Imposing a family symmetry on the Standard Model in order to reduce the number of its free parameters, due to the Schur’s Lemma, requires an explicit breaking of this symmetry. To avoid the need for this symmetry to break, additional Higgs doublets can be introduced. In such an extension of the Standard Model, we investigate family symmetries of the Yukawa Lagrangian. We find that adding a second Higgs doublet (2HDM) does not help, at least for finite subgroups of the $U\left(3\right)$ group up to the order of 1025.

We present a method to identify symmetry groups of the Yukawa sector of the three-Higgs-doublet model and to determine the implication that the symmetry has on the lepton masses and mixing. The method can accommodate different hypotheses about the group representation assignments, and thus support the exploration of candidate symmetry groups. For one particular representation selection scheme we apply the computer-implemented method to scan all discrete groups of order less than 1035. It can be proven that none of these groups defines a flavor symmetry that implies masses and neutrino mixing angles consistent with the experimental lepton data, although several cases are found that are partially or approximately consistent.

In order to explain the fermions masses and mixing parameters appearing in the lepton sector of the Standard Model, one proposes the extension of its symmetry. A discrete, non-abelian subgroup of $U(3)$ is added to the gauge group $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ . Apart from that, one assumes the existence of one extra Higgs doublet. This article focuses mainly on the mathematical theorems and computational techniques which brought us to the results.

The Standard Model does not explain the hierarchy problem. Before the discovery of nonzero lepton mixing angle {\theta}13 high hopes in explanation of the shape of the lepton mixing matrix were combined with non abelian symmetries. Nowadays, assuming one Higgs doublet, it is unlikely that this is still valid. Texture zeroes, that are combined with abelian symmetries, are intensively studied. The neutrino mass matrix is a natural way to study such symmetries.

We address the possible impact of New Physics on neutrino oscillation experiments. This can modify the neutrino production, propagation and/or detection, making the full cross section non-factorizable in general. Thus, for example, the neutrino flux may not be properly described assuming an unitary MNS matrix and/or neutrinos may propagate differently depending of their Dirac or Majorana character. Interestingly enough, present limits on New Physics still allow for observable effects at future neutrino experiments.