Asymptotic safety: from perturbatively exact models to particle physics

The behaviourr of quantum field theories at different energy scales is encoded by its renormalisation group (RG) flow. To be physical and fundamental, their high-energy limit needs to be controlled by an ultraviolet fixed point – a property called asymptotic safety. Recent developments have proven that in strictly four spacetime dimensions and beyond the well-known case of asymptotic freedom, a family of theories with weakly interacting UV fied points exists in a perturbatively exact setting. This thesis is dedicated to further investigate this phenomenon. Following a thorough introduction, this work consist of three main parts. The first part is concerned with technical aspects of computing perturbative renormalisation group equations. We revisit literature results and correct several long-standing errors. Moreover, we present a novel software package to facilitate these calculations, and highlight its advantages over existing tools. The second part is focused on the phenomenon of perturbatively exact asymptotic safety. We systematise the search for such theories to identify new classes of models. For an important benchmark model with unitary gauge symmetry and Dirac fermions, we extend the RG analysis to the highest available loop order. We determine the size of the conformal window and the mechanisms limiting it. Furthermore, we find two novel classes of asymptotically safe theories involving orthogonal or symplectic gauge groups and Majorana fermions. We also discover new large-N equivalences amongst seemingly different quantum field theories, leading to new dualities, and a triality amongst asymptotically safe theories. In the third part, realistic models of particle physics are constructed. The Standard Model is coupled to an extended sector via electroweak, Yukawa and Higgs portal interactions. We demonstrate that safety until the Planck scale and stability of the Higgs potential is achieved in these theories. Moreover, we show that both discrepancies of electron and muon anomalous magnetic moments can be accounted for simultaneously without explicit lepton flavour violation.