In perturbative quantum field theory, the process of renormalization turns
mathematically ill-defined integrals into physical meaningful expressions.
Counterterms are utilized to absorb divergences in Green's functions and
to compute observables as beta functions and anomalous dimensions. In this
talk, we discuss the relation of this classical picture to the
Bogoliubov-Parasyuk-Hepp-Zimmermann (BPHZ) prescription and a
Hopf-algebraic language of renormalization in the context of quantum
electrodynamics with a linear gauge fixing. Our analysis includes a
combinatorial description of renormalization group methods to derive
constraints for the next-to leading log term in the self-energy of the
electron.