Scaling phenomena from non-linear evolution in high energy DIS

TL;DR

The paper investigates geometric scaling in high-energy deep inelastic scattering arising from nonlinear evolution in the color-dipole framework. By numerically solving a BK-like equation with fixed coupling and Glauber–Mueller initial conditions, it demonstrates that the dipole amplitude exhibits scaling with the variable $\tau=Q^2/Q_s^2(x)$ and extracts the saturation scale $Q_s(x)$, finding $Q_s(x)\sim Q_{s0}x^{-q}$ with $q\approx 0.35$ for protons and $q_N\approx 0.32$ for nuclei. The analysis shows scaling remains robust across a broad $x$ and $Q^2$ range, extends into parts of the linear regime with ~10% accuracy, and reveals an A-dependent but universal-like energy scaling $Q_{s}(x)=Q_{s0}(A)x^{-q_N}$ with $q_N$ close to Braunn2’s value of $2/9$. These results provide a coherent picture of saturation in high-density QCD and yield predictions for nuclear targets relevant to RHIC and future colliders.

Abstract

The numerical solutions of the non-linear evolution equation are shown to display the ``geometric'' scaling recently discovered in the experimental data. The phenomena hold both for proton and nucleus targets for all $x$ below $10^{-2}$ and $0.25 {\rm GeV^{2}}\le Q^2 \le 2.5\times10^3 {\rm GeV^{2}}$. The scaling is practically exact (few percent error) in the saturation region. In addition, an approximate scaling is found in the validity domain of the linear evolution where it holds with about 10% accuracy. Basing on the scaling phenomena we determine the saturation scale $Q_s(x)$ and study both its $x$-dependence and the atomic number dependence for the nuclei.

Figures (7)

• Figure 1: Solutions of the equation (\ref{['EQ']}) as a function of distance. The different curves correspond to solutions at $x=10^{-7}$ (the upper curve), $10^{-6}$ and so on down to $x=10^{-2}$ (the lowest curve)
• Figure 2: The derivative functions $Nr$ (dashed line) and $Ny$ (solid line) as functions of the distance $r_\perp$.
• Figure 3: The scaling as a function of the distance $r_\perp$. The positive curves are $Nr/Nr_{min}$ (dashed line) and $Ny/Ny_{min}$ (solid line). The dotted line is $20\times Ra$.
• Figure 4: The saturation scale $Q_s$ is plotted as a function of $x$. (a) - the scales obtained from the equations (\ref{['scale1']}) (solid line) and (\ref{['scale2']}) (dashed line), (b) - the equation (\ref{['Qs3']}) is used to determine the scale. (c) - the result obtained from the equation (\ref{['def4']}).
• Figure 5: The scaling as a function of distance the $r_\perp$). The positive curves are $Nr_{Au}/Nr_{Au\,min}$ and $Ny_{Au}/Ny_{Au\,min}$. The dotted line is $20\times Ra_{Au}$.