Algebraic renormalization and Feynman integrals in configuration spaces

TL;DR

This paper advances the mathematical understanding of Feynman integrals in configuration spaces by developing algebro-geometric renormalization methods and relating results to mixed Tate motives, applicable to both massless and massive cases.

Contribution

It introduces a unified algebro-geometric framework for renormalizing Feynman integrals in configuration spaces, extending previous methods to massive cases and incorporating gauge theories.

Findings

Renormalized Feynman integrals are periods of mixed Tate motives.

A new compactification method handles all graphs with fixed vertices.

The renormalization process is described via Birkhoff factorization in Rota-Baxter algebras.

Abstract

This paper continues our previous study of Feynman integrals in configuration spaces and their algebro-geometric and motivic aspects. We consider here both massless and massive Feynman amplitudes, from the point of view of potential theory. We consider a variant of the wonderful compactification of configuration spaces that works simultaneously for all graphs with a given number of vertices and that also accounts for the external structure of Feynman graph. As in our previous work, we consider two version of the Feynman amplitude in configuration space, which we refer to as the real and complex versions. In the real version, we show that we can extend to the massive case a method of evaluating Feynman integrals, based on expansion in Gegenbauer polynomials, that we investigated previously in the massless case. In the complex setting, we show that we can use algebro-geometric methods to renormalize the Feynman amplitudes, so that the renormalized values of the Feynman integrals are given by periods of a mixed Tate motive. The regularization and renormalization procedure is based on pulling back the form to the wonderful compactification and replace it with a cohomologous one with logarithmic poles. A complex of forms with logarithmic poles, endowed with an operator of pole subtraction, determines a Rota Baxter algebra on the wonderful compactifications. We can then apply the renormalization procedure via Birkhoff factorization, after interpreting the regularization as an algebra homomorphism from the Connes-Kreimer Hopf algebra of Feynman graphs to the Rota-Baxter algebra. We obtain in this setting a description of the renormalization group. We also extend the period interpretation to the case of Dirac fermions and gauge bosons.