Hypergeometric Series Representations of Feynman Integrals by GKZ Hypergeometric Systems

TL;DR

This paper demonstrates that most Feynman integrals can be expressed using GKZ hypergeometric functions, providing a new series representation method that facilitates numerical evaluation and analysis.

Contribution

It introduces a GKZ hypergeometric series framework for Feynman integrals, derived via triangulations of Newton polytopes, enhancing their analytical and numerical treatment.

Findings

Feynman integrals can be represented by Horn hypergeometric functions.

Series representations converge rapidly for suitable kinematics.

The approach enables practical numerical applications of Feynman integrals.

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 $\Delta_G$ 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.