Triangle diagram, Distance Geometry and Symmetries of Feynman Integrals

TL;DR

This paper analyzes the general triangle Feynman diagram using Symmetries of Feynman Integrals, revealing geometric structures and deriving known expressions through a novel approach.

Contribution

It introduces a simple basis for the SFI equations of the triangle diagram and connects the geometry to Distance Geometry of a tetrahedron, providing new insights.

Findings

Derived the SFI equations in a simple basis.

Identified the singular locus and expressed diagram values there.

Revisited the massless triangle and magic connection.

Abstract

We study the most general triangle diagram through the Symmetries of Feynman Integrals (SFI) approach. The SFI equation system is obtained and presented in a simple basis. The system is solved providing a novel derivation of an essentially known expression. We stress a description of the underlying geometry in terms of the Distance Geometry of a tetrahedron discussed by Davydychev-Delbourgo [1], a tetrahedron which is the dual on-shell diagram. In addition, the singular locus is identified and the diagram's value on the locus's two components is expressed as a linear combination of descendant bubble diagrams. The massless triangle and the associated magic connection are revisited.