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

We solve numerically the Schwinger-Dyson equation in the Landau gauge for a given, finite at k = 0 gluon propagator (i.e. the infrared exponent of its dressing function, alpha(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, alpha(ghost),(dagger) is < 0), satisfies the familiar relation alpha(gluon) + 2 alpha(ghost) = 0 and has therefore alpha(ghost) = -1/2, and another one which is finite at the origin with alpha(ghost) = 0 and violates the relations. It is most important. There are regular ones -alpha(F) = 0 - for any coupling below some value, while there is only one singular solution -alpha(F) < 0 -, obtained for a single critical value of the coupling. For all momenta k < 1.5 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.