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<p>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-<i>ϵ</i> 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-<i>ϵ</i> 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.</p>

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Critical points and syzygies for Feynman integrals

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月2日

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  • Ben Page,
  • Qian Song

摘要

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.