Generalized effective potential in nonlinear theories of fourth order

Legendre transformations are used to construct a generalized effective potential $\Gamma(\varphi,G,H,S)$ which depends on the vacuum expectation value of the field, the two- and three-point connected Green's functions $G$ and $H$, and the vacuum expectation $S=\langle0|S_{\text{cl}}|0\rangle$ of the classical action. An expansion is obtained for $\Gamma(\varphi,G,H,S)$ analogous to the loop expansion of the effective action $\Gamma(\varphi)$.