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

Abstract

We report on three improvements in the context of Feynman integral reduction and $\epsilon$-factorised differential equations: Firstly, we show that with a specific choice of prefactors, we trivialise the $\epsilon$-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 $\epsilon$ and compatible with a particular filtration. Thirdly, we prove that such a differential equation can always be transformed to an $\epsilon$-factorised form. This provides a systematic algorithm to obtain an $\epsilon$-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.

9 pages, v2: version to be published