A model for Dirac neutrino mass matrix with only four parameters

The exchange symmetry between the muon neutrino and the tau neutrino for the neutrino mass matrix has been very useful in understanding the near maximal atmospheric neutrino mixing angle. However, this symmetry can not be imposed at the Lagrangian level, since the charged lepton partners, muon and tau do not satisfy this symmetry. We extend the Standard model to include three right handed singlet neutrinos, and impose the most general symmetry between $\nu_{\mu R}$ and $\nu_{\tau R}$ sectors followed by a CP transformation of the leptonic sector at the Lagrangian level. This symmetry does not affect the charged leptons. With the additional assumption of the hermiticity of the ensuing Dirac neutrino mass matrix, we get a 4 parameter neutrino mass matrix in good agreement with the available neutrino data for the inverted neutrino mass hierarchy. The model also predicts the values of the three neutrino masses, and the leptonic CP violating phase which can be tested in the upcoming neutrino experiments.