Small-x physics beyond the Kovchegov equation

TL;DR

The paper addresses unitarity-limited high-energy scattering at small $x$ by contrasting the Kovchegov equation with Balitsky–JIMWLK and proposing a two-boundary (two-saturation) framework to incorporate correlations missing in the mean-field approach. It develops and analyzes BFKL evolution in the presence of two absorptive barriers, showing improved frame (boost) invariance and revealing a strong suppression of Kovchegov scaling due to boundary effects, including a modified effective exponent $\lambda_d$. Through diffusion with barriers and saddle-point analyses, the authors derive how the two boundaries alter the saturation momentum and the energy dependence of scattering, highlighting the potential importance of correlations near the saturation boundary. They further extend the framework to running coupling, outlining how $\alpha(\rho)$ modifies the boundary geometry and the diffusion dynamics, and argue for numerical JIMWLK studies to test the relevance of these correlations relative to Kovchegov and two-boundary predictions.

Abstract

We note the differences between the Kovchegov equation and the Balitsky-JIMWLK equations as methods of evaluating high energy hard scattering near the unitarity limit. We attempt to simulate some of the correlations absent in the Kovchegov equation by introducing two boundaries rather than the single boundary which effectively approximates the unitarity limit guaranteed in the Kovchegov equation. We solve the problem of BFKL evolution in the presence of two boundaries and note that the resulting T-matrix now is the same in different frames, which was not the case in the single boundary case. The scaling behavior of the solution to the Kovchegov equation is apparently now lost.

Figures (8)

• Figure 1: BFKL evolution in the presence of one saturation boundary.
• Figure 2: BFKL evolution in the presence of two saturation boundaries.
• Figure 3: Graphs corresponding to terms in the Kovchegov equation.
• Figure 4: The saturation boundaries $Q_c$ and $\tilde{Q}_c$ separate the saturated regime (shaded areas) from the weak interaction region. In (a) dipole $x$ evolves from $y=0$ up to $y=Y$ and scatters on the elementary dipole $x'$ while in (b) dipole $x'$ evolves from $y=0$ up to $y=Y$ and scatters on the elementary dipole $x$.
• Figure 5: BFKL evolution in two successive steps $0 \to Y/2 \to Y$ in the presence of the internal saturation boundaries $Q_c$ and $\hat{Q}_c$.