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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
Full Text:
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