Flavouring Fundamental Partial Compositeness

One of the main challenges of Composite Higgs Models (CHMs) is the introduction of Standard Model (SM) fermion masses. To this end, the composite dynamics can be coupled to the SM elementary sector either via linear or bilinear interactions. In the latter case, the physical SM fermions are elementary, and get masses via Yukawa couplings with the composite Higgs. Instead, the introduction of linear couplings requires the existence of new composite coloured resonances, mixing with the elementary SM fermions. As a result, the mass eigenstates will be superpositions between the elementary and composite fields: this is the "Partial Compositeness" (PC) scenario. The composite fermions, called partners, are generated by the TC dynamics, and are vector-like, allowing to introduce gauge preserving masses, as opposed to SM chiral fermions. In this thesis work we will discuss one specific realization of partial compositeness, which contemplates the introduction of both TC-fermions and TC-scalars. In this framework one can introduce renormalizable Yukawa couplings involving one TC-fermion, one TC-scalar and one SM fermion. The naturalness problem becomes of secondary importance because this class of models allows for an ultraviolet implementation of the Partial Compositeness scenario. If one insists on naturalness one can view the new scalars as again composite states of a more natural theory. In this context, called "Minimal Fundamental Partial Compositeness", we focus in particular on the structure of the flavour symmetries of the fundamental Lagrangian and determine under which conditions the UV theory reproduces the SM Yukawa sector, including the CKM interactions. Additionally, we address the issue of generating the correct hierarchy among quarks and leptons