Elusiveness of infrared critical exponents in Landau gauge Yang-Mills theories

Abstract

We solve a truncated system of coupled Dyson-Schwinger equations for the gluon and ghost propagators in $\mathrm{SU}{(N}_{c})$ Yang-Mills theories in Faddeev-Popov quantization on a four-torus. This compact space-time manifold provides an efficient mean to solve the gluon and ghost Dyson-Schwinger equations without any angular approximations. We verify that analytically two powerlike solutions in the very far infrared seem possible. However, only one of these solutions can be matched to a numerical solution for nonvanishing momenta. For a bare ghost-gluon vertex this implies that the gluon propagator is only weakly infrared vanishing, ${D}_{\mathrm{gl}}{(k}^{2})\ensuremath{\propto}{(k}^{2}{)}^{2\ensuremath{\kappa}\ensuremath{-}1},$ $\ensuremath{\kappa}\ensuremath{\approx}0.595,$ and the ghost propagator is infrared singular, ${D}_{\mathrm{gh}}{(k}^{2})\ensuremath{\propto}{(k}^{2}{)}^{\ensuremath{-}\ensuremath{\kappa}\ensuremath{-}1}.$ For nonvanishing momenta our solutions are in agreement with the results of recent SU(2) Monte Carlo lattice calculations. The running coupling possesses an infrared fixed point. We obtain $\ensuremath{\alpha}{(0)=8.92/N}_{c}$ for all gauge groups $\mathrm{SU}{(N}_{c}).$ Above one GeV the running coupling rapidly approaches its perturbative form.