We study, to all orders in perturbative QCD, the universal behavior of the
saturation momentum Qs​(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
N=4 SYM theory, we derive the large L expansion of Qs​(L) up to
order αs3/2​. In QCD with massless quarks, where conformal symmetry
is broken by the running of the strong coupling constant, the one-loop QCD
β-function fully accounts for the universal terms in the Qs​(L)
expansion. Therefore, the universal coefficients of this series are known
exactly to all orders in αs​.Comment: 13 pages, 4 figures, 3 appendice