  We study the non-perturbative behavior of two versions of the QCD effective
charge, one obtained from the pinch technique gluon self-energy, and one from
the ghost-gluon vertex. Despite their distinct theoretical origin, due to a
fundamental identity relating various of the ingredients appearing in their
respective definitions, the two effective charges are almost identical in the
entire range of physical momenta, and coincide exactly in the deep infrared,
where they freeze at a common finite value. Specifically, the dressing function
of the ghost propagator is related to the two form factors in the Lorentz
decomposition of a certain Green's function, appearing in a variety of
field-theoretic contexts. The central identity, which is valid only in the
Landau gauge, is derived from the Schwinger-Dyson equations governing the
dynamics of the aforementioned quantities. The renormalization procedure that
preserves the validity of the identity is carried out, and various relevant
kinematic limits and physically motivated approximations are studied in detail.
A crucial ingredient in this analysis is the infrared finiteness of the gluon
propagator, which is inextricably connected with the aforementioned freezing of
the effective charges. Some important issues related to the consistent
definition of the effective charge in the presence of such a gluon propagator
are resolved. We finally present a detailed numerical study of a special set of
Schwinger-Dyson equations, whose solutions determine the non-perturbative
dynamics of the quantities composing the two effective charges.
