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

Source: [Effective actions of local composite operators --- case of $\\varphi^4$ theory, itinerant electron model, and QED](<https://www.sota2.com/research/paper/68fa31d01e209131ce88272a>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-24T19:06:29.581Z

Authors: K. Okumura

Published: 1994-04-18T03:44:26.000Z

## 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.

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/hep-th/9404102>)

[PDF](<https://arxiv.org/pdf/hep-th/9404102.pdf>)