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

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.