Angles of the Strong CP Problem

The Strong CP problem is one of the most challenging and long-standing puzzles in physics beyond the Standard Model. In this thesis we review the formulation of the problem and thoroughly examine some of the most promising solutions put forth over the years. We first delve into both perturbative and non-perturbative aspects related to topological angles, including the issue of their perturbative renormalization. Then we identify and scrutinize two distinct classes of solutions addressing the Strong CP problem: UV solutions and those based on a Peccei-Quinn symmetry. Among the latter, we offer a brief overview of the QCD axion and its associated quality problem. To address this issue we propose an explicit model for a heavy axion, dubbed Grand Color axion, and study its phenomenology in detail. Next, we shift our focus to UV solutions. We start by illustrating the logic behind models of spontaneous P/CP violation, specializing then to the particular case of Nelson-Barr models. We meticulously investigate these models, pinpointing a naturalness issue inherently structured in this class of solutions. Finally, we present a set of scenarios designed to tackle this new problem and implement these ideas in a concrete UV realization.