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 $g\to 0$ for $\frac{{\gamma }_0}{{\beta }_0}>0\mathrm{}$ and $\frac{{\gamma }_0}{{\beta }_0}<0\mathrm{}$ are exhibited. Here ${\gamma }_0$ and ${\beta }_0$ are coefficients of lowest-order terms of the anomalous dimension and of the $\beta$ function. For SU(3) as the color group and quark triplets, the corresponding flavor conditions are $N_F\le 9$ and $10\le N_F\le 16$, respectively. It was shown previously that for $\frac{{\dot{\gamma }}_0}{{\beta }_0}>0\mathrm{}$ 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 $g\to 0$. 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 ${\left(g^2\right)}^{-\frac{{\gamma }_0}{{\beta }_0}}$, 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 $\frac{{\gamma }_0}{{\beta }_0}<0\mathrm{}$. In particular, the condition is fulfilled that $D_{+}$ approaches its free-field value for $g^2\to 0$.

• Received 4 October 1979

DOI:https://doi.org/10.1103/PhysRevD.21.1661