Perturbations to $\mu-\tau$ reflection symmetry due to renormalization group running effects

$\mu-\tau$ reflection symmetry is an attractive flavour symmetry in lepton mixing, which accommodates maximal values of atmospheric mixing angle ($\theta_{23}=\pi/4$) and Dirac CP phase ($\delta=\pi/2/3\pi/2$). Another significance of this symmetry is that it does not constrain $\theta_{13}$ to be zero. As the recent results from $T2K$ and $NO\nu A$ experiments indicate a near-maximal value of the Dirac CP phase, the role of $\mu-\tau$ reflection symmetry becomes more prominent. In this work, we study RG running effects as a perturbation to the $\mu-\tau$ reflection symmetry. Assuming the symmetry to be preserved at the seesaw scale, we study the deviations of mass eigenvalues and lepton mixing parameters at the electroweak scale due to RG running. We derive the one-loop RGEs of the mass eigenvalues and mixing parameters and solve them numerically. Numerical analysis shows that the deviations from $\mu-\tau$ reflection symmetry are consistent with $3\sigma$ range of global oscillation data.