Characterizing Cohen-Macaulay One-Loop Feynman Integrals

TL;DR

This paper investigates the algebraic structure of one-loop Feynman integrals through hypergeometric systems, identifying conditions under which these systems are Cohen-Macaulay, which impacts their mathematical and physical properties.

Contribution

It provides necessary and sufficient conditions for Cohen-Macaulayness of one-loop Feynman integrals, generalizing previous results and linking algebraic properties to combinatorial and graphical methods.

Findings

Cohen-Macaulay property is determined by an integer linear program.

Established conditions for massive one-loop integrals.

Provided a graphical description of solutions.

Abstract

We study the generalized hypergeometric systems, in the sense of Gel'fand, Kapranov, and Zelevinsky, associated with one-loop Feynman integrals, and determine when their rank is independent of space-time dimension and propagator powers. This is equivalent to classifying when the associated affine semigroup ring is Cohen-Macaulay. For massive one-loop integrals, we prove necessary and sufficient conditions for Cohen-Macaulayness, generalizing previous results on normality for these rings. We show that for Feynman integrals, the Cohen-Macaulay property is fully determined by an integer linear program built from the Newton polytope of the integrand and find a graphical description of its solutions. Furthermore, we provide a sufficient condition for Cohen-Macaulayness of general one-loop integrals.