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

Abstract

We study a surprising phenomenon in which Feynman integrals in $D=4-2\epsilon$ space-time dimensions as $\epsilon \to 0$ can be fully characterized by their behavior in the opposite limit, $\epsilon \to \infty$. More concretely, we consider vector bundles of Feynman integrals over kinematic spaces, whose connections have a polynomial dependence on $\epsilon$ 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 $\epsilon$ or $1/\epsilon$. 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 ${\alpha }^{\prime }\to 0$ and ${\alpha }^{\prime }\to \infty$ limits of intersection numbers coincide for scattering amplitudes of massless quantum field theories.

arXiv comment41 pages, reference added

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