  We present the detailed derivation of the longitudinal part of the
three-gluon vertex from the Slavnov-Taylor identities that it satisfies, by
means of a nonperturbative implementation of the Ball-Chiu construction; the
latter, in its original form, involves the inverse gluon propagator, the ghost
dressing function, and certain form factors of the ghost-gluon kernel. The main
conceptual subtlety that renders this endeavor nontrivial is the infrared
finiteness of the gluon propagator, and the resulting need to separate the
vertex into two pieces, one that is intimately connected with the emergence of
a gluonic mass scale, and one that satisfies the original set of Slavnov-Taylor
identities, but with the inverse gluon propagator replaced by its "kinetic"
term. The longitudinal form factors obtained by this construction are presented
for arbitrary Euclidean momenta, as well as special kinematic configurations,
parametrized by a single momentum. A particularly preeminent feature of the
components comprising the tree-level vertex is their considerable suppression
for momenta below 1 GeV, and the appearance of the characteristic
"zero-crossing" in the vicinity of 100-200 MeV. Special combinations of the
form factors derived with this method are compared with the results of recent
large-volume lattice simulations as well as Schwinger-Dyson equations, and good
overall agreement is found. A variety of issues related to the distribution of
the pole terms responsible for the gluon mass generation are discussed in
detail, and their impact on the structure of the transverse parts is
elucidated. In addition, a brief account of several theoretical and
phenomenological possibilities involving these newly acquired results is
presented.
