Rota-Baxter algebras, singular hypersurfaces, and renormalization on Kausz compactifications

TL;DR

This paper explores the use of Rota-Baxter algebras and Kausz compactifications to analyze and regularize divergent Feynman integrals, revealing new mixed Tate structures in the process.

Contribution

It introduces a novel regularization method for Feynman amplitudes using Rota-Baxter algebras on Kausz compactifications, highlighting their mixed Tate properties.

Findings

Kausz compactification is a Tate motive.

Boundary divisors are mixed Tate configurations.

New regularization differs from traditional renormalization methods.

Abstract

We consider Rota-Baxter algebras of meromorphic forms with poles along a (singular) hypersurface in a smooth projective variety and the associated Birkhoff factorization for algebra homomorphisms from a commutative Hopf algebra. In the case of a normal crossings divisor, the Rota-Baxter structure simplifies considerably and the factorization becomes a simple pole subtraction. We apply this formalism to the unrenormalized momentum space Feynman amplitudes, viewed as (divergent) integrals in the complement of the determinant hypersurface. We lift the integral to the Kausz compactification of the general linear group, whose boundary divisor is normal crossings. We show that the Kausz compactification is a Tate motive and that the boundary divisor and the divisor that contains the boundary of the chain of integration are mixed Tate configurations. The regularization of the integrals that we obtain differs from the usual renormalization of physical Feynman amplitudes, and in particular it may give mixed Tate periods in some cases that have non-mixed Tate contributions when computed with other renormalization methods.