Neutrino mu-tau reflection symmetry and its breaking in the minimal seesaw

In this paper, we attempt to implement the neutrino $\mu$-$\tau$ reflection symmetry (which predicts ${\theta }_{23}^{}=\pi \mathrm{/}4$ and $\delta =\pm \pi \mathrm{/}2$ as well as trivial Majorana phases) in the minimal seesaw (which enables us to fix the neutrino masses). For some direct (the preliminary experimental hints towards ${\theta }_{23}^{}\mathrm{\ne }\pi \mathrm{/}4$ and $\delta \mathrm{\ne }-\pi \mathrm{/}2$) and indirect (inclusion of the renormalization group equation effect and implementation of the leptogenesis mechanism) reasons, we particularly study the breakings of this symmetry and their phenomenological consequences.