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$μ$-$τ$ reflection symmetry and radiative corrections

The $μ$-$τ$ reflection symmetry is compatible with current neutrino oscillation data and easily realized under family symmetries. We prove that this symmetry preserves $θ_{23}=45^\circ$, $δ=\pm90^\circ$, $ρ,σ=0,90^\circ$, and can be embedded into the seesaw mechanism. The $μ$-$τ$ reflection symmetry preserved at a high energy scale $Λ_\text{FS}$ will be broken by radiative corrections and result in deviations of $θ_{23}$ from $45^\circ$ and $δ$ from $\pm90^\circ$ at the electroweak scale. We develop an analytical method to derive the corrections to all the mixing parameters. We perform a numerical analysis in the MSSM for $δ=-90^\circ$ at $Λ_\text{FS}$, and observe that $θ_{23}>45^\circ$ in the normal mass ordering, $θ_{23}<45^\circ$ in the inverted mass ordering, and the sizable correction to $δ$ prefers a negative sign. These deviations have definite directions and can be tested in the future neutrino oscillation experiments.