The Tri-Bimaximal (TBM) mixing predicts a vanishing θ13. This can be attributed to the inherited µ − τ symmetry of TBM mixing. We break its µ − τ symmetry by adding a complex magic matrix with one variable to TBM neutrino mass matrix with one vanishing eigenvalue. We present two such textures and study their phenomenological implications.

We study the phenomenological implications of breaking the Tri-Bimaximal (TBM) mixing in such a way that either first or second column of TBM mixing matrix remains invariant. We present two such textures and confront them with the experimental data. We give the predictions of these textures for atmospheric mixing angle θ 23 and Dirac-type CP violating phase δ that will be measured in the future experiments.

The TBM neutrino mass matrix is invariant under the TM1, TM2, and μ − τ exchange symmetry. The prediction of vanishing θ 13 by TBM is due to its invariance under μ − τ exchange symmetry. We can construct neutrino mass textures where TBM mass matrix is the main contributing matrix and a μ − τ symmetry breaking term is added as an additional contribution. The overall matrix will be invariant under either TM1 or TM2 symmetry only. We present eight such textures which are invariant under either TM1 or TM2 symmetry but are not invariant under the μ − τ exchange symmetry. These textures give definite predictions on the values of m β , m ββ , CP violating phase δ, and the octant of the mixing angle θ 23.

Non-Abelian discrete symmetry group [Formula: see text] has been used to construct quark mass matrices in the flavor basis. Interestingly, the case when [Formula: see text] is nondiagonal and [Formula: see text] is diagonal, is very much compatible with the presently refined data regarding the quark masses and mixing. However, the other case, when [Formula: see text] is nondiagonal and [Formula: see text] is diagonal, looks to be incompatible with the data.