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Hypergeometric Series Representations of Feynman Integrals by GKZ Hypergeometric Systems

René Pascal Klausen

TL;DR

The paper establishes a GKZ-hypergeometric framework for Feynman integrals by encoding each graph via the Lee-Pomeransky polynomial $G$ and its Newton polytope $oldsymbol{ riangle_G}$. For regular unimodular triangulations, generalized Feynman integrals $J_ ext{A}(oldsymbol{ u}, z)$ admit explicit Horn-type series representations as finite sums of GKZ $ ext{Gamma}$-series with boundary-determined coefficients, and these representations extend to the $oldsymbol{ u}$-expansion and various parametric forms. It further shows how non-generic limits and $oldsymbol{ u}$-expansions can be managed through limit procedures, leading to a practical algorithm for computing Feynman amplitudes in many kinematic regions, as demonstrated in the full massive sunset example. The study also discusses the role of unimodularity in triangulations, the treatment of non-unimodular cases via subtriangulations or dilations, and outlines open questions about extending unimodularity and applying GKZ to broader parametric representations, with implications for numerical efficiency and analytic structure of amplitudes.

Abstract

We show that almost all Feynman integrals as well as their coefficients in a Laurent series in dimensional regularization can be written in terms of Horn hypergeometric functions. By applying the results of Gelfand-Kapranov-Zelevinsky (GKZ) we derive a formula for a class of hypergeometric series representations of Feynman integrals, which can be obtained by triangulations of the Newton polytope $Δ_G$ corresponding to the Lee-Pomeransky polynomial $G$. Those series can be of higher dimension, but converge fast for convenient kinematics, which also allows numerical applications. Further, we discuss possible difficulties which can arise in a practical usage of this approach and give strategies to solve them.

Hypergeometric Series Representations of Feynman Integrals by GKZ Hypergeometric Systems

TL;DR

The paper establishes a GKZ-hypergeometric framework for Feynman integrals by encoding each graph via the Lee-Pomeransky polynomial and its Newton polytope . For regular unimodular triangulations, generalized Feynman integrals admit explicit Horn-type series representations as finite sums of GKZ -series with boundary-determined coefficients, and these representations extend to the -expansion and various parametric forms. It further shows how non-generic limits and -expansions can be managed through limit procedures, leading to a practical algorithm for computing Feynman amplitudes in many kinematic regions, as demonstrated in the full massive sunset example. The study also discusses the role of unimodularity in triangulations, the treatment of non-unimodular cases via subtriangulations or dilations, and outlines open questions about extending unimodularity and applying GKZ to broader parametric representations, with implications for numerical efficiency and analytic structure of amplitudes.

Abstract

We show that almost all Feynman integrals as well as their coefficients in a Laurent series in dimensional regularization can be written in terms of Horn hypergeometric functions. By applying the results of Gelfand-Kapranov-Zelevinsky (GKZ) we derive a formula for a class of hypergeometric series representations of Feynman integrals, which can be obtained by triangulations of the Newton polytope corresponding to the Lee-Pomeransky polynomial . Those series can be of higher dimension, but converge fast for convenient kinematics, which also allows numerical applications. Further, we discuss possible difficulties which can arise in a practical usage of this approach and give strategies to solve them.

Paper Structure

This paper contains 18 sections, 18 theorems, 85 equations, 3 figures.

Table of Contents

  1. Feynman Integrals
  2. Parametric Representations of Feynman Integrals
  3. Mathematical Interlude: Convex Polytopes
  4. Properties of Feynman Integrals
  5. General Hypergeometric Functions
  6. GKZ Hypergeometric Functions
  7. Feynman Integrals as Hypergeometric Functions
  8. Hypergeometric Series Representations of Generalized Feynman Integrals
  9. The Limit from Generalized Feynman Integrals to Non-Generic Feynman Integrals
  10. The $\epsilon$ Expansion of Hypergeometric Series Representations
  11. Applications to other Parametric Representations
  12. Non-Unimodular Triangulations of Feynman Polytopes
  13. Advanced Example: Full Massive Sunset
  14. Conclusion and Outlook
  15. Appendix
  16. ...and 3 more sections

Key Result

Theorem 1.1

The Feynman integral from equation (eq:FI1) can also be written as where the integral depends only on the sum of Symanzik polynomials $G=U+F$, which we call the Lee-Pomeransky polynomial $G$. The equality of representations is in the sense of meromorphic extension.

Figures (3)

  • Figure 1: The $1$-loop $2$-point function with one mass.
  • Figure 2: The two possible regular triangulations of the Newton polytope $\Delta_G=\operatorname{Conv}(\mathrm A)$ corresponding to the Lee-Pomeransky polynomial $G= z_1 x_1+z_2x_2+z_3x_1 x_2+z_4x_1^2$.
  • Figure 3: The $2$-loop $2$-point function (sunset graph) with three different masses.

Theorems & Definitions (41)

  • Definition 1.1: Feynman integral
  • Remark
  • Theorem 1.1: Lee-Pomeransky representation Lee.PomeranskyCriticalPointsNumber2013
  • proof
  • Definition 1.2: Generalized Feynman Integrals
  • Remark
  • Theorem 1.2: Representation as Fox $H$-function
  • Corollary 1.3
  • proof
  • Remark
  • ...and 31 more