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Logarithmic forms and differential equations for Feynman integrals. (English) Zbl 1435.81078

Summary: We describe in detail how a \(d\) log 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 that reappeared in the context of Feynman integrals. The \(d\) log representation allows us to generalize the differential equations to higher loops and motivates the study of certain mixed-dimension integrals.

MSC:

81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
81U05 \(2\)-body potential quantum scattering theory
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