadronic scattering.

Abstract

We analyze the general nonlinear evolution equations for multi gluon correlators derived in hep-ph/9709432 by restricting ourselves to a double logarithmic region. In this region our evolution equation becomes local in transverse momentum space and amenable to simple analysis. It provides a complete nonlinear generalization of the GLR equation. We find that the full double log evolution at high density becomes strikingly different from its linear doubly logarithmic DGLAP counterpart. An effective mass is induced by the nonlinear corrections which at high densities slows down the evolution considerably. We show that at small values of impact parameter the gluonic density grows as a logarithm of energy. At higher values of impact parameter the growth is faster, since the density of gluons is lower and nonlinearities are less important.

Figures (2)

• Figure 1: The diagrams contributing to the chromoelectric field at order $\alpha_s$. The full circles denote the background charge density $\rho$. The empty circles denote the average of the charge density induced by the fluctuations $<\delta\rho>=\sigma$. The black bars denote contractions corresponding to the mean fluctuation of the induced charge density $<\delta\rho\delta\rho>=\chi$.
• Figure 2: The average fluctuation of the induced chromoelectric field $\chi^{ab}_{ij}$. The notations are the same as in Fig.1.