Characterizing Cohen-Macaulay One-Loop Feynman Integrals

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.

💡 Research Summary

The paper investigates the algebraic structure underlying one‑loop scalar Feynman integrals by interpreting them as solutions of Gel’fand‑Kapranov‑Zelevinsky (GKZ) hypergeometric systems. Starting from the Lee‑Pomeransky representation, each integral is written as an Euler‑Mellin integral whose integrand is the sum of the two Symanzik polynomials, (U) and (F). The exponents of the monomials in this polynomial are collected into an integer matrix (A) (the “A‑matrix”) and a parameter vector (\beta) encodes the space‑time dimension (D) and the propagator powers (\nu_i). The GKZ system (H_A(\beta)=I_A+Z_A(\beta)) consists of a toric ideal (I_A) generated by binomials reflecting linear relations among the columns of (A), together with Euler operators that enforce the homogeneity dictated by (\beta).

A central observation, originally proved in earlier work, is that the rank (dimension of the solution space) of a GKZ system is independent of (\beta) if and only if the affine semigroup ring (\mathbb{C}