Hopf algebraic Renormalization of Kreimer's toy model

Abstract

This masters thesis reviews the algebraic formulation of renormalization using Hopf algebras as pioneered by Dirk Kreimer and applies it to a toy model of quantum field theory given through iterated insertions of a single primitive divergence into itself. Using this example in a subtraction scheme, we exhibit the renormalized Feynman rules to yield Hopf algebra morphisms into the Hopf algebra of polynomials and as a consequence study the emergence of the renormalization group in connection with combinatorial Dyson-Schwinger equations. In particular we relate the perturbative expansion of the anomalous dimension to the coefficients of the Mellin transform of the integral kernel specifying the primitve divergence. A theorem on the Hopf algebra of rooted trees relates different Mellin transforms by automorphisms of this Hopf algebra.

arXiv comment70 pages

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Cited by in corpus (11)

• Feynman integrals and hyperlogarithms
• Renormalization and Mellin transforms
• Dyson-Schwinger Equations in Minimal Subtraction
• Renormalization, Hopf algebras and Mellin transforms
• Gauge Symmetries and Renormalization
• Families of algebraic structures
• Free Rota-Baxter family algebras and Free (tri)dendriform family algebras
• An Asymptotic Expansion for the Number of 2-Connected Chord Diagrams
• Bogoliubov type recursions for renormalisation in regularity structures
• Hopf algebra and renormalization: A brief review
• Bialgebra of specified graphs and external structures