  A loop or coupling expansion of a so-called n-particle irreducible (nPI)
generating functional provides a well-defined approximation scheme in terms of
self-consistently dressed propagators and n-point vertices. A self-consistently
complete description determines the functional for arbitrarily high n to a
given order in the expansion. We point out an equivalence hierarchy for nPI
effective actions, which allows one to obtain a self-consistently complete
result in practice. The method is applied to a SU(N) gauge theory with fermions
up to four-loop or O(g^6) corrections. For non-equilibrium we discuss the
connection to kinetic theory. The leading-order on-shell results in g can be
obtained from the three-loop effective action approximation, which already
includes in particular all diagrams enhanced by the Landau Pomeranchuk Migdal
effect. Furthermore, we compare the effective action approach with
Schwinger-Dyson (SD) equations. By construction, SD equations are expressed in
terms of loop diagrams including both classical and dressed vertices, which
leads to ambiguities of whether classical or dressed ones should be employed at
a given truncation order. We point out that these problems are absent using
effective action techniques. We show that a wide class of truncations of SD
equations cannot be obtained from the nPI effective action. In turn, our
results can be used to resolve SD ambiguities of whether classical or dressed
vertices should be employed at a given truncation order.
