Modifications to the neutrino mixing given by the mu-tau reflection symmetry

The $\mu$-$\tau$ reflection symmetry serves as a unique basis for understanding the observed neutrino mixing as it can lead us to the interesting results $\theta^{}_{23} = \pi/4$ and $\delta = -\pi/2$ which stand close to the current experimental results. But a precise measurement for $\theta^{}_{23}$ and $\delta$ will probably force us to modify the neutrino mixing $U^{(0)}$ given by such a symmetry. Hence we perform a study for modifications to $U^{(0)}$ in the forms of $U^{(1)\dagger} U^{(0)}$ and $U^{(0)} U^{(1)}$ in the scenarios (1) $U^{(1)}= R^{(1)}_{ij}$ (with $R^{(1)}_{ij}$ denoting a real orthogonal rotation in the $ij$ sector for $ij = 12, 23$ and 13) where the rotation angle $\theta^{(1)}_{ij}$ is allowed to take arbitrary values in the range $[0, \pi/2]$ (2) $U^{(1)}= R^{(1)}_{23} R^{(1)}_{13} R^{(1)}_{12}$ with $\theta^{(1)}_{ij}$ being small (comparable to or even much smaller than the measured value of $\theta^{}_{13}$).