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

Two negative signs naturally appear in the $U_{\mu 1}$ and $U_{\tau 2}$ elements of the Tri-bimaximal (TBM) matrix for positive values of the mixing angles $\theta_{12}$ and $\theta_{23}$. 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 $\theta_{12}$ and $\theta_{23}$. 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 TBM matrices is accompanied by two different $\mu-\tau$ symmetric mass matrices which can separately be related to the groups. To accommodate non-zero value of $\theta_{13}$ and deviate $\theta_{23}$ towards first octant, we then perturb the TBM mixing ansatz with the help of charged lepton correction. The diagonalizing matrices for charged lepton mass matrices also possess sign ambiguity and respect the grouping of TBM matrices. They are parametrized in terms of the Wolfenstein parameter $\lambda$ and satisfy unitarity condition up to second order in $\lambda$.