Feynman Integrals of Grassmannians

TL;DR

This paper embeds Feynman integrals into Grassmannian subvarieties, deriving GKZ-systems and expressing integrals as hypergeometric functions, with detailed examples and parametric representations for 2-loop diagrams.

Contribution

It introduces a novel geometric framework for Feynman integrals using Grassmannians and GKZ-systems, providing explicit solution methods and parametric forms.

Findings

Feynman integrals can be expressed as hypergeometric functions of GKZ-systems.

Explicit fundamental solutions near regular singularities are constructed.

Parametric representations for 2-loop self-energy diagrams are developed.

Abstract

We embed Feynman integrals in the subvarieties of Grassmannians through homogenization of the integrands in projective space, then obtain GKZ-systems satisfied by those scalar integrals. The Feynman integral can be written as linear combinations of the hypergeometric functions of a fundamental solution system in neighborhoods of regular singularities of the GKZ-system, whose linear combination coefficients are determined by the integral on an ordinary point or some regular singularities. Taking some Feynman diagrams as examples, we elucidate in detail how to obtain the fundamental solution systems of Feynman integrals in neighborhoods of regular singularities. Furthermore we also present the parametric representations of Feynman integrals of the 2-loop self-energy diagrams which are convenient to embed in the subvarieties of Grassmannians.