CAN THE QCD RUNNING COUPLING HAVE A CAUSAL ANALYTICITY STRUCTURE

Solving the QCD renormalization group equation at the 2-loop and 3-loo p 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 tha t perturbative freezing can lead, in certain cases, to an analyticity structure that is consistent with causality. We analyze the Analytic P erturbation Theory (APT) approach which is intended to remove the ''La ndau singularities'', and show that at 2-loops it is uniquely defined in terms of the Lambert W function, and that, depending on the value o f 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-perturb ative infrared coupling with a limit of 1/beta(0) (for beta(1) > beta( 0)(2). The possibility of a causal perturbative coupling is in accorda nce 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 greater than or equal to 10.