On overlapping Feynman (sub)graphs

TL;DR

This paper analyzes how subgraphs in Feynman diagrams can overlap, establishing non-overlap theorems and justifying the skeleton expansion in scalar field theories for complex vertex functions.

Contribution

It introduces a general framework for understanding subgraph overlaps and provides a justification for the skeleton expansion in scalar field theories.

Findings

Derived non-overlap theorems for 2-, 3-, and 4-leg subgraphs.

Provided a justification for the skeleton expansion for vertices with more than five legs.

Discussed extensions of the skeleton expansion to other graph classes.

Abstract

We discuss, on general grounds, how two subgraphs of a given Feynman graph can overlap with each other. For this, we use the notion of connecting and returning lines that describe how any subgraph is inserted within the original graph. This, in turn, allows us to derive "non-overlap" theorems for one-particle-irreducible subgraphs with $2$, $3$ and $4$ external legs. As an application, we provide a simple justification of the skeleton expansion for vertex functions with more than five legs, in the case of scalar field theories. We also discuss how the skeleton expansion can be extended to other classes of graphs.