Effective actions of local composite operators --- case of $\varphi^4$ theory, itinerant electron model, and QED

A compact graph rule for the effective action $\Gamma[\phi]$ of a local composite operator is given in this paper. This long-standing problem of obtaining $\Gamma[\phi]$ in this case is solved directly without using the auxiliary field. The rule is first deduced with help of the inversion method, which is a technique for making the Legendre transformation perturbatively. It is then proved by using a topological relation and also by the sum-up rule. Explicitly derived are the rules for the effective action of $\langle \varphi(x)^2 \rangle$ in the $\varphi^4$ theory, of the number density $\langle n_{{\bf r}\sigma} \rangle$ in the itinerant electron model, and of the gauge invariant operator $\langle \bar{\psi}\gamma^\mu\psi \rangle$ in QED.