A combinatorial matrix approach for the generation of vacuum Feynman graphs multiplicities in $\phi^4$ theory

TL;DR

This paper introduces a combinatorial matrix method linked to permutation groups to efficiently generate and count vacuum Feynman graphs in $$ theory, offering a new computational approach.

Contribution

It presents a novel combinatorial matrix framework that relates to permutation groups for generating and counting Feynman vacuum graphs in $$ theory.

Findings

Provides an explicit relation between combinatorial matrices and permutation groups.

Offers an efficient method for generating all Feynman vacuum graphs.

Facilitates computation of graph multiplicities in quantum field theory.

Abstract

From the standard procedure for constructing Feynman vacuum graphs in $\phi^4$ theory from the generating functional $Z$, we find a relation with sets of certain combinatorial matrices, which allows us to generate the set of all Feynman graphs and the respective multiplicities in an equivalent combinatoric way. These combinatorial matrices are explicitly related with the permutation group, which facilitates the construction of the vacuum Feynman graphs. Various insights in this combinatoric problem are proposed, which in principle provide an efficient way to compute the Feynman vacuum graphs and its multiplicities.