The running coupling of Yang-Mills theory has been investigated in a number of approaches in recent years . The focus of these studies is the question how to extend our knowledge of the coupling from the perturbative, large momentum region towards small scales of the order of ΛQCD. Perturbation theory alone, plagued by the problem of the Landau pole, is clearly insufficient for this task. In this respect it seems remarkable that the mere improvement of the perturbation series by analyticity constraints leads to a well defined running coupling that freezes out in the infrared. Infrared fixed points of the couplings of Yang-Mills theory have also been found in two functional approaches to QCD, the functional (or ’exact’) renormalization group (FRG) and the framework of Dyson-Schwinger equations (DSEs) . In these approaches nonperturbative running couplings are defined in terms of dressing functions of propagators and dressing functions of the primitively divergent vertices of the theory. In Landau gauge, the couplings from the ghost-gluon vertex, α gh−gl, the three-gluon vertex, α 3g, and the four-gluon vertex, α 4g, are given by : α gh−gl (p 2) = g2 4π G2 (p 2)Z(p 2), (1) α 3g (p 2) = g2 4π [Γ3g (p 2)] 2 Z 3 (p 2), (2) α 4g (p 2) = g2 4π [Γ4g (p 2)] Z 2 (p 2). (3) Here g 2 /4π is the coupling at the renormalization point µ 2, whereas Z(p 2) denotes the dressing function of the gluon propagator Dµν and G(p 2) the dressing of the ghost propagator D G, i.e. D G (p 2) = − G(p2) p 2, Dµν(p 2) = δµν − pµpν p
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