Modifications to the neutrino mixing from 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 $U^{(0)}$ from such a symmetry. Here we perform a study for modifications to $U^{(0)}$ in the forms of $U^{(1)\dagger} U^{(0)}$ and $U^{(0)} U^{(1)}$ where $U^{(1)}= R^{(1)}_{ij}$ (for $ij = 12, 23$ and 13) with $R^{(1)}_{ij}$ denoting a real orthogonal rotation on the $ij$ plane.