Mu-tau neutrino refraction and collective three-flavor transformations in supernovae

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We study three-flavor collective neutrino transformations in the dense-neutrino region above the neutrino sphere of a supernova core. We find that two-flavor conversions driven by the atmospheric mass difference and the 13-mixing angle capture the full effect if one neglects the second-order difference between the muon and tau neutrino refractive index. Including this "mu-tau matter term" provides a resonance at a density of approximately 3 x 10⁷ g cm-3 that typically causes significant modifications of the overall electron neutrino and antineutrino survival probabilities. This effect is surprisingly sensitive to deviations from maximal 23-mixing, being different for each octant.