Dual Graph Polynomials and a 4-face Formula

TL;DR

This paper investigates dual graph polynomials in graphs with a 4-face, proving duality-admissibility up to 18 loops and showing invariance of the $c_2$ invariant across different Feynman period representations.

Contribution

It introduces new results on dual graph polynomials for graphs with a 4-face and establishes the invariance of the $c_2$ invariant across multiple Feynman representations.

Findings

Proved duality-admissibility for all graphs up to 18 loops.

Showed $c_2$ invariant is the same across all four Feynman period representations.

Analyzed graphs with no triangles but with a 4-face.

Abstract

We study the dual graph polynomials and the case when a Feynman graph has no triangles but has a 4-face. This leads to the proof of the duality-admissibility of all graphs up to 18 loops. As a consequence, the $c_2$ invariant is the same for all 4 Feynman period representations (position, momentum, parametric and dual parametric) for any physically relevant graph.