Twisted Feynman Integrals: from generating functions to spin-resummed post-Minkowskian dynamics

TL;DR

The paper introduces twisted Feynman integrals, defined by an exponential deformation $e^{i\langle \alpha,\ell\rangle}$, and develops a geometric framework that treats loops as cycles in a graph with twists interpreted as magnetic-flux-like data. It shows twisted integrals belong to exponential periods, extends standard tools (graph polynomials, Baikov parametrisation, differential equations) to the twisted setting, and demonstrates that leading singularities via Baikov do not capture the full function space. A three-loop example illustrates the equivalence classes of twisted integrals under different realisations, reinforcing the geometric view. The work lays groundwork for applications in tensor reduction, Fourier transforms of Feynman integrals, and spin-resummed post-Minkowskian dynamics, while proposing avenues for numerical evaluation and deeper geometric analyses of twisted spaces.

Abstract

We propose to call a class of deformed Feynman integrals as twisted Feynman integrals, where the integrand has an additional exponential factor linear in loop momenta. Such integrals appear in various contexts: tensor reduction of Feynman integrals, Fourier transform of Feynman integrals, and spin-resummed dynamics in post-Minkowskian gravity. First, we construct a mathematical framework that manifests the geometric interpretation of twisted Feynman integrals. Next, we generalise the standard mathematical tools for studying Feynman integrals for application to their twisted cousins, and explore their mathematical properties. In particular, it is found that (i) twisted Feynman integrals fall under the class of exponential periods, and (ii) the leading singularity approach using the (generalised) Baikov parametrisation applied to twisted Feynman integrals fail to detect the geometry underlying their function space.

Figures (5)

• Figure 1: A one-loop diagram for spin-resummed dynamics that requires one-loop master integrals of the form Eq. (\ref{['eq:TFI_2PM_ex']}). Wavy lines denote massless mediator fields, dotted lines denote background worldline of massive particles, solid lines denote worldline fluctuations, and empty/shaded boxes denote MHV/$\overline{\text{MHV}}$-type couplings with the massless mediator field. Figure reproduced from Ref. Kim:2024grz.
• Figure 2: A one-loop Feynman graph and its twist, with momentum and position labels.
• Figure 3: A twisted Feynman graph is a Feynman graph put in a "magnetic field."
• Figure 4: Three realisations of the same twisted Feynman graph.
• Figure 5: Three realisations of the same twisted Feynman graph.