Neutrino oscillations, neutrino masses and supersymmetry

The neutrino mass matrix is constraint considering in a three-generation scenario, which includes or excludes LSND and considers the atmospheric neutrinos of Super-Kamiokande and the solar neutrino experiments. In this way one is able to construct the neutrino mass matrix, up to a mass scale and up to the Majorana neutrino CP eigenvalues. By using the expectation value of the electron neutrino mass obtained in the neutrinoless double beta decay, one can derive an upper limit of 2.53 [eV] for the sum of the masses of the three neutrinos. In a second part we use the R violating minimal supersymmetric model (R-MSSM) to construct the neutrino mass matrix. The contributions on the tree level yield by mixing with the neutralinos (photino, zinc, higgsino 1 and higgsino 2) a separable 3 . 3 mass matrix, which has two mass eigenvalues, which are zero and a third eigenvalue different from zero. This yields a hierachichal structure of two neutrinos with small masses and a third neutrino with a larger mass. If one includes the loop diagrams of quarks and SUSY quarks and of leptons and SUSY leptons one obtains for the three neutrinos a sum of tree separable matrices. Such a separable mass matrix of rank 3 has 9 independent parameters. One can fit five parameters from the neutrino oscillations: the three mixing angles of the matrix transforming the neutrinos from the mass eigenstates to the weak eigenstates and the two differences of the squared masses. One therefore has to reduce the nine free parameters of the mass matrix in the R-MSSM from nine to five. This is possible imposing a constraint on the cubic couplings employing an additional U(1) flavour symmetry. This approach has been very successful in studying the Yukawa couplings to fix the masses of the quarks and the non-neutrino leptons. In this way it is possible to reduce the R-parity violating coupling constants of the the R-MSSM from six to one. This reduces the free parameters of the neutrino mass matrix obtained in the SUSY model to four. We find in a three-family mixing scheme an averaged Majorana neutrino mass, which in principle can be obtained from the neutrinoless double beta decay, of \ < m > \ = 0.009 to 0.045[eV]. These predictions can perhaps be tested by the next generation of neutrinoless double beta decay experiments.