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From infinity to four dimensions: higher residue pairings and Feynman integrals. (English) Zbl 1435.81079
Summary: We study a surprising phenomenon in which Feynman integrals in \(D = 4 - 2\epsilon\) space-time dimensions as \(\epsilon \rightarrow 0\) can be fully characterized by their behavior in the opposite limit, \(\epsilon \rightarrow \infty\). 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 \(\alpha' \rightarrow 0\) and \(\alpha' \rightarrow \infty\) limits of intersection numbers coincide for scattering amplitudes of massless quantum field theories.

MSC:
81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
81U05 \(2\)-body potential quantum scattering theory
Software:
Fuchsia; Macaulay2
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