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

TL;DR

This paper introduces systematic improvements in Feynman integral reduction and differential equations, enabling more efficient and systematic derivation of epsilon-factorised forms crucial for high-precision calculations in quantum field theory.

Contribution

It presents a systematic algorithm for transforming Feynman integrals into epsilon-factorised differential equations, improving reduction efficiency and basis selection.

Findings

Trivialisation of epsilon-dependence in IBP identities.

Direct basis extraction with Laurent polynomial differential equations.

Universal transform to epsilon-factorised form.

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.