  We study, to all orders in perturbative QCD, the universal behavior of the
saturation momentum $Q_s(L)$ controlling the transverse momentum distribution
of a fast parton propagating through a dense QCD medium with large size $L$.
Due to the double logarithmic nature of the quantum evolution of the saturation
momentum, its large $L$ asymptotics is obtained by slightly departing from the
double logarithmic limit of either next-to-leading log (NLL) BFKL or leading
order DGLAP evolution equations. At fixed coupling, or in conformal
$\mathcal{N}=4$ SYM theory, we derive the large $L$ expansion of $Q_s(L)$ up to
order $\alpha_s^{3/2}$. In QCD with massless quarks, where conformal symmetry
is broken by the running of the strong coupling constant, the one-loop QCD
$\beta$-function fully accounts for the universal terms in the $Q_s(L)$
expansion. Therefore, the universal coefficients of this series are known
exactly to all orders in $\alpha_s$.
