The $\mu$-$\tau$ symmetry imposed on the neutrino mass matrix in the flavor basis is known to be quite predictive. We integrate this very specific neutrino symmetry into a more general framework based on the supersymmetric $SO(10)$ grand unified theory. As in several other models, the fermion mass spectrum is determined by Hermitian mass matrices resulting from the renormalizable Yukawa couplings of the 16-plet of fermions with the Higgs fields transforming as 10, $\overline{126}$, and 120 representations of the $SO(10)$ group. The $\mu$-$\tau$ symmetry is spontaneously broken through the 120-plet. Consequences of this scheme are considered for fermion masses using both a type-I and a type-II seesaw mechanism. This scenario is shown to lead to a generalized $CP$ invariance of the mass matrices and vanishing $CP$ violating phases if the Yukawa couplings are invariant under the $\mu$-$\tau$ symmetry. Small explicit breaking of the $\mu$-$\tau$ symmetry is then shown to provide a very good understanding of all of the fermion masses and mixing. Detailed fits to the fermion spectrum are presented in several scenarios. One obtains a very good fit to all observables in the context of the type-I seesaw mechanism, but the type-II seesaw model also provides a good description except for the overall scale of the neutrino masses. Three major predictions on the leptonic mixing parameters in the type-I seesaw case are (i) the atmospheric mixing angle ${\theta }_{23}^l$ close to maximal, (ii) ${\theta }_{13}^l$ close to the present upper bound, and (iii) a negative but very small Dirac $CP$ violating phase in the neutrino oscillations.

• Received 12 May 2009

DOI:https://doi.org/10.1103/PhysRevD.79.115017