Scale symmetry is an important concept in quantum and statistical physics. It arises at fixed points of the renormalisation group, often alongside full conformal symmetry, and implies that theories are massless with correlation functions given by universal numbers. New phenomena arise when scale symmetry is broken spontaneously, leading to a Goldstone boson, the dilaton, and the appearance of a mass scale that is not determined by the fundamental parameters of the theory. In this talk, I discuss scalar, fermionic, and Yukawa theories in three dimensions, each with lines of strongly-coupled conformal fixed points that terminate with spontaneous scale symmetry breaking. Interrelations between models, dualities, and aspects of dilaton physics are worked out from first principles. Further implications for CFTs and model building are indicated.

Free or interacting UV fixed points play a key role in the fundamental definition of QFT. In this talk, I give a broad overview of weakly and strongly interacting fixed points in 3d and 4d QFTs including models of particle physics with or without supersymmetry, and fermionic theories. Further, I explain methods and ideas to search for fixed points in 4d quantum gravity. Implications from the viewpoint of CFTs and higher-spin gauge theories through the AdS/CFT conjecture are also discussed.

It is commonly believed that for a quantum field theory to be fundamental, its high-energy behaviour must be governed by an ultraviolet (UV) fixed point. A very satisfying example is given by asymptotic freedom of QCD where the UV fixed point is non-interacting. In this talk, I discuss the existence of interacting UV fixed points in various quantum field theories, in particular in four dimensions, both with and without gravity. Implications for particle physics are also evaluated.