A Gaussian effective theory for gluon saturation

TL;DR

The paper develops a Gaussian approximation to the Colour Glass Condensate effective theory, producing a tractable weight functional for color sources that interpolates between the BFKL regime at high transverse momentum and gluon saturation at low momentum. This approach yields infrared-finite, gauge-invariant observables and derives analytic expressions for the unintegrated gluon distribution and dipole-hadron scattering amplitude, exhibiting geometric scaling near the saturation scale $Q_s( au)$. Saturation emerges from color-neutral correlations encoded in the Gaussian kernel, with a simple saturation criterion defining $Q_s$, and the framework reproduces the correct limiting behavior in both dilute and dense regimes. The results offer a practical, analytic tool for CGC phenomenology in DIS and heavy-ion collisions and provide a clean benchmark for more complete BK/JIMWLK evolutions and lattice studies.

Abstract

We construct a Gaussian approximation to the effective theory for the Colour Glass Condensate which describes correctly the gluon distribution both in the low density regime at high transverse momenta (above the saturation scale $Q_s$), and in the high density regime below $Q_s$, and provides a simple interpolation between these two regimes. At high momenta, the effective theory reproduces the BFKL dynamics, while at low momenta, it exhibits gluon saturation and, related to it, colour neutrality over the short distance scale $1/Q_s \ll 1/Λ_{QCD}$. Gauge--invariant quantities computed within this approximation are automatically infrared finite.

Figures (7)

• Figure 1: Momentum dependence of $\lambda^{\rm red}_{\rm y}(k_\perp)= (\pi/k_\perp^2)\bar{\lambda}_{\rm y}(k_\perp)$, plotted as a function of $k_\perp/Q_s(\tau)$ for two values of $\gamma$. Solid (red) line: $\gamma=0.64$, and dashed (green) line $\gamma=1$.
• Figure 2: Longitudinal structure of $\lambda^{\rm red}_{\rm y}(k_\perp)= (\pi/k_\perp^2)\bar{\lambda}_{\rm y}(k_\perp)$. This is plotted as a function of y for two transverse momenta: $k_\perp^{\rm high}=3 \, Q_s^2(\tau)$ (green line, dashed), and $k_\perp^{\rm low}=0.02\, Q_s^2(\tau)$ (red line, solid). We have used $\gamma=0.64$, $c=4.84$, $\bar{\alpha}_s=0.2$, and $\tau=10$. The corresponding separation rapidities are $\tau_s(k_\perp^{\rm high})\simeq 11$ and $\tau_s(k_\perp^{\rm low})\simeq 5$.
• Figure 3: $\bar{\varphi}_\tau(k_\perp)$ as a function of $k_\perp/Q_s(\tau)$ in linear scale (red line, solid), compared with its high momentum approximation $(Q_s^2(\tau)/k_\perp^2)^{\gamma}$ (green line, dashed). We have used $\bar{\alpha}_s=0.1$ and $\gamma=0.64$.
• Figure 4: $\bar{\varphi}_\tau(k_\perp)$ as a function of $k_\perp/Q_s(\tau)$ in log--log scale for two values of $\gamma$ : $\gamma=0.64$ (red line, thick solid), and $\gamma=1$ (green line, thick dashed). Comparison is made with $1/k_\perp^{2\gamma},\ \gamma=0.64$ (magenta line, thin dotted), $1/k_\perp^2$ (blue line, thin dashed), and $(1/\bar{\alpha}_s)\ln( Q_s^2(\tau)/k_\perp^2)$ (cyan line, thin dot--dashed).
• Figure 5: Energy dependence of $\bar{\varphi}_\tau(k_\perp)$ for $\gamma=0.64$. We have plotted $\varphi_{\tau}(k_\perp)$ as a function of $k_\perp/Q_s(\tau_0)$ (with $\tau_0$ some value of reference) for six values of $\tau$. The lines, from the bottom to the top, correspond successively to $\tau=\tau_0,\, \tau_0 + 2,\, \cdots,\, \tau_0+10.$ The increase with $\tau$ is exponential at high momenta (giving equidistant curves in this log--log plot), but only logarithmic at low momenta.