Multiplicative Renormalization in Causal Perturbation Theory

TL;DR

This paper develops a multiplicative renormalization framework within Epstein--Glaser perturbative algebraic quantum field theory by integrating Connes--Kreimer's Hopf algebra approach, enabling regularized Feynman rules and explicit examples.

Contribution

It combines Connes--Kreimer and Epstein--Glaser renormalization schemes, establishing a multiplicative renormalization method with algebraic Birkhoff decomposition and explicit $Z$-factor definitions.

Findings

Constructed the renormalized Feynman rules and counterterm map.

Showed the Hadamard singular part satisfies the Rota--Baxter property.

Applied the framework to $oldsymbol{ ext{phi}^3_6}$-theory.

Abstract

We construct multiplicative renormalization for the Epstein--Glaser renormalization scheme in perturbative Algebraic Quantum Field Theory: To this end, we fully combine the Connes--Kreimer renormalization framework with the Epstein--Glaser renormalization scheme. In particular, in addition to the already established position-space renormalization Hopf algebra, we also construct the renormalized Feynman rules and the counterterm map via an algebraic Birkhoff decomposition. This includes a discussion about the appropriate target algebra of regularized distributions and the renormalization scheme as a Rota--Baxter operator thereon. In particular, we show that the Hadamard singular part satisfies the Rota--Baxter property and thus relate factorization in Epstein--Glaser with multiplicativity in Connes--Kreimer. Next, we define $Z$-factors as the images of the counterterm map under the corresponding combinatorial Green's functions. This allows us to define the multiplicatively renormalized Lagrange density, for which we show that the corresponding Feynman rules are regular. Finally, we exemplify the developed theory by working out the specific case of $\phi^3_6$-theory.