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New algorithms for Feynman integral reduction and epsilon-factorized differential equations

Iris Bree1, Federico Gasparotto2, Antonela Matijašić1, Pouria Mazloumi1, Dmytro Melnichenko1, Sebastian Pögel3, Toni Teschke1, Xing Wang4, Stefan Weinzierl1 et al. (ϵ Collaboration)

Stefan Weinzierl1, Konglong Wu5, and Xiaofeng Xu6 (ϵ Collaboration)

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Phys. Rev. D 113, 116019 – Published 15 June, 2026

DOI: https://doi.org/10.1103/mjpn-61yv

Abstract

In this paper, we give a detailed account of the algorithm outlined in [I. Bree et al. (ϵ Collaboration), companion Letter, Phys. Rev. Lett. 136, 241602 (2026).] for Feynman integral reduction and ϵ-factorized differential equations. The algorithm consists of two steps. In the first step, we use a new geometric order relation in the integration-by-parts reduction to obtain a basis of master integrals, whose differential equations on the maximal cut are of a Laurent polynomial form in the regularization parameter ϵ and compatible with a filtration. This step works entirely with rational functions. In a second step, we provide a method to ϵ-factorize the aforementioned Laurent differential equations. The second step may introduce algebraic and transcendental functions. We illustrate the versatility of the algorithm by applying it to different examples with a wide range of complexity.

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See Also

Geometric Bookkeeping Guide to Feynman Integral Reduction and ϵ-Factorized Differential Equations

Iris Bree, Federico Gasparotto, Antonela Matijašić, Pouria Mazloumi, Dmytro Melnichenko, Sebastian Pögel, Toni Teschke, Xing Wang, Stefan Weinzierl, Konglong Wu, and Xiaofeng Xu (ϵ Collaboration)
Phys. Rev. Lett. 136, 241602 (2026)

Article Text

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Outline

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Phys. Rev. D 113, 116019– Published 15 June, 2026
  • Received 20 November 2025
  • Accepted 11 May 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

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