Cohen-Macaulay Property of Feynman Integrals

TL;DR

This paper establishes the Cohen-Macaulay property for certain Feynman integrals' associated hypergeometric systems, enabling more efficient and dimension-independent solution methods.

Contribution

It proves the Cohen-Macaulay property for toric ideals linked to specific Feynman integrals, enhancing understanding of their solution structure.

Findings

Number of independent solutions is dimension-independent.

Dynamical singularities do not depend on space-time dimension.

Series representations can be computed algorithmically.

Abstract

The connection between Feynman integrals and GKZ $A$-hypergeometric systems has been a topic of recent interest with advances in mathematical techniques and computational tools opening new possibilities; in this paper we continue to explore this connection. To each such hypergeometric system there is an associated toric ideal, we prove that the latter has the Cohen-Macaulay property for two large families of Feynman integrals. This implies, for example, that both the number of independent solutions and dynamical singularities are independent of space-time dimension and generalized propagator powers. Furthermore, in particular, it means that the process of finding a series representation of these integrals is fully algorithmic.