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

TL;DR

This paper introduces twisted Feynman integrals with exponential factors, exploring their mathematical properties and geometric interpretations, relevant to tensor reduction, Fourier transforms, and spin-resummed gravity dynamics.

Contribution

It develops a mathematical framework for twisted Feynman integrals, generalizes tools for their study, and reveals their unique properties and geometric aspects.

Findings

Symanzik polynomials become graded and non-homogeneous.

Twisted Feynman integrals are classified as exponential periods.

Function space geometry cannot be deduced from leading singularities.

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) Symanzik polynomials are no longer homogeneous and become graded, (ii) twisted Feynman integrals fall under the class of exponential periods, and (iii) the geometry of the function space cannot be inferred from the leading singularity computed through the (generalised) Baikov parametrisation of twisted Feynman integrals.