Two Categories of Approximately mu-tau Symmetric Neutrino Mass Textures
  Our approximately \mu-\tau symmetric neutrino mass textures fall into two different categories, whose behaviors in the \mu-\tau symmetric limit are characterized by either \sin(theta_{13})->0 (referred to as C1)), or \sin(theta_{12})->0 (referred to as C2)). We present ten phenomenologically viable neutrino mass textures: two for the normal mass hierarchy, three for the inverted mass hierarchy, and five for the quasi degenerate mass pattern. Tiny \mu-\tau symmetry breaking ensures that \sin^2(theta_{13}) << 1 for C1), and \Delta m^2_\odot/\Delta m^2_{atm} (\equiv R) << 1 for C2). A correlation among small quantities is provided by \cos 2(theta_{23}) \sim \sin(theta_{13}) for C1), and by either \cos(2theta_{23}) \sim R, or \cos(2theta_{23})\sin(theta_{13}) \sim R for C2). It is further shown that \tan(2theta_{12}) \sim \cos(2theta_{23})/\sin(theta_{13}) is satisfied for C2). We find specific properties for each mass ordering, which are discussed in this article. 
Kenichi Fuki and Masaki Yasue