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Register a new userDiagonalization of a Hermitian matrix and its application to neutrino mass matrix
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We carry out diagonalization of a $3times3$ Hermitian matrix of which Real component and Imaginary part are commutative and apply it to Majorana neutrino mass matrix $M=M_ u M_ u^dagger$ which satisfies the same condition. It is shown in a model-independent way for the kind of matrix M of which Real component and Imaginary part are commutative that $delta = pmpi/2$ which implies the maximal strength of CP violation in neutrino oscillations. And we obtain the prediction $cos (2theta_{23})=0$ for this kind of M. It is shown that the kind of Hermitian Majorana neutrino mass matrix M has only five real parameters and furthermore, only one free real parameter (D or A) if using the measured values of three mixing angles and mass differences as input.
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Using a mechanism which allows naturally small Dirac neutrino masses and its linkage to a dark gauge $U(1)_D$ symmetry, a realistic Dirac neutrino mass matrix is derived from $S_3$. The dark sector naturally contains a fermion singlet having a small seesaw mass. It is thus a good candidate for freeze-in dark matter from the decay of the $U(1)_D$ Higgs boson.
In the flavor basis where the mass eigenstates of three charged leptons are identified with their flavor eigenstates, one may diagonalize a 3 X 3 Majorana neutrino mass matrix M_nu by means of the standard parametrization of the 3 X 3 neutrino mixing matrix V. In this treatment the unphysical phases of M_nu have to be carefully factored out, unless a special phase convention for neutrino fields is chosen so as to simplify M_nu to M_nu without any unphysical phases. We choose this special flavor basis and establish some exact analytical relations between the matrix elements of M_nu M_nu^dag and seven physical parameters --- three neutrino masses (m_1, m_2, m_3), three flavor mixing angles (theta_12, theta_13, theta_23) and the Dirac CP-violating phase (delta). Such results allow us to derive the conditions for the mu-tau flavor symmetry with theta_23 = pi/4 and maximal CP violation with delta = +/- pi/2, which should be useful for discussing specific neutrino mass models. In particular, we show that theta_23 = pi/4 and delta = +/- pi/2 keep unchanged when constant matter effects are taken into account for a long-baseline neutrino oscillation experiment.
We investigate the possibility of expressing the charged leptons and neutrino mass matrices as linear combinations of generators of a single finite group. Constraints imposed on the resulting mixing matrix by current data restrict the group types, but allow a non zero value for the theta_13 mixing angle
Using the residual symmetry approach, we propose a complex extension of the scaling ansatz on $M_ u$ which allows a nonzero mass for each of the three light neutrinos as well as a nonvanishing $theta_{13}$. Leptonic Dirac CP violation must be maximal while atmospheric neutrino mixing need not to be exactly maximal. Each of the two Majorana phases, to be probed by the search for $0 u betabeta$ decay, has to be zero or $pi$ and a normal neutrino mass hierarchy is allowed.
Neutrino mass sum rules have recently gained again more attention as a powerful tool to discriminate and test various flavour models in the near future. A related question which was not yet discussed fully satisfactorily was the origin of these sum rules and if they are related to any residual or accidental symmetry. We will address this open issue here systematically and find previous statements confirmed. Namely, that the sum rules are not related to any enhanced symmetry of the Lagrangian after family symmetry breaking but that they are simply the result of a reduction of free parameters due to skillful model building.
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