Generalised Graph Laplacians and Canonical Feynman Integrals with Kinematics

To any graph with external half-edges and internal masses, we associate canonical integrals which depend non-trivially on particle masses and momenta, and are always finite. They are generalised Feynman integrals which satisfy graphical relations obtained from contracting edges in graphs, and a coproduct involving both ultra-violet and infra-red subgraphs. Their integrands are defined by evaluating bi-invariant forms, which represent stable classes in the cohomology of the general linear group, on a generalised graph Laplacian matrix which depends on the external kinematics of a graph.