  This paper derives the Feynman rules for the diagrammatic perturbation
expansion of the effective action around an arbitrary solvable problem. The
perturbation expansion around a Gaussian theory is well known and composed of
one-line irreducible diagrams only. For the expansions around an arbitrary,
non-Gaussian problem, we show that a more general class of irreducible diagrams
remains in addition to a second set of diagrams that has no analogue in the
Gaussian case. The effective action is central to field theory, in particular
to the study of phase transitions, symmetry breaking, effective equations of
motion, and renormalization. We exemplify the method on the Ising model, where
the effective action amounts to the Gibbs free energy, recovering the
Thouless-Anderson-Palmer mean-field theory in a fully diagrammatic derivation.
Higher order corrections follow with only minimal effort compared to existing
techniques. Our results show further that the Plefka expansion and the
high-temperature expansion are special cases of the general formalism presented
here.
