Partial $\mu-\tau$ symmetry from kinetic normalization

We consider models with broken $\mu$-$\tau$ permutation symmetry through higher dimensional operators renormalizing the lepton kinetic terms in the action. We study the consequences on the structure of the neutrino mass matrix and find in particular that the allowed region for the lightest mass in the normal hierarchy plotted as a function of the atmospheric mixing element $|V_{\mu3}|$ is very restricted with the atmospheric mixing angle $\theta_{23}$ to lie in the second octant. On the other hand, the corresponding allowed region in the inverted hierarchy regime is less restrictive.