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

TL;DR

This paper reveals that Feynman integrals in four-dimensional quantum field theory can be understood through their behavior at large dimensions using higher residue pairings, connecting geometric intersection theory with physics.

Contribution

It introduces a novel approach to analyze Feynman integrals via intersection numbers and higher residue pairings, linking large-dimensional limits to physical properties.

Findings

Feynman integrals are characterized by their behavior as  o ext{infinity}

Intersection numbers localize on critical points of a Morse function

Large- and small- limits of intersection numbers coincide in certain cases

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 $\alpha' \to 0$ and $\alpha' \to \infty$ limits of intersection numbers coincide for scattering amplitudes of massless quantum field theories.