Numerators in Parametric Representations of Feynman Diagrams

TL;DR

This paper improves the parametric representation of Feynman diagrams by simplifying formulas using Dodgson identities, especially in the massless case, aiding the analysis of singularities and transcendental numbers.

Contribution

It introduces a simplified approach to include numerators in parametric Feynman diagram representations using Dodgson identities, enhancing computational efficiency.

Findings

Simplified formulas for numerators in parametric representations.

Reduction of maximal power of Symanzik polynomial in massless case.

Enhanced analysis of singularities and transcendental numbers.

Abstract

The parametric representation has been used since a long time for the evaluation of Feynman diagrams. As a dimension independent intermediate representation, it allows a clear description of singularities. Recently, it has become a choice tool for the investigation of the type of transcendent numbersappearing in the evaluation of Feynman diagrams. The inclusion of numerators has however stagnated since the ground work of Nakanishi. I here show howto greatly simplify the formulas through the use of Dodgson identities. In the massless case in particular, reduction to the completion to a vacuum graph allows for a strong reduction of the maximal power of the Symanzik polynomial in the denominator.