Constraining Neutrino Mass Matrix from Modified BM with Softly Broken $\mu-\tau$ Symmetry

The bimaximal (BM) neutrino mixing matrix was formulated in order to accommodate the data of the experimental results which indicate that both solar and atmospheric neutrino oscillation in vacuum are near maximal. But, after the T2K and Daya Bay Collaborations reported that the mixing angle $\theta_{13}$ is nonzero and relatively large, many authors have modified the neutrino mixing matrix in order to accommodate experimental data. We modified the BM mixing matrix by introducing a simple perturbation matrix into BM mixing matrix. The modified BM mixing matrix can proceed the mixing angles which are compatible with the globat fit analysis data and by imposing the $\mu-\tau$ symmetry into mass matrix from modified BM, we have the neutrino mass in normal hierarchy: $m_{1}<m_{2}<m_{3}$. Using the neutrino masses that obtained from neutrino mass matrix in the scheme of modified BM and imposing the constraint exact $\mu-\tau$ symmetry into neutrino mass matrix, we cannot have compatible squared-mass differences for both $\Delta m_{21}^{2}$ and $\Delta m_{32}^{2}$ as dictated by experimental results. We break softly the $\mu-\tau$ symmetry by introducing a small parameter $\lambda$ into neutrino mass matrix which then can proceed neutrino masses are in agreement with the squared mass difference as dictated by experimental results.The predicted neutrino effective mass: $\left|m_{ee}\right|=0.0155 {\rm eV}$ in this paper can be tested in the future neutrinoless double beta decay