The geometric bookkeeping guide for $\varepsilon$-factorised differential equations

Abstract

Precision predictions for high-energy experiments rely on accurately evaluating multi-loop, multi-scale Feynman integrals in dimensional regularisation. The method of differential equations is by now the standard tool for this task, but its full power is only realised when the system can be brought into an $\varepsilon$-factorised form. In this talk, we present an algorithmic framework that systematically constructs $\varepsilon$-factorised differential equations for arbitrary integral families, independent of their underlying geometry. We work in the setting of twisted cohomology and study the space of differential forms associated with a given family of Feynman integrals in the Baikov representation. Our approach consists of two steps. First, we introduce a particular ordering for the Laporta algorithm that orders Feynman integrals within a sector according to their geometric properties. We observe that this order relation yields a basis whose differential equation is in a Laurent polynomial form in the dimensional regulator $\varepsilon$. In the second step, we systematically construct transformation matrices such that the resulting system is in the $\varepsilon$-factorised form.