Composite Higgs and Axions: Towards a Natural Theory

We review some attempts to solve the Naturalness problem - namely the the lightness of the Higgs mass in relation to the scale of new physics - and the strong CP problem, that is the measured CP-invariance in the strong sector despite the presence of the so-called term. In particular, in the first part we study the so-called composite Higgs models as a solution to the Naturalness problem, focussing our attention on a particular model, the minimal linear sigma model. In the second part we review one of the most convincing explanation to the strong CP problem, the so-called axions. There are essentially three ways to introduce axions in a beyond Standard Model Lagrangian; after a short summary, we focus on the so-called KSVZ model and verify why it is safe from possible fine-tuning problems. Finally, we investigate the possibility to solve both the problems in an unique framework, the so-called axion minimal linear sigma model. These models, unlike the KSVZ model, suffers fine-tuning problems that suggest to associate the arising Nambu-Goldstone boson to a more massive axion-like particle, rather than an axion.