Hepp's bound for Feynman graphs and matroids

TL;DR

This paper introduces a new matroid invariant derived from Feynman integrals, providing efficient computation methods and insights into the relationship between tropical and transcendental integrals, aiding in approximating unknown Feynman periods.

Contribution

It defines a rational matroid invariant as a tropicalization of Feynman periods, with formulas for computation and validation against known identities.

Findings

Invariant equals the volume of the polar of the matroid polytope

Efficient formulas for computing the invariant

Strong correlation between tropical and transcendental integrals

Abstract

We study a rational matroid invariant, obtained as the tropicalization of the Feynman period integral. It equals the volume of the polar of the matroid polytope and we give efficient formulas for its computation. This invariant is proven to respect all known identities of Feynman integrals for graphs. We observe a strong correlation between the tropical and transcendental integrals, which yields a method to approximate unknown Feynman periods.