Deviation of atmospheric mixing from maximal and structure in the leptonic flavor sector

I attempt to quantify how far from maximal one should expect the atmospheric mixing angle to be given a neutrino mass matrix that leads, at zeroth order, to a ${\nu }_3$ mass eigenstate that is 0% ${\nu }_e,$ 50% ${\nu }_{\mu },$ and 50% ${\nu }_{\tau }.$ This is done by assuming that the solar mass-squared difference is induced by an “anarchical” first order perturbation, an approach then can naturally lead to experimentally allowed values for all oscillation parameters. In particular, both $|\cos 2{\theta }_{\mathrm{atm}}|$ (the measure for the deviation of atmospheric mixing from maximal) and $|U_{e3\mathrm{}}|$ are of order $\sqrt{\Delta m_{\mathrm{sol}}^2/\Delta m_{\mathrm{atm}}^2}$ in the case of a normal neutrino mass hierarchy or of order $\Delta m_{\mathrm{sol}}^2/\Delta m_{\mathrm{atm}}^2$ in the case of an inverted one. Hence, if any of the textures analyzed here has anything to do with reality, next-generation neutrino experiments can see a nonzero $\cos 2{\theta }_{\mathrm{atm}}$ in the case of a normal mass hierarchy, while in the case of an inverted mass hierarchy only neutrino factories should be able to see a deviation of ${\sin }^{2}2{\theta }_{\mathrm{atm}}$ from 1.