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A geometrical angle on Feynman integrals

A. I. Davydychev, R. Delbourgo

TL;DR

Davydychev and Delbourgo establish a direct link between one-loop $N$-point Feynman integrals and the geometry of an $N$-dimensional simplex, recasting Feynman parameter integrals as integrals over a quadratic form tied to the simplex in constant-curvature space. The framework yields explicit geometric interpretations for the $N$-point functions: for $N=n$ the integral equals a ratio of a solid angle to the simplex content, and for the 4-point case in $4$D the result is related to the volume of a non-Euclidean tetrahedron, computable by splitting into birectangular pieces and expressed via dilogarithms and Clausen functions. The paper also develops reduction identities that express higher-$N$ or higher-propagator-power integrals in terms of lower-point functions, using the $c$-matrix formalism and its dual, which recovers known results such as Nickel’s reductions and provides a unified geometric interpretation. Overall, the approach provides a powerful, geometrically transparent method for analyzing the analytic structure and reductions of one-loop diagrams with many external legs, with potential extensions to higher loops. The height $H_0$ and the dual Gram structures play central roles in connecting parametric integrals to non-Euclidean simplex volumes and their decompositions.

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.

A geometrical angle on Feynman integrals

TL;DR

Davydychev and Delbourgo establish a direct link between one-loop -point Feynman integrals and the geometry of an -dimensional simplex, recasting Feynman parameter integrals as integrals over a quadratic form tied to the simplex in constant-curvature space. The framework yields explicit geometric interpretations for the -point functions: for the integral equals a ratio of a solid angle to the simplex content, and for the 4-point case in D the result is related to the volume of a non-Euclidean tetrahedron, computable by splitting into birectangular pieces and expressed via dilogarithms and Clausen functions. The paper also develops reduction identities that express higher- or higher-propagator-power integrals in terms of lower-point functions, using the -matrix formalism and its dual, which recovers known results such as Nickel’s reductions and provides a unified geometric interpretation. Overall, the approach provides a powerful, geometrically transparent method for analyzing the analytic structure and reductions of one-loop diagrams with many external legs, with potential extensions to higher loops. The height and the dual Gram structures play central roles in connecting parametric integrals to non-Euclidean simplex volumes and their decompositions.

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.

Paper Structure

This paper contains 24 sections, 199 equations.

Table of Contents

  1. Introduction
  2. Basic simplices in $N$ dimensions
  3. $N$-point function and the related simplex
  4. Normals, dihedral angles and the dual matrix
  5. Splitting the basic simplex
  6. Choice of the momenta $p_i$
  7. From Feynman parameters to non-Euclidean geometry
  8. Moving from linear to quadratic hypersurface
  9. The case $\sum\nu_i=n$
  10. The general case
  11. Splitting the $N$-dimensional solid angle
  12. Two-point function
  13. Three-point function
  14. Geometrical picture and the three-dimensional case
  15. The four-dimensional case
  16. ...and 9 more sections