IR finiteness of the ghost dressing function from numerical resolution of the ghost SD equation

Abstract

We solve numerically the Schwinger-Dyson ghost equation in the Landau gauge for a given, finite at k = 0 gluon propagator (i.e. the infrared exponent of its dressing function, αgluon, is 1) and under the usual assumption of constancy of the ghost-gluon vertex ; we show that there exist two possible types of ghost dressing function solutions, as we have previously inferred from analytical considerations: one which is singular at zero momentum (the infrared exponent of its dressing function, αghost,† is < 0), satisfies the familiar relation αgluon + 2αghost = 0 and has therefore αghost = −1/2, and another one which is finite at the origin with αghost = 0 and violates the relation. It is most important that the type of solution which is realized depends on the value of the coupling constant. There are regular ones — αF = 0 — for any coupling below some value, while there is only one singular solution — αF < 0 —, obtained for a single critical value of the coupling. For all momenta k < 1.5GeV where they can be trusted, our lattice data exclude neatly the singular one, and agree very well with the regular solution we obtain at a coupling constant compatible with the bare lattice value.