The 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 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 power-like solutions in the very far infrared seem possible. However, only one of these solutions can be matched to a numerical solution for non-vanishing momenta. For a bare ghost-gluon vertex this implies that the gluon propagator is only weakly infrared vanishing, $D_{gl}(k^2) \propto (k^2)^{2\kappa -1}$, $\kappa \approx 0.595$, and the ghost propagator is infrared singular, $D_{gh}(k^2) \propto (k^2)^{-\kappa -1}$. For non-vanishing 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 $\alpha(0) = 8.92/N_c$ for all gauge groups SU($N_c$). Above one GeV the running coupling rapidly approaches its perturbative form.