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Geometric Bookkeeping Guide to Feynman Integral Reduction and ϵ-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 Xu1,6 (ϵ Collaboration)

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Phys. Rev. Lett. 136, 241602 – Published 15 June, 2026

DOI: https://doi.org/10.1103/pyt8-d7rt

Abstract

We report on three improvements in the context of Feynman integral reduction and ϵ-factorized differential equations. First, we show that with a specific choice of prefactors, we trivialize the ϵ dependence of the integration-by-parts identities. Second, we observe that with a specific choice of order relation in the Laporta algorithm, we directly obtain a basis of master integrals, whose differential equation on the maximal cut is in Laurent polynomial form with respect to ϵ and compatible with a particular filtration. Third, we prove that such a differential equation can always be transformed to an ϵ-factorized form. This provides a systematic algorithm to obtain an ϵ-factorized differential equation for any Feynman integral. Furthermore, the choices for the prefactors and the order relation significantly improve the efficiency of the reduction algorithm.

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See Also

New algorithms for Feynman integral reduction and epsilon-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. D 113, 116019 (2026)

Article Text

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Outline

Information

Phys. Rev. Lett. 136, 241602– Published 15 June, 2026
  • Received 12 June 2025
  • Revised 11 February 2026
  • Accepted 25 March 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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