Recursive Graphical Construction of Feynman Diagrams and Their Multiplicities in phi^4- and in phi^2A-Theory

TL;DR

This paper introduces a recursive graphical method to systematically construct Feynman diagrams and their multiplicities in phi^4- and phi^2A-theories, automating the process with computer algebra.

Contribution

It presents a novel recursive approach to generate all connected vacuum diagrams and their multiplicities, including external lines, in scalar field theories, automated via computer algebra.

Findings

All connected vacuum diagrams are generated order by order in the coupling constant.

The recursive method is successfully applied to phi^4- and phi^2A-theories.

Diagram multiplicities are accurately computed and diagrams are generated automatically.

Abstract

The free energy of a field theory can be considered as a functional of the free correlation function. As such it obeys a nonlinear functional differential equation which can be turned into a recursion relation. This is solved order by order in the coupling constant to find all connected vacuum diagrams with their proper multiplicities. The procedure is applied to a multicomponent scalar field theory with a phi^4-self-interaction and then to a theory of two scalar fields phi and A with an interaction phi^2 A. All Feynman diagrams with external lines are obtained from functional derivatives of the connected vacuum diagrams with respect to the free correlation function. Finally, the recursive graphical construction is automatized by computer algebra with the help of a unique matrix notation for the Feynman diagrams.