Can the QCD running coupling have a causal analyticity structure?
  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. 
Einan Gardi, Georges Grunberg and Marek Karliner