GKZ-hypergeometric systems for Feynman integrals

TL;DR

This paper derives GKZ-hypergeometric systems from Mellin-Barnes representations for various Feynman diagrams, enabling systematic solutions that recover known results in quantum field theory.

Contribution

It introduces a method to obtain GKZ-hypergeometric systems for Feynman integrals and demonstrates how to solve them using triangulation and canonical series solutions.

Findings

Derived GKZ-hypergeometric systems for multiple Feynman diagrams.

Constructed canonical series solutions matching known results.

Provided a systematic approach for solving Feynman integrals using hypergeometric systems.

Abstract

Basing on the systems of linear partial differential equations derived from Mellin-Barnes representations and Miller's transformation, we obtain GKZ-hypergeometric systems of one-loop self energy, one-loop triangle, two-loop vacuum, and two-loop sunset diagrams, respectively. The codimension of derived GKZ-hypergeometric system equals the number of independent dimensionless ratios among the external momentum squared and virtual mass squared. Taking GKZ-hypergeometric systems of one-loop self energy, massless one-loop triangle, and two-loop vacuum diagrams as examples, we present in detail how to perform triangulation and how to construct canonical series solutions in the corresponding convergent regions. The series solutions constructed for these hypergeometric systems recover the well known results in literature.