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

TL;DR

This paper presents a direct, graph-based method to compute the effective action of local composite operators across various models, including $$ theory, electron systems, and QED, without auxiliary fields.

Contribution

It introduces a compact graph rule for the effective action of local composite operators, derived via the inversion method and topological relations, applicable to multiple quantum field theories.

Findings

Derived explicit rules for $^2$ in $^4$ theory

Established effective action rules for electron density in itinerant models

Formulated gauge-invariant operator effective actions in QED

Abstract

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.