Interacting ultraviolet completions of four-dimensional gauge theories

Ultraviolet fixed points provide an important means to define continuum limits of quantum field theories. In this contribution, we explain how interacting ultraviolet fixed points may arise in general four-dimensional gauge theories coupled weakly to scalar and fermionic matter fields. Our results derive from a systematic classification of all high- and low-energy fixed points in general gauge theories within perturbation theory. It is shown why and how Yukawa interactions provide a unique mechanism for the generation of perturbative ultraviolet fixed points. Phase diagrams for simple gauge theories are discussed to illustrate the main results. Some implications for particle physics beyond the Standard Model are indicated.