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

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, α_gluon, 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, α_ghost, (We shall use α_G and α_F as shorthands for α_gluonand α_ghost respectively; let us recall that we denote the gluon by a G and the ghost by a F, for ``fantôme''.) is <0), satisfies the familiar relation α_gluon+2α_ghost = 0 and has therefore α_ghost = −1/2, and another one which is finite at the origin with α_ghost = 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 — α_F = 0 — for any coupling below some value, while there is only one singular solution — α_F <0 —, obtained for a single critical value of the coupling. For all momenta k <.5 GeV 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.