Gauge field propagator and the number of fermion fields

Abstract

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 $g\ensuremath{\rightarrow}0$ for $\frac{{\ensuremath{\gamma}}_{0}}{{\ensuremath{\beta}}_{0}}>0$ and $\frac{{\ensuremath{\gamma}}_{0}}{{\ensuremath{\beta}}_{0}}<0$ are exhibited. Here ${\ensuremath{\gamma}}_{0}$ and ${\ensuremath{\beta}}_{0}$ are coefficients of lowest-order terms of the anomalous dimension and of the $\ensuremath{\beta}$ function. For SU(3) as the color group and quark triplets, the corresponding flavor conditions are ${N}_{F}\ensuremath{\le}9$ and $10\ensuremath{\le}{N}_{F}\ensuremath{\le}16$, respectively. It was shown previously that for $\frac{{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\gamma}}}_{0}}{{\ensuremath{\beta}}_{0}}>0$ 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 $g\ensuremath{\rightarrow}0$. 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 ${({g}^{2})}^{\ensuremath{-}\frac{{\ensuremath{\gamma}}_{0}}{{\ensuremath{\beta}}_{0}}}$, 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 $\frac{{\ensuremath{\gamma}}_{0}}{{\ensuremath{\beta}}_{0}}<0$. In particular, the condition is fulfilled that ${D}_{+}$ approaches its free-field value for ${g}^{2}\ensuremath{\rightarrow}0$.