$\mu-\tau$ Symmetry, Nonzero $\theta_{13}$, and CP Violation

If we impose the $\mu-\tau$ symmetry, as a constraint into neutrino mass matrix, one find that the Jarlskog rephasing invariant: $J_{\rm CP}=0$ which implies that CP violation cannot be accommodated in the $\mu-\tau$ symmetry scheme. By introducing a small parameter $x$ that perturb the neutrino mass matrix with $\mu-\tau$ symmetry with trace of neutrino mass matrix remain constant, we can obtain $J_{CP}\neq 0$ and consequently $\theta_{13}\neq 0$.