Feynman integrals as A-hypergeometric functions

TL;DR

This paper demonstrates that Feynman integrals can be represented as solutions to GKZ hypergeometric systems, enabling new computational approaches using algebraic and geometric methods.

Contribution

It introduces a novel framework linking Feynman integrals to A-hypergeometric functions via GKZ systems and develops algorithms for their evaluation.

Findings

Feynman integrals are solutions of GKZ systems.

Canonical series algorithms can evaluate integrals with arbitrary powers.

The approach connects algebraic geometry with quantum field theory calculations.

Abstract

We show that the Lee-Pomeransky parametric representation of Feynman integrals can be understood as a solution of a certain Gel'fand-Kapranov-Zelevinsky (GKZ) system. In order to define such GKZ system, we consider the polynomial obtained from the Symanzik polynomials $g=\mathcal{U}+\mathcal{F}$ as having indeterminate coefficients. Noncompact integration cycles can be determined from the coamoeba---the argument mapping---of the algebraic variety associated with $g$. In general, we add a deformation to $g$ in order to define integrals of generic graphs as linear combinations of their canonical series. We evaluate several Feynman integrals with arbitrary non-integer powers in the propagators using the canonical series algorithm.

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