Hopf algebraic Renormalization of Kreimer's toy model

TL;DR

This thesis explores the algebraic structure of renormalization in quantum field theory using Hopf algebras, focusing on a simplified toy model to analyze the emergence of the renormalization group and Dyson-Schwinger equations.

Contribution

It applies Hopf algebraic renormalization techniques to a toy model, illustrating the connection between algebraic structures and renormalization processes in quantum field theory.

Findings

Derived explicit Hopf algebra morphisms for the toy model

Linked the anomalous dimension expansion to Mellin transform coefficients

Established automorphisms relating Mellin transforms within the Hopf algebra of rooted trees

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.