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Multiplicative Renormalization in Causal Perturbation Theory

Jonah Epstein, Arne Hofmann, David Prinz

TL;DR

This work addresses how to realize multiplicative renormalization within causal perturbation theory by unifying Epstein–Glaser extension with the Connes–Kreimer Hopf-algebraRenormalization framework. It develops a position-space, algebraic Birkhoff decomposition using a Hadamard-based Rota–Baxter renormalization scheme, defines Z-factors from the counterterm map acting on combinatorial Green's functions, and proves that the renormalized Feynman rules coincide with those obtained from the renormalized Lagrange density. The paper establishes locality and multiplicativity as a consequence of the Rota–Baxter property and demonstrates the framework concretely through massless $\phi^3_6$ theory, including explicit two-loop analyses. The results bridge the analytic, causal perturbation theory with the combinatorial CK approach, yielding a local, multiplicative renormalization in AQFT with clear analytic–algebraic separation and potential broad applicability. Overall, it provides a rigorous path to multiplicative renormalization in position space that preserves locality and aligns the EG extension with CK renormalization.

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 $φ^3_6$-theory.

Multiplicative Renormalization in Causal Perturbation Theory

TL;DR

This work addresses how to realize multiplicative renormalization within causal perturbation theory by unifying Epstein–Glaser extension with the Connes–Kreimer Hopf-algebraRenormalization framework. It develops a position-space, algebraic Birkhoff decomposition using a Hadamard-based Rota–Baxter renormalization scheme, defines Z-factors from the counterterm map acting on combinatorial Green's functions, and proves that the renormalized Feynman rules coincide with those obtained from the renormalized Lagrange density. The paper establishes locality and multiplicativity as a consequence of the Rota–Baxter property and demonstrates the framework concretely through massless theory, including explicit two-loop analyses. The results bridge the analytic, causal perturbation theory with the combinatorial CK approach, yielding a local, multiplicative renormalization in AQFT with clear analytic–algebraic separation and potential broad applicability. Overall, it provides a rigorous path to multiplicative renormalization in position space that preserves locality and aligns the EG extension with CK renormalization.

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 -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 -theory.

Paper Structure

This paper contains 25 sections, 25 theorems, 138 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Epstein--Glaser and the extension of distributions
  3. Renormalisation maps and their construction
  4. Extension maps
  5. Connes--Kreimer and multiplicative renormalization
  6. The renormalization Hopf algebra
  7. The algebraic Birkhoff decomposition
  8. Position space renormalization data
  9. Multiplicative renormalization
  10. Example: Massless phi-3-6-theory
  11. Lagrange density
  12. Feynman rules
  13. Combinatorial Green's functions
  14. Coproduct identities
  15. Counterterms, renormalized Feynman rules and Z-factors
  16. ...and 10 more sections

Key Result

Lemma 2.4

Let $I \subset \mathbb{N}$ be finite. If $J\subset I$, define $J^c \coloneq I \backslash J$ and define the open sets Then

Figures (1)

  • Figure :

Theorems & Definitions (89)

  • Definition 2.1: Propagators
  • Definition 2.2
  • Example 2.3
  • Lemma 2.4
  • Definition 2.5
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • Definition 2.10
  • Remark 2.11
  • ...and 79 more