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

Abstract

We study a surprising phenomenon in which Feynman integrals in D = 4 − 2ε space-time dimensions as ε → 0 can be fully characterized by their behavior in the opposite limit, ε → ∞. More concretely, we consider vector bundles of Feynman integrals over kinematic spaces, whose connections have a polynomial dependence on ε 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 ε or 1/ε. 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 α′ → 0 and α′ → ∞ limits of intersection numbers coincide for scattering amplitudes of massless quantum field theories.