Feynman integral relations from parametric annihilators

TL;DR

The paper links shift relations among Feynman integrals to parametric annihilators of the Lee-Pomeransky polynomial through a twisted Mellin transform, showing that IBP and dimension-shift relations are encompassed in this framework. By leveraging D-module theory and the Loeser–Sabbah theorem, it proves that the number of master integrals equals the Euler characteristic of the complement of {\mathcal{G}=0}, {\mathcal{G}=\mathcal{U}+\mathcal{F}}, in the appropriate torus, providing a topological criterion for finiteness and a practical computational handle. It develops multiple tools (graph polynomials, Grothendieck ring techniques, linearly reducible graphs, and sunrise analyses) and showcases algorithmic implementations (Macaulay2, Oaku–Takayama, and Azurite) that agree with established IBP-based counts. The work offers a geometric and algebraic lens on master integrals, with potential for automated counting and basis construction across a wide range of Feynman graphs, including dimensionally regulated settings.

Abstract

We study shift relations between Feynman integrals via the Mellin transform through parametric annihilation operators. These contain the momentum space IBP relations, which are well-known in the physics literature. Applying a result of Loeser and Sabbah, we conclude that the number of master integrals is computed by the Euler characteristic of the Lee-Pomeransky polynomial. We illustrate techniques to compute this Euler characteristic in various examples and compare it with numbers of master integrals obtained in previous works.

Figures (8)

• Figure 2.1: The one-loop bubble graph with momentum flow.
• Figure 3.1: A 1-scale subgraph $\gamma$ of $G$ is replaced by a single edge in $G'$.
• Figure 3.2: The series (S) and parallel (P) operations consist of replacing a sequential or parallel pair of massless edges with a single edge.
• Figure 4.1: Some linearly reducible propagators ($P_i$) and form factors ($F_i$). All internal edges are massless, and the form factors have two massless external legs ($p_1^2=p_2^2=0$) and one massive leg $p_3^2\neq 0$, indicated by the label $m$.
• Figure 4.2: The propagator graphs $\mathrm{WS}_{L}'$ ($\mathrm{WS}_{L}"$) with $L-1$ loops are obtained from cutting a rim (spoke) of the wheel $\mathrm{WS}_{L}$ with $L$ loops.

Theorems & Definitions (91)

• Definition 1
• Proposition 2
• Example 3
• Remark 4: meromorphicity
• Example 5
• Definition 6
• Lemma 7
• proof
• Definition 8
• Definition 9