Motivic Galois theory for one-loop Feynman integrals in momentum space

TL;DR

This paper introduces a motivic framework for one-loop Feynman integrals in momentum space, enhancing the understanding of their algebraic and geometric structures, especially with graph cuts included.

Contribution

It develops a functorial motivic approach for one-loop Feynman integrals that incorporates graph cuts and computes their motivic local systems and Galois actions.

Findings

Computed weight-graded pieces of motivic local systems as Tate twists of quadratic Artin motives.

Derived a formula for the de Rham motivic Galois group's (co)action in terms of cut quotient graphs.

Framework naturally includes graphs with cuts, extending previous Feynman representation approaches.

Abstract

We develop a motivic framework for Feynman integrals of one-loop graphs in momentum space. Its advantage compared to the already existing framework in Feynman representation is that it naturally includes graphs with cuts. To each such graph, we associate a motivic local system over the space of generic kinematics. Our construction is functorial with respect to the natural operations on graphs: edge contraction and cutting. We compute the weight-graded pieces of the motivic local systems. They are Tate twists of quadratic Artin motives associated with maximally cut quotient graphs. We also derive a formula for the (co)action of the de Rham motivic Galois group, expressed in terms of cut quotient graphs.