Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 007, 25 pages      arXiv:2207.05028      https://doi.org/10.3842/SIGMA.2025.007

Numerators in Parametric Representations of Feynman Diagrams

Marc P. Bellon
Sorbonne Université, CNRS, Laboratoire de Physique Théorique et Hautes Energies, Paris, France

Received May 23, 2024, in final form January 24, 2025; Published online February 06, 2025

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.

Key words: Feynman integrals; parametric representation; Dodgson identities.

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