Scale invariance and the hierarchy problem: From dynamical symmetry breaking to a pseudo-Goldstone Higgs boson

In the Standard Model of particle physics, the masses of fermions and gauge bosons are generated by the Higgs mechanism at the classical level. The Higgs boson is an elementary scalar field and its mass is not protected against quantum correction from new scales thereby giving rise to the hierarchy problem. In this thesis, we explore classical scale invariance as a potential solution to the hierarchy problem. Our main result is the proposal of Custodial Naturalness, which is based on a combination of classical scale invariance and an enhanced custodial symmetry in the scalar sector, imposed at some high scale. Both symmetries are radiatively broken, and dimensional transmutation generates an intermediate scale that spontaneously breaks the enhanced custodial symmetry. The Standard Model Higgs boson arises as a pseudo-Nambu-Goldstone-boson associated with this breaking. We present several models in which Custodial Naturalness is realized. The number of free parameters in the minimal model is the same as in the Standard Model, and extensions can populate the neutrino portal or introduce dark matter candidates. Further, we also discuss outer automorphisms. The transformations of fields under outer automorphisms are not symmetries of the Lagrangian. We show that they can be thought of as symmetries of the beta functions. Further, we explore connections to naturalness and discuss scale transformations as outer automorphisms.