Generalised graph Laplacians and canonical Feynman integrals with kinematics

TL;DR

This paper introduces a new class of finite, generalized Feynman integrals associated with graphs, which depend on particle masses and momenta, and satisfy specific graphical relations and coproduct structures.

Contribution

It develops a framework linking graph Laplacians, cohomology, and Feynman integrals, providing a novel mathematical foundation for analyzing particle interactions.

Findings

Defines canonical integrals for graphs with external legs and masses

Establishes relations from graph contractions and coproduct structures

Connects integrands to cohomology of the general linear group

Abstract

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.