Universal features of JIMWLK and BK evolution at small x

TL;DR

The study numerically probes universal features of small-$x$ QCD evolution by implementing JIMWLK and BK equations on a lattice, focusing on IR safety, scaling, and phase-space structure. It demonstrates that both equations exhibit a saturation-driven, nearly scale-invariant approach toward a universal form, with the saturation scale $Q_s( au)$ governing the dynamics. A central finding is that running coupling is essential to control UV diffusion and realize reliable continuum limits, while the resulting evolution remains approximately geometric in form and shows rapid memory loss of initial $A$-dependence. The results establish strong quantitative agreement between JIMWLK and BK under chosen initial conditions and highlight the need for improved running-coupling implementations to refine the $ au$-dependence of $Q_s$.

Abstract

In this paper we present the results of numerical studies of the JIMWLK and BK equations with a particular emphasis on the universal scaling properties and phase space structure involved. The results are valid for near zero impact parameter in DIS. We demonstrate IR safety due to the occurrence of a rapidity dependent saturation scale Q_s(τ). Within the set of initial conditions chosen both JIMWLK and BK equations show remarkable agreement. We point out the crucial importance of running coupling corrections to obtain consistency in the UV. Despite the scale breaking induced by the running coupling we find that evolution drives correlators towards an asymptotic form with near scaling properties. We discuss asymptotic features of the evolution, such as the τ- and A-dependence of Q_s away from the initial condition.

Figures (19)

• Figure 1: Gluon distributions grow towards small $x$. To the left: qualitative growth as from typical DGLAP fits of DIS at HERA ($e A$) for different $Q^2$. To the right: Growth of gluon (parton) numbers with $Q^2$ and $\ln(1/x)$. Going to sufficiently small $x$ at fixed $Q^2$ partons start to overlap.
• Figure 2: $q\Bar q$ pairs interacting with a nuclear target at small $x$. (a) shows the situation in the targets infinite momentum frame with fully Lorentz-contracted target fields, (b) shows the situation in the target rest frame in which the $\gamma^* q\Bar q$ vertex is far outside the target at a distance proportional to $1/x = e^\tau$.
• Figure 3: RG corrections to the average over background field configurations. The shaded areas contain the contributions subsumed into the averaging procedure at successive values of $\tau=\ln(1/x)$.
• Figure 4: Generic evolution trend for a one scale dipole correlator $N_{\bm{x y}}$ plotted against $\vert \bm{x}-\bm{y}\vert$ as $x\to 0$: the curves move towards the left as $\tau=\ln(1/x)$ increases, the asymptotic form being a step function at the origin
• Figure 5: Evolution of the dipole operator from the JIMWLK equation on an $256^2$ lattice, shown on a linear (left) and logarithmic (right) distance scale. The general pattern as predicted from the existence of the fixed point and drawn qualitatively in Fig. \ref{['fig:generic-evol']} shows up clearly. On the right we see a scaling form emerge: a stable shape of the curve that merely shifts to the left unaltered.