  High-energy completeness of quantum electrodynamics (QED) can be induced by
an interacting ultraviolet fixed point of the renormalization flow. We provide
evidence for the existence of two of such fixed points in the subspace spanned
by the gauge coupling, the electron mass and the Pauli spin-field coupling.
Renormalization group trajectories emanating from these fixed points correspond
to asymptotically safe theories that are free from the Landau pole problem. We
analyze the resulting universality classes defined by the fixed points,
determine the corresponding critical exponents, study the resulting phase
diagram, and quantify the stability of our results with respect to a systematic
expansion scheme. We also compute high-energy complete flows towards the
long-range physics. We observe the existence of a renormalization group
trajectory that interconnects one of the interacting fixed points with the
physical low-energy behavior of QED as measured in experiment. Within pure QED,
we estimate the crossover from perturbative QED to the asymptotically safe
fixed point regime to occur somewhat above the Planck scale but far below the
scale of the Landau pole.
