Motivic Galois coaction and one-loop Feynman graphs

TL;DR

This paper explores the motivic Galois coaction on one-loop Feynman graphs, providing explicit formulas and analyzing variations of mixed Hodge structures to understand the algebraic structure of Feynman amplitudes.

Contribution

It explicitly computes the motivic Galois coaction for one-loop Feynman graphs, including the box graph, and analyzes degenerate configurations involving blow-ups.

Findings

Explicit formula for the coaction on the four-edge cycle graph (box graph).

Extension of coaction computation to degenerate configurations via blow-ups.

Provides a framework for understanding motivic Feynman amplitudes in one-loop graphs.

Abstract

Following the work of Brown, we can canonically associate a family of motivic periods -- called the motivic Feynman amplitude -- to any convergent Feynman integral, viewed as a function of the kinematic variables. The motivic Galois theory of motivic Feynman amplitudes provides an organizing principle, as well as strong constraints, on the space of amplitudes in general, via Brown's "small graphs principle". This serves as motivation for explicitly computing the motivic Galois action, or, dually, the coaction of the Hopf algebra of functions on the motivic Galois group. In this paper, we study the motivic Galois coaction on the motivic Feynman amplitudes associated to one-loop Feynman graphs. We study the associated variations of mixed Hodge structures, and provide an explicit formula for the coaction on the four-edge cycle graph -- the box graph -- with non-vanishing generic kinematics, which leads to a formula for all one-loop graphs with non-vanishing generic kinematics in four-dimensional space-time. We also show how one computes the coaction in some degenerate configurations -- when defining the motive of the graph requires blowing up the underlying family of varieties -- on the example of the three-edge cycle graph.