Logarithmic forms and differential equations for Feynman integrals

TL;DR

The paper develops a d log integrand framework that turns Feynman integral differential equations into tractable, residue-based computations via localization in loop or embedding space. It shows how one-loop D-gon integrals satisfy motivic-like differential relations and how generalized unitarity guides a canonical-like, dimension-shifting structure. Extending to higher loops, it uncovers mixed-dimension integrals (weight mixing) and derives explicit differential equations for the two-loop off-shell ladder that mirror known ladder relations. Residue theorems further constrain the symbol alphabet, linking cut geometry to a minimal set of letters. The approach unifies geometric, motivic, and unitarity perspectives and points toward scalable, integrand-level methods for evaluating complex Feynman integrals.

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.

Figures (3)

• Figure 1: Sketch of the geometry of the various points and surfaces involved in the localization of the toy integral. In the left figure $\Sigma$ denotes the integration cycle, the green disks $D_\epsilon P$ denote the excision of the singularities $P$. The two points $(z_{\pm},\overline z_{\pm})$ outside of the integration cycle $\Sigma$ are the solutions to Eq.\ref{['eq:toycut']}. The right figure shows the relative positions of all the points and $\Sigma$ in the subspace ${\rm Im}(z) = {\rm Im}(\overline z) = 0$.
• Figure 2: Illustration of the geometry of the intersection of the integration cycle $\Sigma_4$ and the light-cones $(YX_{\pm})=0$. $\Sigma_2$ is the cycle that the differential of the $d \log$ integral localizes to.
• Figure 3: Spurious IR singularities from individual boxes. The double circle indicates a composite residue of the three propagators as well as the Jacobian.