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

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.