  Solving the QCD renormalization group equation at the 2-loop and 3-loop
orders we obtain explicit expressions for the coupling as a function of the
scale in terms of the Lambert W function. We study the nature of the ``Landau
singularities'' in the complex Q^2 plane and show that perturbative freezing
can lead, in certain cases, to an analyticity structure that is consistent with
causality. We analyze the Analytic Perturbation Theory (APT) approach which is
intended to remove the ``Landau singularities'', and show that at 2-loops it is
uniquely defined in terms of the Lambert W function, and that, depending on the
value of the first two beta function coefficients beta_0 and beta_1, it is
either consistent with perturbative freezing (for beta_1 < -beta_0^2) with an
infrared limit of -beta_0/beta_1 or leads to a non-perturbative infrared
coupling with a limit of 1/beta_0 (for beta_1 > -beta_0^2). The possibility of
a causal perturbative coupling is in accordance with the idea that a purely
perturbative Banks-Zaks phase with an infrared fixed-point exists in QCD if the
number of flavours (N_f) is increased. The causality condition implies that the
perturbative phase is realized for N_f \geq 10.
