On the ambiguities in the tri-bimaximal mixing matrix and corresponding charged lepton corrections

Abstract

Two negative signs naturally appear in the Uμ1 and Uτ2 elements of the tri-bimaximal (TBM) matrix for positive values of the mixing angles θ12 and θ23. Apart from this, in other TBM matrices negative signs are shifted to other elements in each case. They account for positive as well as negative values of θ12 and θ23. We discuss the sign ambiguity in the TBM matrix and find that the TBM matrices, in fact, can be divided into two groups under certain circumstances. Interestingly, this classification of the TBM matrices is accompanied by two different μ − τ symmetric mass matrices which can separately be related to the groups. To accommodate the non-zero value of θ13 and deviate θ23 toward the first octant, we then perturb the TBM mixing ansatz with the help of charged lepton correction. The diagonalizing matrices for the charged lepton mass matrices also possess sign ambiguity and respect the grouping of the TBM matrices. They are parameterized in terms of the Wolfenstein parameter λ and satisfy the unitarity condition up to the second order in λ.