Precision neutrino data confronts $\mu\leftrightarrow\tau$ symmetry

Neutrino oscillation data indicate that $\theta_{23}$ is close to $\pi/4$ and $\theta_{13}$ is very small. A simple $\mu\leftrightarrow\tau$ exchange symmetry of the neutrino mass matrix predicts $\theta_{23}=-\pi/4$ and $\theta_{13}=0$. Since the experimental measurements differ from these predictions, this symmetry is obviously broken. This breaking is given by two parameters: $\varepsilon_1$ parametrizing the inequality bewteen $12$ and $13$ elements and $\varepsilon_2$ parametrizing the inequality bewteen $22$ and $33$ elements. We show that the magnitude of $\theta_{13}$ is essentially controlled by $\varepsilon_1$ whereas the deviation of $\theta_{23}$ from maximality is controlled by $\varepsilon_2$. The measured value of $\theta_{13}$ requires $\mu\leftrightarrow\tau$ symmetry to be badly broken for both normal hierarchy and inverted hierarchy, though the level of breaking depends sensitively on the hierarchy. In this paper we obtain constraints on the parameters of neutrino mass matrix, including the symmetry breaking parameters, using the precision oscillation data. We find that this precision data constrains all elements of neutrino mass matrix to be in very narrow ranges. We also consider $\mu\leftrightarrow -\tau$ exchange symmetry in the case of inverted hierarchy and find that it provides an explanation of neutrino mixing angles with some fine-tuning.