What Does mu-tau Symmetry Imply about Neutrino Mixings?

Abstract

The requirement of the mu-tau symmetry in the neutrino sector that yields the maximal atmospheric neutrino mixing is shown to yield either sin(θ_{13})=0 (referred to as C1)) or sin(θ_{12})=0 (referred to as C2)), where θ_{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(2θ_{23}) \simsin(θ_{13}) for C1) and cos(2θ_{23}) \sim Δm^2_\odot/Δm^2_{atm}(\equiv R) for C2), which indicates that the smallness of cos(2θ_{23}) is a good measure of the mu-tau symmetry breaking, where Δm^2_{atm} (Δm^2_\odot) stands for the square mass differences of atmospheric (solar) neutrinos. We further find that the relation R \sim sin^2(θ_{13}) arises from contributions of O(sin^2(θ_{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(2θ_{12})=O(1) for C2). To satisfy R \sim sin^2(θ_{13}) for C1), neutrinos exhibit the inverted mass hierarchy, or the quasi degenerate mass pattern with | m_{1,2,3}| \sim O(\sqrt{Δm^2_{atm}}), and, to realize sin^2(2θ_{12})=O(1) for C2), there should be an additional small parameter ηwhose size is comparable to that of the mu-tau symmetry breaking parameter ε, giving tan(2θ_{12}) \sim ε/ηwith η\sim εto be compatible with the observed large mixing.