  In this article we study the general structure and special properties of the
Schwinger-Dyson equation for the gluon propagator constructed with the pinch
technique, together with the question of how to obtain infrared finite
solutions, associated with the generation of an effective gluon mass.
Exploiting the known all-order correspondence between the pinch technique and
the background field method, we demonstrate that, contrary to the standard
formulation, the non-perturbative gluon self-energy is transverse
order-by-order in the dressed loop expansion, and separately for gluonic and
ghost contributions. We next present a comprehensive review of several subtle
issues relevant to the search of infrared finite solutions, paying particular
attention to the role of the seagull graph in enforcing transversality, the
necessity of introducing massless poles in the three-gluon vertex, and the
incorporation of the correct renormalization group properties. In addition, we
present a method for regulating the seagull-type contributions based on
dimensional regularization; its applicability depends crucially on the
asymptotic behavior of the solutions in the deep ultraviolet, and in particular
on the anomalous dimension of the dynamically generated gluon mass. A
linearized version of the truncated Schwinger-Dyson equation is derived, using
a vertex that satisfies the required Ward identity and contains massless poles
belonging to different Lorentz structures. The resulting integral equation is
then solved numerically, the infrared and ultraviolet properties of the
obtained solutions are examined in detail, and the allowed range for the
effective gluon mass is determined. Various open questions and possible
connections with different approaches in the literature are discussed.
