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What Does μ-τ Symmetry Imply about Neutrino Mixings?

TOKAI-HEP/TH-0601 with Kenichi Fuki, published in Phys. Rev. D 73 No.5 (2006) 055014 1-8 (Received date: January, 2006).

$The requirement of the mu-tau symmetry in the neutrino sector that yields the maximal atmospheric neutrino mixing is shown to yield either sin(\theta_{13})=0 (referred to as C1)) or sin(\theta_{12})=0 (referred to as C2)), where \theta_{12(13)} stands for the solar (reactor) neutrino mixing angle. We study general properties possessed by approximately mu-tau symmetric textures. It is argued that the tiny mu-tau symmetry breaking generally leads to cos(2\theta_{23}) \simsin(\theta_{13}) for C1) and cos(2\theta_{23}) \sim \Delta m^2_\odot/\Delta m^2_{atm}(\equiv R) for C2), which indicates that the smallness of cos(2\theta_{23}) is a good measure of the mu-tau symmetry breaking, where \Delta m^2_{atm} (\Delta m^2_\odot) stands for the square mass differences of atmospheric (solar) neutrinos. We further find that the relation R \sim sin^2(\theta_{13}) arises from contributions of O(sin^2(\theta_{13})) in the estimation of the neutrino masses (m_{1,2,3}) for C1), and that possible forms of textures are strongly restricted to realize sin^2(2\theta_{12})=O(1) for C2). To satisfy R \sim sin^2(\theta_{13}) for C1), neutrinos exhibit the inverted mass hierarchy, or the quasi degenerate mass pattern with \vert m_{1,2,3}\vert \sim O(\sqrt{\Delta m^2_{atm}}), and, to realize sin^2(2\theta_{12})=O(1) for C2), there should be an additional small parameter \eta whose size is comparable to that of the mu-tau symmetry breaking parameter \varepsilon, giving tan(2\theta_{12}) \sim \varepsilon/\eta with \eta \sim \varepsilon to be compatible with the observed large mixing.$

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