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.

Vector Space of Feynman Integrals and Multivariate Intersection Numbers

“…When at least one of the forms is non-logarithmic, the formula (3.75) is only valid asymptotically in the limit γ → ∞. In those cases one can still calculate intersection numbers as a series expansion in 1/γ, which was successfully applied to the computation of differential equations for certain Feynman integrals in [17]. The recursive algorithm for the computation of the multivariate intersection numbers presented in Sec.…”

“…The intersection theory-based decomposition has also been recently applied to the study of Feynman integrals in d = 4 ± 2 space-time dimensions, from which an unexpected relation between the behaviors around → 0 and → ∞ emerged [17] and was used to investigate the properties of canonical systems of differential equations [18]. A further interesting step for the construction of canonical integrals with intersection theory has been reported in [19].…”

Intersection Theory and Higgs Physics

“…[21,22]) and to the ε-expansions of Feynman integrals (see ref. [23]), but at least from the speakers perspective the most promising potential development would be that of a completely general computer implementation of the reduction algorithm described in the above, since there is much reason to believe that such an implementation would be much more efficient that those based on IBPs, as the huge linear systems that have to be solved as an intermediate step in the IBP based approach, are avoided completely.…”

Deriving canonical differential equations for Feynman integrals from a single uniform weight integral

“…Specifically, there are two approaches that seem promising in this regard. Firstly, it has been argued [4] (see also [40]) that a pre-canonical form should exist for all Feynman integrals, of the following type, d g(x) = [dA 0 (x) + dA 1 (x)] g(x) , (5.1) where the matrices A 0 and A 1 only involve logarithms (i.e., the fuchsian property of the differential equations is manifest for all singular points). In the polylogarithmic case, 'integrating out' the A 0 part can be done using algebraic functions only, but in the elliptic case (and beyond) this leads to special functions.…”