Gribov's horizon and the ghost dressing function

We study a relation recently derived by K. Kondo at zero momentum between the Zwanziger's horizon function, the ghost dressing function and Kugo's functions u and w. We agree with this result as far as bare quantities are considered. However, assuming the validity of the horizon gap equation, we argue that the solution w(0)=0 is not acceptable since it would lead to a vanishing renormalized ghost dressing function. On the contrary, when the cutoff goes to infinity, u(0)->infinity, w(0)->-infinity such that u(0)+w(0)->-1. Furthermore w and u are not multiplicatively renormalizable. Relaxing the gap equation allows w(0)=0 with u(0)->-1. In both cases the bare ghost dressing function, F(0,Lambda), goes logarithmically to infinity at infinite cutoff. We show that, although the lattice results provide bare results not so different from the F(0,Lambda)=3 solution, this is an accident due to the fact that the lattice cutoffs lie in the range 1-3 GeV-1. We show that the renormalized ghost dressing function should be finite and nonzero at zero momentum and can be reliably estimated on the lattice up to powers of the lattice spacing; from published data on a 80(4) lattice at beta=5.7 we obtain F-R(0,mu=1.5 GeV)similar or equal to 2.2.