Solution to the evolution equation for high parton density QCD

TL;DR

The paper tackles the nonlinear evolution of parton densities in high-density QCD by solving the GLR-type equation within the color-dipole framework. It develops a two-region analytic solution: a generating-function solution to the right of the critical line and a z-variable solution to the left, with careful matching on the critical line to ensure a consistent description over all x. Key results include explicit expressions for the saturation scale Q_cr(x,b_t), the behavior of the dipole density N and gluon distribution xG_A across regions, and a scaling relation connecting nucleus and nucleon DIS observables. These findings clarify saturation dynamics, reveal how the mean transverse momentum is set by Q_cr at low x, and provide analytic predictions relevant for heavy-ion phenomenology and future DIS studies.

Abstract

In this paper a solution is given to the nonlinear equation which describes the evolution of the parton cascade in the case of the high parton density. The related physics is discussed as well as some applications to heavy ion-ion collisions.

Figures (5)

• Figure 1: Different regions for the solution to the hdQCD evolution equation in the kinematic plot of DIS. The equation $N({\mathbf{x}},b_t,y) \,=\,1$ gives the critical line.
• Figure 2: Pictorial representation of the nonlinear evolution equation.
• Figure 3: "Fan" ( Fig. 3a ) and enhanced ( Fig. 3b ) diagrams.
• Figure 4: Dipole number density $N(z)$ as a function of the critical line parameter $z$. Solid line is the numerical solution with $\tilde{N}'(0)=0.01$, while dashed line corresponds to $\tilde{N}'(0)=0.4$. The bold dashed line shows the asymptotic solution of Eq. (\ref{['ZETAN']}) with $\tilde{N}'(0)=0.4$.
• Figure 5: The ratio of the dipole cross section to the geometrical estimates $\frac{\sigma^A_{dipole}}{\pi R^2_A}$ for low $x = 10^{-4}$ ( $y = \ln(0.01/x)$ ) as a function of $Q^2$ for different nuclei. All other curves correspond to A=40, 150, 300 going from the bottom to the top; $\alpha_S = 0.25, Q^2_0 = 1\,GeV^2$ .