Phenomenology of Discrete Flavour Symmetries

The flavour puzzle is an open problem both in the Standard Model and in its possible supersymmetric or grand unified extensions. In this thesis, we discuss possible explanations of the origin of fermion mass hierarchies and mixings by the use of non-Abelian discrete flavour symmetries. We present two realisations in which the flavour symmetry contains either the double-valued group $T'$ or the permutation group $S_4$: the spontaneous breaking of the flavour symmetry produces realistic fermion mass hierarchies, the lepton mixing matrix close to the so-called tribimaximal pattern ($\sin^2\theta_{12}=1/3$, $\sin^2\theta_{23}=1/2$ and $\theta_{13}=0$) and the quark mixing matrix comparable to the Wolfenstein parametrisation. The exact tribimaximal scheme deviates from the experimental best-fit angles for values at most of the $1\sigma$ level. In the $T'$- and $S_4$-based models, the symmetry breaking accounts for such discrepancies, by introducing corrections to the tribimaximal pattern of the order of $\lambda^2$, being $\lambda$ the Cabibbo angle. On the experimental side, the present measurements do not exclude $\theta_{13} \sim \lambda$ and therefore, if it is found that $\theta_{13}$ is close to its present upper bound, this could be interpreted as an indication that the agreement with the tribimaximal mixing is accidental. Then a scheme where instead the bimaximal mixing ($\sin^2\theta_{12}=1/2$, $\sin^2\theta_{23}=1/2$ and $\theta_{13}=0$) is the correct first approximation modified by terms of $\mathcal{O}(\lambda)$ could be relevant. This recalls the well-known empirical quark-lepton complementarity, for which $\theta_{12}+\lambda\sim \pi/4$. We present a flavour model based on the spontaneous breaking of the $S_4$ discrete group which naturally leads to the bimaximal mixing at the leading order and, after the introduction of the breaking terms, to $\theta_{13} \sim \lambda$ and $\theta_{12}+\mathcal{O}(\lambda)\sim \pi/4$, which we call ``weak'' complementarity relation. Masses and mixings are evaluated at a very high energy scale and for a comparison with experimental measurements we illustrate a general analysis on the stability under the renormalisation group running to evolve these observables to low energies. We consider also the constraints on flavour violating processes arising from introducing a flavour symmetry: in particular we concentrate on the lepton sector, analysing some lepton flavour violating decays and the discrepancy between the theoretical prediction and the experimental measurement of the anomalous magnetic moment of the muon. We develop the study both in the Standard Model scenario and in its minimal supersymmetric extension, using at first an effective operator approach and then a complete loop computation. Interesting hints for the scale of New Physics and for the forthcoming experimental results from LHC are found. Finally we discuss the impact of an underlining flavour symmetry on leptogenesis in order to explain the baryon asymmetry of the universe.