From Infinity to Four Dimensions: Higher Residue Pairings and Feynman Integrals

TL;DR

This work reframes multi-loop Feynman integrals in $D=4-2\varepsilon$ as twisted-periods on graph moduli spaces, enabling a vector-bundle view over the kinematic space whose connection is polynomial in $\varepsilon$. By organizing the $\varepsilon$-dependence through Saito's higher residue pairings, the authors show that the exact differential equations governing the integrals can be obtained from a finite truncation of the opposite limit $\varepsilon\to\infty$, with critical points localizing contributions. The paper provides explicit constructions and computations for a one-loop massless box and a two-loop massive sunrise, deriving the connection matrices from leading, subleading, and subsubleading residues and checking integrability against existing tools. The results uncover a moduli-space localization phenomenon in scattering amplitudes, bridge to Landau–Ginzburg dualities, and suggest a robust, geometry-driven method to analyze Feynman integrals across regulator regimes, including potential ε-form simplifications in suitable bases.

Abstract

We study a surprising phenomenon in which Feynman integrals in $D=4-2\varepsilon$ space-time dimensions as $\varepsilon \to 0$ can be fully characterized by their behavior in the opposite limit, $\varepsilon \to \infty$. More concretely, we consider vector bundles of Feynman integrals over kinematic spaces, whose connections have a polynomial dependence on $\varepsilon$ and are known to be governed by intersection numbers of twisted forms. They give rise to differential equations that can be obtained exactly as a truncating expansion in either $\varepsilon$ or $1/\varepsilon$. We use the latter for explicit computations, which are performed by expanding intersection numbers in terms of Saito's higher residue pairings (previously used in the context of topological Landau-Ginzburg models and mirror symmetry). These pairings localize on critical points of a certain Morse function, which correspond to regions in the loop-momentum space that were previously thought to govern only the large-$D$ physics. The results of this work leverage recent understanding of an analogous situation for moduli spaces of curves, where the $α' \to 0$ and $α' \to \infty$ limits of intersection numbers coincide for scattering amplitudes of massless quantum field theories.