Logarithmic forms and differential equations for Feynman integrals

TL;DR

This paper introduces a dlog-based approach to derive simple differential equations for Feynman integrals, connecting their geometry to special points in kinematic space and extending to higher loops.

Contribution

It presents a novel dlog representation method for Feynman integrals that simplifies differential equations and relates them to geometric structures, applicable to higher-loop cases.

Findings

Derived differential equations using localization and unitarity.

Connected the alphabet of equations to geometric cut equations.

Reproduced motivic formulas at one loop and extended to higher loops.

Abstract

We describe how a dlog representation of Feynman integrals leads to simple differential equations. We derive these differential equations directly in loop momentum or embedding space making use of a localization trick and generalized unitarity. For the examples we study, the alphabet of the differential equation is related to special points in kinematic space, described by certain cut equations which encode the geometry of the Feynman integral. At one loop, we reproduce the motivic formulae described by Goncharov \cite{Goncharov:1996tate} that reappeared in the context of Feynman parameter integrals in \cite{Spradlin:2011wp,Arkani-Hamed:2017ahv}. The dlog representation allows us to generalize the differential equations to higher loops and motivates the study of certain mixed-dimension integrals.