Recursion Relation for the Feynman Diagrams of the Effective Action for the Third Legendre Transformation

TL;DR

This paper derives a recursion relation for Feynman diagrams in the third Legendre transformation of bosonic field theory with cubic interactions, enabling systematic computation up to five loops.

Contribution

It introduces a novel recursion relation for Feynman diagrams in the third Legendre transformation, demonstrating three-particle irreducibility via induction.

Findings

Recursion relation for Feynman diagrams derived

Effective action diagrams computed up to five loops

Three-particle irreducibility established

Abstract

We derive a recursion relation of the Feynman diagrams of the effective action for the third Legendre transformation in case of the bosonic field theory with cubic interaction. We apply the recursion relation to obtain the Feynman diagrams of the effective action for the third Legendre transformation up to the five-loop order. The three-particle irreducibility of the Feynman diagrams of the effective action for the third Legendre transformation is shown by induction.