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.
Highlights
Where dmz is the measure form and ∂i = ∂/∂zi
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
We found that intersection numbers of the cohomology classes [φ±] ∈ H±mdW can be expressed in terms of Grothendieck residues around the critical points Crit(W ) as follows: φ−|φ+ dW = τ m ResdW =0 φ−∇−1 1∇−2 1 . . . ∇−m1 φ+ dmz, (2.45)
Summary
We briefly review the geometric setup underlying the remainder of the paper.
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