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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.

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In this article, we present a sufficient condition for the exponential exp(−f ) to have a tail decay stronger than any Gaussian, where f is defined on a locally convex space X and grows faster than a squared seminorm on X. In particular, our result proves that exp(−p(x) 2+ε + αq(x) 2 ) is integrable for all α, ε > 0 w.r.t. a Radon Gaussian measure on a nuclear space X, if p and q are continuous seminorms on X with compatible kernels. is can be viewed as an adaptation of Fernique's theorem and, for example, has applications in quantum field theory.

The functional renormalisation group equation is derived in a mathematically rigorous fashion in a framework suitable for the Osterwalder-Schrader formulation of quantum field theory. To this end, we devise a very general regularisation scheme and give precise conditions for the involved regulators guaranteeing physical boundary conditions. Furthermore, it is shown how the classical limit is altered by the regularisation process leading to an inevitable breaking of translation invariance.We also give precise conditions for the convergence of the obtained theories upon removal of the regularisation.