  We have computed the five-loop corrections to the scale dependence of the
renormalized coupling constant for Quantum Chromodynamics (QCD), its
generalization to non-Abelian gauge theories with a simple compact Lie group,
and for Quantum Electrodynamics (QED). Our analytical result, obtained using
the background field method, infrared rearrangement via a new
diagram-by-diagram implementation of the R* operation and the Forcer program
for massless four-loop propagators, confirms the QCD and QED results obtained
by only one group before. The numerical size of the five-loop corrections is
briefly discussed in the standard MSbar scheme for QCD with n_f flavours and
for pure SU(N) Yang-Mills theory. Their effect in QCD is much smaller than the
four-loop contributions, even at rather low scales.
