The $\mu-\tau$ reflection symmetry of Dirac neutrinos and its breaking effect via quantum corrections

Given the Dirac neutrino mass term, we explore the constraint conditions which allow the corresponding mass matrix to be invariant under the \mu-\tau reflection transformation, leading us to the phenomenologically favored predictions \theta_{23} = \pi/4 and \delta = 3\pi/2 in the standard parametrization of the 3\times 3 lepton flavor mixing matrix. If such a flavor symmetry is realized at a superhigh energy scale \Lambda_{\mu\tau}, we investigate how it is spontaneously broken via the one-loop renormalization-group equations (RGEs) running from \Lambda_{\mu\tau} down to the Fermi scale \Lambda_{\rm F}. Such quantum corrections to the neutrino masses and flavor mixing parameters are derived, and an analytical link is established between the Jarlskog invariants of CP violation at \Lambda_{\mu\tau} and \Lambda_{\rm F}. Some numerical examples are also presented in both the minimal supersymmetric standard model and the type-II two-Higgs-doublet model, to illustrate how the octant of \theta_{23}, the quadrant of \delta and the neutrino mass ordering are correlated with one another as a result of the RGE-induced \mu-\tau reflection symmetry breaking effects.