Traveling wave fronts and the transition to saturation

TL;DR

This paper develops a traveling-wave framework to analyze nonlinear QCD evolution and saturation in high-energy scattering, applying it to the Balitsky-Kovchegov equation with both fixed and running couplings. By mapping BK dynamics to pulled-front propagation, it derives the saturation scale $Q_s(Y)$ and near-front gluon densities, including subleading rapidity corrections. For fixed coupling, saturation is approached via diffusive front formation with a Gaussian in $\log(k^2/Q_s^2)$; for running coupling, the approach is governed by anomalous diffusion, with an Airy-front profile and a sqrt($Y$) growth of $Q_s$. The results reproduce known leading terms and provide a unified, physically transparent method with potential extensions to full saturation dynamics.

Abstract

We propose a general method to study the solutions to nonlinear QCD evolution equations, based on a deep analogy with the physics of traveling waves. In particular, we show that the transition to the saturation regime of high energy QCD is identical to the formation of the front of a traveling wave. Within this physical picture, we provide the expressions for the saturation scale and the gluon density profile as a function of the total rapidity and the transverse momentum. The application to the Balitsky-Kovchegov equation for both fixed and running coupling constants confirms the effectiveness of this method.

Figures (3)

• Figure 1: The front velocity as a function of the steepness $\gamma_0$ of the initial condition at $x\rightarrow\infty$ ($u(x,t\!=\!0)\sim e ^{-\gamma_0 x}$). Continuous line: the actual front velocity. Dashed line: the phase velocity for a wave with wave number $\gamma_0$. The abscissa points $\gamma_-$, $\gamma_+$, $\gamma_c$ correspond to initial conditions resp. below, larger than and equal to the critical value (resp. cases (i), (ii) and (iii), see text).
• Figure 2: Picture of the evolution of a front for three typical initial conditions. Dashed line: the initial profile $u(x,t=0)$. Full line: the evolved front after at large time $t$. Upper plot (case (i)): the exponential slope $\gamma_0$ of the initial condition is $\gamma_-<\gamma_c$. The front propagates at the phase velocity $v_-=v_\varphi(\gamma_-)$. Middle plot (case (iii)): $\gamma_0=\gamma_c$. The front propagates at velocity $v_g$. Lower plot (case (ii)): $\gamma_0=\gamma_+>\gamma_c$. The front still propagates at velocity $v_g$, the size of the front is finite of order $t^\alpha$, and its slope is given by $\gamma_c$.
• Figure 3: Evolution of the reduced front profile in the case of fixed coupling (left) and running coupling (right). The reduced front profile $(k^2/Q_s^2)^{\gamma_c}\,{\cal N}(k/Q_s(Y),Y)$ is plotted against $\log(k^2/Q_s^2)$ for different rapidities. The various lines correspond to rapidities from 2 (lower curves, full line) up to 10 (upper curves). Note the similarity of the wave fronts, but the quicker time evolution (in $\sqrt{t}$) for fixed coupling, by contrast with the slow time evolution (in $t^{1/3}$) for the running coupling case.