Gauge field propagator and the number of fermion fields

The structure of the transverse gluon propagator D of massless quantum chromodynamics is considered in the Landau gauge. The essential differences in the weak-coupling limit g→0 for γ₀β₀>0 and γ₀β₀<0 are exhibited. Here γ₀ and β₀ are coefficients of lowest-order terms of the anomalous dimension and of the β function. For SU(3) as the color group and quark triplets, the corresponding flavor conditions are NF≤9 and 10≤NF≤16, respectively. It was shown previously that for ̇ \.γ₀β₀>0 there are inconsistencies with the postulates of local quantum field theory and the requirement that the positive-norm contribution D₊ to D should approach its free-field value for g→0. In the present paper, it is investigated in detail how this requirement is violated assuming that the other postulates hold, including invariance under the renormalization group. Using a specific, simple projection into a subspace of positive norm, it is shown that D₊ diverges like (g²)^-γ₀β₀, while the free-field value and higher-order terms of D are entirely due to contributions from negative-norm states. In contradistinction, the required dominance of positive-norm states in the weak-coupling limit prevails for γ₀β₀<0. In particular, the condition is fulfilled that D₊ approaches its free-field value for g²→0.