An Introduction to Motivic Feynman Integrals

TL;DR

This paper introduces how motivic Galois theory applies to the algebraic structures of Feynman integrals, illustrating with examples from scalar massless $\

Contribution

It provides a step-by-step introduction to representing Feynman integrals as motives and reviews current research on their algebraic and categorical structures.

Findings

Motivic Galois theory offers a new perspective on Feynman integrals.

Detailed analysis of primitive log-divergent Feynman graphs in $\

The approach connects quantum field theory with algebraic geometry and motives.

Abstract

This article gives a short step-by-step introduction to the representation of parametric Feynman integrals in scalar perturbative quantum field theory as periods of motives. The application of motivic Galois theory to the algebro-geometric and categorical structures underlying Feynman graphs is reviewed up to the current state of research. The example of primitive log-divergent Feynman graphs in scalar massless $\phi^4$ quantum field theory is analysed in detail.