Numerical analysis of the Balitsky-Kovchegov equation with running coupling: dependence of the saturation scale on nuclear size and rapidity

TL;DR

This study numerically analyzes the Balitsky-Kovchegov equation with a running coupling to quantify how the saturation scale Q_s depends on rapidity Y and nuclear size A. It confirms that running coupling slows the Y-growth of Q_s, preserves a scaling form, and yields a larger small-distance anomalous dimension (γ ≈ 0.85) compared to fixed coupling (γ ≈ 0.65). The Y-dependence follows Q_s^2(Y) ∝ exp(Δ'√(Y+X)) with Δ' ≈ 3.2, and the A-dependence weakens with Y, scaling roughly as 1/√Y at large Y. These results are robust to the running-coupling implementation and initial conditions, highlighting the qualitative differences between fixed and running coupling in BK evolution and informing saturation-phenomenology and future work including impact-parameter effects.

Abstract

We study the effects of including a running coupling constant in high-density QCD evolution. For fixed coupling constant, QCD evolution preserves the initial dependence of the saturation momentum $Q_s$ on the nuclear size $A$ and results in an exponential dependence on rapidity $Y$, $Q^2_s(Y) = Q^2_s(Y_0) \exp{[ \barα_s d (Y-Y_0) ]}$. For the running coupling case, we re-derive analytical estimates for the $A$- and $Y$-dependences of the saturation scale and test them numerically. The $A$-dependence of $Q_s$ vanishes $\propto 1/ \sqrt{Y}$ for large $A$ and $Y$. The $Y$-dependence is reduced to $Q_s^2(Y) \propto \exp{(Δ^\prime\sqrt{Y+X})}$ where we find numerically $Δ^\prime\simeq 3.2$. We study the behaviour of the gluon distribution at large transverse momentum, characterizing it by an anomalous dimension $1-γ$ which we define in a fixed region of small dipole sizes. In contrast to previous analytical work, we find a marked difference between the fixed coupling ($γ\simeq 0.65$) and running coupling ($γ\sim 0.85$) results. Our numerical findings show that both a scaling function depending only on the variable $r Q_s$ and the perturbative double-leading-logarithmic expression, provide equally good descriptions of the numerical solutions for very small $r$-values below the so-called scaling window.

Figures (7)

• Figure 1: Diagrams for gluon emission in the evolution of a dipole and its $N_c\to \infty$ limit.
• Figure 2: Solutions of the BK equation for GBW initial condition (dotted line) for rapidities $Y=6$, 12 and 18 with $\bar{\alpha}_0=0.4$. Left plot: Evolution with fixed ($K0$, solid lines) and running coupling ($K1$, dashed lines). Right plot: evolution with running coupling for kernel modifications $K1$ (solid lines), $K2$ (dashed lines) and $K3$ (dashed-dotted lines).
• Figure 3: Scaling solutions of BK for $Y=0$, 20, 30 and 40 (plots on the left) and $Y=0$, 40, 60 and 80 (plots on the right). Upper-left: evolution for fixed (solid) and running coupling ($K1$, dashed lines) for GBW initial conditions. Upper-right: solutions for the kernels $K1$ (solid), $K2$ (dashed) and $K3$ (dashed-dotted lines). Lower-left: scaling function for $K1$ with two different values of frozen coupling, $\bar{\alpha}_0=0.4$ (solid) and $\bar{\alpha}_0=0.2$ (dashed lines). Lower-right: scaling solutions with running coupling ($K1$) for two different initial conditions, GBW (solid) and MV (dashed lines). In all plots the initial conditions correspond to the dotted lines and $\bar{\alpha}_0=0.4$ unless otherwise stated.
• Figure 4: The rapidity dependence of the parameter $\gamma$, characterizing the anomalous dimension $1-\gamma$, as determined by a fit of (\ref{['eq:f1']}) to the BK solutions for different initial conditions: GBW (squares), MV (circles), and AS with $c=1.17$ (stars) and $c=0.84$ (triangles). Left plot: results for fixed coupling with $\bar{\alpha}_0=0.2$. Right plot: results for running coupling with $\bar{\alpha}_0=0.4$ and two versions of the kernel $K1$ (empty symbols) and $K2$ (filled symbols).
• Figure 5: Plot on the left: solutions of the BK equation (solid lines) with GBW initial condition and fixed coupling $\bar{\alpha}_s=0.2$ compared to fits (dashed lines) to the DLL expression (\ref{['eq:dll']}), for rapidities $Y=0$, 4, 8, 12, 16 and 20 (curves from right to left). Plots on the right: values of the coefficients $a(Y)$ and $b(Y)$ (circles) in the DLL expression versus $Y$, compared to fits (curves) to the functional form suggested by DLL.