Neutrino Mass Matrix Subject to $\mu -\tau$ Symmetry and Invariant under a Cyclic Permutation

Neutrino masses arise via a seesaw mechanism and its mass hierarchy, with assumption that heavy Majorana neutrino mass matrix subject to $\mu -\tau$ symmetry and invariant under a cyclic permutation, are evaluated. Within this scenario, the neutrino masses: $\mid m_1\mid =\mid m_2\mid <\mid m_3\mid$ are obtained, which are incompatible with the experimental data. By modifying neutrino mass matrix with the zero sum rule condition, the neutrino masses in inverted hierarchy: $\mid m_3\mid <\mid m_1\mid <\mid m_2\mid$ are obtained.