The geometric bookkeeping guide to Feynman integral reduction and $\varepsilon$-factorised differential equations

Iris Bree; Federico Gasparotto; Antonela Matijašić; Pouria Mazloumi; Dmytro Melnichenko; Sebastian Pögel; Toni Teschke; Xing Wang; Stefan Weinzierl; Konglong Wu; Xiaofeng Xu

arXiv · 2506.09124

The geometric bookkeeping guide to Feynman integral reduction and $\varepsilon$-factorised differential equations

Iris Bree·Federico Gasparotto·Antonela Matijašić·Pouria Mazloumi·Dmytro Melnichenko·Sebastian Pögel·Toni Teschke·Xing Wang·Stefan Weinzierl·Konglong Wu·Xiaofeng Xu

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Abstract

We report on three improvements in the context of Feynman integral reduction and $\varepsilon$-factorised differential equations: Firstly, we show that with a specific choice of prefactors, we trivialise the $\varepsilon$-dependence of the integration-by-parts identities. Secondly, 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 $\varepsilon$ and compatible with a particular filtration. Thirdly, we prove that such a differential equation can always be transformed to an $\varepsilon$-factorised form. This provides a systematic algorithm to obtain an $\varepsilon$-factorised 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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Iris Bree, Federico Gasparotto, Antonela Matijašić, Pouria Mazloumi, Dmytro Melnichenko, Sebastian Pögel, Toni Teschke, Xing Wang, Stefan Weinzierl, Konglong Wu, Xiaofeng Xu. 2026-03-26. The geometric bookkeeping guide to Feynman integral reduction and $\varepsilon$-factorised differential equations. https://arxiv.org/abs/2506.09124

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