  In the context of massless quantum electrodynamics (QED) with a linear
covariant gauge fixing, the connection between the counterterm and the
Hopf-algebraic approach to renormalization is examined. The coproduct formula
of Green's functions contains two invariant charges, which give rise to
different renormalization group functions. All formulas are tested by explicit
computations to third loop order. The possibility of a finite electron
self-energy by fixing a generalized linear covariant gauge is discussed. An
analysis of subdivergences leads to the conclusion that such a gauge only
exists in quenched QED.
