On solutions of the Balitsky-Kovchegov equation with impact parameter

TL;DR

The paper numerically solves the BK equation with full impact-parameter dependence and demonstrates that initial b-profile generation of a finite saturation region in dipole size persists only up to two scales, $Q_s(b,Y)$ and $R_H(b,Y)$. It uncovers a power-like tail in impact parameter driven by long-range kernel contributions, leading to exponential growth of the cross section with rapidity and a violation of the Froissart bound, highlighting the absence of confinement effects in the BK framework. The results show that geometric scaling survives only locally in $b$-dependent form, while the evolution broadens the interaction region and induces nontrivial angular dependence for large dipoles. The study thus argues that confinement must be incorporated into the evolution kernel to enforce unitarity at asymptotically high energies.

Abstract

We numerically analyze the Balitsky-Kovchegov equation with the full dependence on impact parameter b. We show that due to a particular b-dependence of the initial condition the amplitude decreases for large dipole sizes r. Thus the region of saturation has a finite extension in the dipole size r, and its width increases with rapidity. We also calculate the b-dependent saturation scale and discuss limitations on geometric scaling. We demonstrate the instant emergence of the power-like tail in impact parameter, which is due to the long range contributions. Thus the resulting cross section violates the Froissart bound despite the presence of a nonlinear term responsible for saturation.

Figures (9)

• Figure 1: The triangle geometry in Eq. (\ref{['eq:BK']}). The points ${\bf x},{\bf y},{\bf z}$ are the dipole ends in the transverse space, and the vectors ${\bf b}_{{\bf x}{\bf z}},{\bf b}_{{\bf x}{\bf y}},{\bf b}_{{\bf z}{\bf y}}$ are the impact parameters of the three dipoles.
• Figure 2: Dipole degrees of freedom: the dipole vector ${\bf r}$ and the impact parameter ${\bf b}$.
• Figure 3: The amplitude $N(r,b,\theta;Y)$ as a function of dipole size $r$ for indicated values of rapidity $Y$ and two values of the fixed impact parameter: $b=0.2$ (left) and $b=5$ (right). The orientation of the dipole $\cos\theta=0$. The red-dashed line is the input distribution (\ref{['eq:GM']}) with profile (\ref{['eq:Sb']}).
• Figure 4: The dependence of the saturation scale $Q_s(b,Y)$ on impact parameter for two fixed values of rapidity. For the comparison the saturation scale at initial rapidity (\ref{['eq:SatIni']}) is shown by the dashed line, normalised to the value of $Q_s(b,Y)$ at $b=0.1$.
• Figure 5: Geometric scaling for $b=0.2$ obtained after rescaling $r$ by the rapidity dependence of the saturation scale $Q_s(b,Y)$. Dotted, dashed and solid lines are for $Y=11,9,7$ respectively. The orientation of the dipole has been fixed such that $\cos \theta=0$.