Triangle diagram, distance geometry and Symmetries of Feynman Integrals

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.

Numerator seagull and extended Symmetries of Feynman Integrals

“…By now, SFI was further developed and numerous diagrams have been analyzed within it [8][9][10][11][12][13][14][15]. Even though these diagrams are relatively basic, the analysis achieved new results including the value of the seagull diagram (to be discussed below) in a 3 mass scale sector and the value of the kite diagram (2-loop 2-leg) throughout its singular locus.…”

Numerator Seagull and Extended Symmetries of Feynman Integrals

“…By now, SFI was further developed and numerous diagrams have been analyzed within it [8][9][10][11][12][13][14][15]. Even though these diagrams are relatively basic, the analysis achieved new results including the value of the seagull diagram (to be discussed below) in a 3 mass scale sector and the value of the kite diagram (2-loop 2-leg) throughout its singular locus.…”