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arXiv · 2605.29789

Magic Relations and Critical Varieties of Feynman Integrals

Giulio Crisanti·Hjalte Frellesvig·Andrzej Pokraka·Sid Smith
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Abstract

Magic relations are a class of integration-by-parts identities where all integrals in the generating sector drop out. Since their presence causes several otherwise successful methods in the Feynman-integral computational pipeline to break down, they are important to detect and understand. In this paper, we take a first step toward a systematic characterization of such identities. Specifically, we observe and argue that the occurrence of magic relations always coincides with the presence of higher-dimensional critical varieties in the generating sector. This provides a practical computational test to check if a family of Feynman integrals can contain magic relations and to find them, which we implement in the ancillary Mathematica file Magic-Test.m. Additionally, we discuss how to count the number of master integrals in the presence of higher-dimensional critical varieties, classify the behavior of magic relations under symmetries, and we discuss their interplay with cuts.

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Giulio Crisanti, Hjalte Frellesvig, Andrzej Pokraka, Sid Smith. 2026-05-28. Magic Relations and Critical Varieties of Feynman Integrals. https://arxiv.org/abs/2605.29789

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