Magic Relations and Critical Varieties of Feynman Integrals
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866913170492227584 |
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| author | Crisanti, Giulio Frellesvig, Hjalte Pokraka, Andrzej Smith, Sid |
| author_facet | Crisanti, Giulio Frellesvig, Hjalte Pokraka, Andrzej Smith, Sid |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_29789 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Magic Relations and Critical Varieties of Feynman Integrals Crisanti, Giulio Frellesvig, Hjalte Pokraka, Andrzej Smith, Sid High Energy Physics - Theory High Energy Physics - Phenomenology Mathematical Physics 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. |
| title | Magic Relations and Critical Varieties of Feynman Integrals |
| topic | High Energy Physics - Theory High Energy Physics - Phenomenology Mathematical Physics |
| url | https://arxiv.org/abs/2605.29789 |