  In this paper we study the nonperturbative structure of the SU(3) four-gluon
vertex in the Landau gauge, concentrating on contributions quadratic in the
metric. We employ an approximation scheme where "one-loop" diagrams are
computed using fully dressed gluon and ghost propagators, and tree-level
vertices. When a suitable kinematical configuration depending on a single
momentum scale $p$ is chosen, only two structures emerge: the tree-level
four-gluon vertex, and a tensor orthogonal to it. A detailed numerical analysis
reveals that the form factor associated with this latter tensor displays a
change of sign (zero-crossing) in the deep infrared, and finally diverges
logarithmically. The origin of this characteristic behavior is proven to be
entirely due to the masslessness of the ghost propagators forming the
corresponding ghost-loop diagram, in close analogy to a similar effect
established for the three-gluon vertex. However, in the case at hand, and under
the approximations employed, this particular divergence does not affect the
form factor proportional to the tree-level tensor, which remains finite in the
entire range of momenta, and deviates moderately from its naive tree-level
value. It turns out that the kinematic configuration chosen is ideal for
carrying out lattice simulations, because it eliminates from the connected
Green's function all one-particle reducible contributions, projecting out the
genuine one-particle irreducible vertex. Motivated by this possibility, we
discuss in detail how a hypothetical lattice measurement of this quantity would
compare to the results presented here, and the potential interference from an
additional tensorial structure, allowed by Bose symmetry, but not encountered
within our scheme.
