Properties of Feynman graph polynomials

TL;DR

This paper explores the mathematical properties of Feynman graph polynomials, focusing on Symanzik polynomials, their derivation from graph structures, and their factorization identities, with applications to multi-loop integrals.

Contribution

It introduces new factorization identities for Symanzik polynomials derived from spanning forests, Laplacian matrices, and matroid theory, enhancing understanding of Feynman integrals.

Findings

Derived factorization identities for Symanzik polynomials

Applied Dodgson's relation to Feynman graph polynomials

Connected graph polynomials to matroid theory and Whitney's theorem

Abstract

In this talk I discuss properties of the two Symanzik polynomials which characterise the integrand of an arbitrary multi-loop integral in its Feynman parametric form. Based on the construction from spanning forests and Laplacian matrices, Dodgson's relation is applied to derive factorisation identities involving both polynomials. An application of Whitney's 2-isomorphism theorem on matroids is discussed.