Aspects of BSM physics: quality problem, axion detection and naturalness

The Standard Model (SM) successfully describes most of the known interactions among elementary particles, yet several fundamental questions remain unanswered. This Ph.D. thesis investigates theoretical and phenomenological aspects of physics Beyond the Standard Model (BSM), focusing on two central issues in high-energy physics: the strong CP problem and the naturalness of the Higgs boson mass. This work addresses these challenges by combining non-perturbative methods in quantum field theory, effective field theory techniques, and phenomenologically motivated applications, with the goal of deepening our understanding of the mechanisms underlying BSM physics. A central theme of the thesis is the study of the QCD axion, a theoretically well-motivated candidate for resolving the strong CP problem and a possible component of dark matter. Using the formalism of instantons and the corresponding ’t Hooft vertex operators, we analyze the non-perturbative contributions to the axion potential, with particular attention to gravitational instanton effects. The thesis also explores connections with condensed matter physics. By employing effective field theories for anisotropic antiferromagnets, we describe the emergence of gapped Goldstone and pseudo-Goldstone modes. Building on these insights, we investigate the potential of antiferromagnetic materials, such as nickel oxide (NiO), as experimental targets for axion dark matter detection, thereby creating a bridge between high-energy theoretical predictions and experimental approaches in solid-state physics. Finally, the work examines the implications of the naturalness principle in a cosmological context, analyzing how naturalness considerations constrain inflationary dynamics and early-universe cosmology, particularly in models featuring super-Planckian field excursions. Taken together, the results presented in this thesis provide a coherent contribution to the study of physics beyond the Standard Model. By integrating rigorous theoretical analysis, phenomenological relevance, and experimental considerations, this work advances the understanding of non-perturbative effects, axion physics, and the interplay between high-energy and condensed matter systems, offering new insights into some of the most compelling questions in particle physics today