  We furnish details and extensions for the vertex paradigm and related ideas.
The vertex paradigm is a method for dealing non-perturbatively with the
Schwinger-Dyson equations (SDE) of asymptotically-free (AF) gauge theories such
as QCD, even in the face of necessary approximations. It provides a useful
truncation for the infinitely-many SDE of the gauge- and renormalization-group
invariant Pinch Technique (PT-RGI). We implement the vertex paradigm by
successive approximations at the one-dressed-loop level, postulating input
tree-level gluon and ghost propagators and a 3-gluon vertex that are
well-behaved in the infrared and also satisfy several crucial PT-RGI Ward
identities that are QED-like and ghost-free. Good IR behavior is assured by
including a (non-running) gauge-invariant dynamical gluon and ghost mass as
part of the input. The non-trivial part of the vertex paradigm is that, with
our inputs, the one-loop output vertex then satisfies the correct Ward identity
from which we can construct the output gluon propagator, by taking account of
Nambu-Goldstone-like massless scalars and related technical problems that arise
whenever there is dynamical gluon mass generation. The one-loop outputs show a
number of desirable features: They are PT-RGI; free of any reference to a
coupling (dimensional transmutation); satisfy the ghost-free Ward identities
connecting them; give exactly the known one-loop UV behavior; and are free of
IR singularities. We give a much simpler illustration of the main principles of
the vertex paradigm in a modified $\phi^3_6$ model that is AF. Our
successive-approximation scheme is not designed to estimate the gluon dynamical
mass, but it shows that there is a lower limit to the mass below which the AF
theory breaks down.
