Non-Unitary $3 \times 3$ Mixing in Majorana Neutrinos and Vector-like Quark Models

Non-unitary $3 \times 3$ mixing matrices are present in many extensions of the Standard Model. One of the most simple extensions is the addition of $n_R$ right-handed neutrinos without imposing Lepton Number Conservation ($n_R$$ν$SM). In the region of the parameter space where the usual seesaw approximation is valid, deviations from $3 \times 3$ unitarity of the leptonic mixing matrix are negligible, while one has three light neutrinos and two or three heavy neutrinos with masses close to the GUT scale. This gap between the mass scales occurs while having order 1 Yukawa couplings, providing a natural explanation for the smallness of light neutrino masses - hence the name seesaw. Other regions of the parameter space, with lighter heavy neutrinos - with an eV or keV mass, for instance - and bigger deviations from unitarity of the leptonic mixing matrix, are still allowed by experiment. In fact, whenever heavy neutrino masses are many orders of magnitude below the GUT scale there are interesting phenomenological implications at low energies. To analyse such regions with precision, one needs an exact parameterisation, since approximations that were valid in the usual seesaw case now fail. This thesis will consist of a study and classification of regions of the 3$ν$SM parameter space in the light of a newly-developed exact parameterisation. Furthermore, it will also include a study of models with vector-like quarks, another simple extension of the SM, that contains a $3 \times 3$ non-unitary quark mixing matrix. Studying the most interesting regions of the parameter space of vector-like quarks models also benefits from the use of the aforementioned exact parameterisation. These regions are phenomenologically rich and may explain certain open questions like the CKM unitarity problem/Cabibbo angle anomaly. A chapter discussing the experimental implications of these models is also included.