From infinity to four dimensions: higher residue pairings and Feynman integrals

DOI10.1007/JHEP02(2020)159zbMATH Open1435.81079arXiv1910.11852WikidataQ114233661 ScholiaQ114233661MaRDI QIDQ2188662

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Publication date: 11 June 2020

Published in: (Search for Journal in Brave)

Abstract: We study a surprising phenomenon in which Feynman integrals in $D=4-2varepsilon$ space-time dimensions as $varepsilono0$ can be fully characterized by their behavior in the opposite limit, $varepsilonoinfty$. 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 $alph{a}^{\prime }o0$ and $alph{a}^{\prime }oinfty$ limits of intersection numbers coincide for scattering amplitudes of massless quantum field theories.

Full work available at URL: https://arxiv.org/abs/1910.11852

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