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

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 $Y$, $Q^2_s(Y) = Q^2_s(Y_0) \exp{[ \bar\alpha_s d (Y-Y_0) ]}$. 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 $Q_s^2(Y) \propto \exp{(\Delta^\prime\sqrt{Y+X})}$ where we find numerically $\Delta^\prime\simeq 3.2$, approximately 12% smaller than analytical estimates. In contrast to previous analytical work, we find a marked difference between the anomalous dimension $1-\gamma$ governing the large transverse momentum behaviour of the gluon distribution for fixed coupling ($\gamma \simeq 0.65$) and for running coupling ($\gamma \sim 0.85$). Moreover, our numerical findings do not indicate the return of the solution of the BK equation to the perturbative double-leading-logarithmic expression, rather showing that the so-called scaling window extends up to the largest accessible transverse momentum scales.