  As has been recently realized, a certain two-point function $\Lambda$ -- and
its associated form factors G and L -- play a prominent role in the PT-BFM
formulation of the Schwinger-Dyson equations used to study gauge-invariantly
the gluon and ghost propagators. After showing that in the (background) Landau
gauge $\Lambda$ fully constrains the QCD ghost sector, we show that G coincides
with the Kugo-Ojima function u, whose infrared behavior has traditionally
served as the standard criterion for the realization of the Kugo-Ojima
confinement mechanism. The determination of the behavior of G for all momenta
through a combination of the available lattice data on the gluon and ghost
propagators, as well as the dynamical equation G satisfies, will be then
discussed. In particular we will show that in the deep infrared the function
deviates considerably from the value associated with the realization of the
Kugo-Ojima confinement scenario; the dependence on the renormalization point of
u, and especially of its value at q=0, will be also briefly discussed.
