Can the QCD running coupling have a causal analyticity structure?

Solving the QCD renormalization group equation at the 2-loop and 3-looporders we obtain explicit expressions for the coupling as a function of thescale in terms of the Lambert W function. We study the nature of the ``Landausingularities'' in the complex Q^2 plane and show that perturbative freezingcan lead, in certain cases, to an analyticity structure that is consistent withcausality. We analyze the Analytic Perturbation Theory (APT) approach which isintended to remove the ``Landau singularities'', and show that at 2-loops it isuniquely defined in terms of the Lambert W function, and that, depending on thevalue of the first two beta function coefficients beta_0 and beta_1, it iseither consistent with perturbative freezing (for beta_1 < -beta_0^2) with aninfrared limit of -beta_0/beta_1 or leads to a non-perturbative infraredcoupling with a limit of 1/beta_0 (for beta_1 > -beta_0^2). The possibility ofa causal perturbative coupling is in accordance with the idea that a purelyperturbative Banks-Zaks phase with an infrared fixed-point exists in QCD if thenumber of flavours (N_f) is increased. The causality condition implies that theperturbative phase is realized for N_f \geq 10.Comment: 26 pages, 7 postscript figure