Yukawa matrix unification in the Minimal Supersymmetric Standard Model

In this dissertation, the Minimal Supersymmetric Standard Model (MSSM) is studied as a low-energy theory stemming from the $SU(5)$ Grand Unified Theory (GUT). We investigate the possibility of satisfying the minimal $SU(5)$ boundary condition $\mathbf{Y}^d=\mathbf{Y}^{e\,T}$ for the full $3\!\times\!3$ down-quark and lepton Yukawa matrices at the GUT scale within the $R$-parity conserving MSSM. We give numerical evidence in favour of the statement: There exist regions in the parameter space of the R-parity conserving MSSM for which the unification of the down-quark and lepton Yukawa matrices takes place, while the predicted values of flavour, electroweak and other collider observables are consistent with experimental constraints. Furthermore, we find evidence that the bottom-tau and strange-muon Yukawa unification is possible with a stable MSSM vacuum in the standard form. We investigate two separate scenarios of the soft supersymmetry breaking terms at the GUT scale. In the first one, it is assumed that the soft terms are flavour-diagonal in the super-CKM basis. In such a case, the trilinear Higgs-squark-squark $A$-terms can generate large threshold corrections to $\mathbf{Y}^d$ at the superpartner decoupling scale. In effect, the condition $\mathbf{Y}^d=\mathbf{Y}^{e\,T}$ imposed at the GUT scale can be satisfied. However, the large trilinear terms make the usual Higgs vacuum metastable (though sufficiently long-lived). In the second scenario, we consider non-vanishing flavour off-diagonal entries in the soft SUSY-breaking mass matrices. We show that a non-trivial flavour structure of the soft SUSY-breaking sector can allow a precise bottom-tau and strange-muon Yukawa coupling unification, while satisfying all phenomenological constraints.