A geometrical angle on Feynman integrals

TL;DR

This paper establishes a geometric interpretation of one-loop Feynman integrals using N-dimensional simplices in non-Euclidean geometry, linking parametric representations to simplex volumes and providing new computational insights.

Contribution

It introduces a geometric framework connecting Feynman integrals to simplex volumes in non-Euclidean space, offering novel methods for calculating N-point functions.

Findings

Four-point function in four dimensions relates to spherical/hyperbolic tetrahedron volume.

Reduction formulas correspond to splitting N-dimensional simplices into rectangular parts.

Provides a geometric interpretation that simplifies Feynman integral calculations.

Abstract

A direct link between a one-loop N-point Feynman diagram and a geometrical representation based on the N-dimensional simplex is established by relating the Feynman parametric representations to the integrals over contents of (N-1)-dimensional simplices in non-Euclidean geometry of constant curvature. In particular, the four-point function in four dimensions is proportional to the volume of a three-dimensional spherical (or hyperbolic) tetrahedron which can be calculated by splitting into birectangular ones. It is also shown that the known formula of reduction of the N-point function in (N-1) dimensions corresponds to splitting the related N-dimensional simplex into N rectangular ones.