Hypergeometric Series Representations of Feynman Integrals by GKZ Hypergeometric Systems

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.

39 pages, 3 figures, changed misleading nomenclature, added footnotes, updated table 1, corrected minor issues in section 3.4