Dodgson polynomial identities

TL;DR

This paper introduces new combinatorial identities for Dodgson polynomials, simplifying quantum electrodynamics Feynman integrals by relating complex sums to powers of the Kirchhoff polynomial.

Contribution

It provides a novel combinatorial interpretation and generalization of Dodgson polynomials, leading to identities that simplify parametric Feynman integrals in quantum field theory.

Findings

Derived two new identities relating Dodgson and Kirchhoff polynomials.

Simplified the parametric integrand for QED photon propagator graphs.

Demonstrated the applicability of identities to renormalized Feynman graphs.

Abstract

Dodgson polynomials appear in Schwinger parametric Feynman integrals and are closely related to the well known Kirchhoff (or first Symanzik) polynomial. In this article a new combinatorial interpretation and a generalisation of Dodgson polynomials are provided. This leads to two new identities that relate large sums of products of Dodgson polynomials to a much simpler expression involving powers of the Kirchhoff polynomial. These identities can be applied to the parametric integrand for quantum electrodynamics, simplifying it significantly. This is worked out here in detail on the example of superficially renormalised photon propagator Feynman graphs, but works much more generally.