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Macaulay Matrix for Feynman Integrals: Linear Relations and Intersection Numbers

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arxiv 2204.12983 v2 pith:5DZKORHZ submitted 2022-04-27 hep-th

Vsevolod Chestnov , Federico Gasparotto , Manoj K. Mandal , Pierpaolo Mastrolia , Saiei J. Matsubara-Heo , Henrik J. Munch , Nobuki Takayama This is my paper
classification hep-th
keywords integralsfeynmanpfaffianrelationsequationsintersectionlinearmacaulay
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abstract

We elaborate on the connection between Gel'fand-Kapranov-Zelevinsky systems, de Rham theory for twisted cohomology groups, and Pfaffian equations for Feynman integrals. We propose a novel, more efficient algorithm to compute Macaulay matrices, which are used to derive Pfaffian systems of differential equations. The Pfaffian matrices are then employed to obtain linear relations for ${\cal A}$-hypergeometric (Euler) integrals and Feynman integrals, through recurrence relations and through projections by intersection numbers.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Resonance and Differential Reduction of Feynman Integrals

    hep-th 2026-06 unverdicted novelty 7.0 of 10

    The paper develops reduction operators from resonance in GKZ systems to contract edges in Feynman graphs for one-loop, sunrise, and banana graphs, closing differential equation systems to master integrals.

  2. Loop integrals in de Sitter spacetime: The parity-split IBP system and $\mathrm{d}\log$-form differential equations

    hep-th 2026-04 unverdicted novelty 7.0 of 10

    A parity-split IBP system for n-propagator families in de Sitter space is identified, along with a conjecture that dlog-form differential equations extend to dS integrands with Hankel functions, verified for the one-l...

  3. Feynman integral reduction with intersection theory made simple

    hep-th 2026-04 unverdicted novelty 7.0 of 10

    Branch representation reduces the variable count for intersection-theory-based Feynman integral reduction to at most 3L-3 for L-loop integrals regardless of leg number.

  4. Feynman Integral Reduction without Integration-By-Parts

    hep-th 2024-12 unverdicted novelty 5.0 of 10

    Contour equivalence in Feynman parameterization yields universal reduction formulas for one-loop integrals without integration-by-parts.

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