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 $Y$, $Q_s^2(Y)=Q_s^2(Y_0)\exp [{\overline{\alpha }}_{s}d(Y-Y_0)]$. For the running coupling case, we rederive 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 $Q_s^2(Y)\propto \exp ({\Delta }^{\prime }\sqrt{Y+X})$, where we find numerically ${\Delta }^{\prime }\simeq 3.2$. We study the behavior of the gluon distribution at large transverse momentum, characterizing it by an anomalous dimension $1-\gamma$, 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 ($\gamma \simeq 0.65$) and running coupling ($\gamma \sim 0.85$) 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.

• Received 20 August 2004

DOI:https://doi.org/10.1103/PhysRevD.71.014003