Abstract
We investigate a novel theoretical structure underlying the computation of integration-by-parts relations between Feynman integrals via syzygy-based methods. Building on insights from intersection theory, we analyze the large-ϵ limit of dimensional regularization on the maximal cut, showing that total derivatives vanish on the critical locus of the logarithm of the Baikov polynomial — the locus known to govern the number of master integrals. We introduce “critical syzygies” as a distinguished subset of syzygies that captures this behavior. We show that, when the critical locus is isolated, critical syzygies generate a sufficient set of total derivatives in the large-ϵ limit. We study their structure analytically at one loop and develop a numerical approach for their construction at two loops. Our results demonstrate that critical syzygies are a valuable tool for integral reduction in cutting-edge two-loop examples, offering a novel geometric perspective on integration-by-parts relations.
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Acknowledgments
We thank Giulio Gambuti, Harald Ita, Pavel Novichkov and Vasily Sotnikov for insightful discussions. We thank Harald Ita and Pavel Novichkov for comments on the draft. The work of Qian Song was supported by the European Research Council (ERC) under the European Union’s Horizon Europe research and innovation program grant agreement 101078449 (ERC Starting Grant MultiScaleAmp). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.
Data Availability Statement. This article has no associated data or the data will not be deposited.
Code Availability Statement. This article has no associated code or the code will not be deposited.
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Page, B., Song, Q. Critical points and syzygies for Feynman integrals. J. High Energ. Phys. 2026, 4 (2026). https://doi.org/10.1007/JHEP02(2026)004
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DOI: https://doi.org/10.1007/JHEP02(2026)004