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

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 _ and _ refractive index. Including this ``mu-tau matter term'' provides a resonance at a density of 310^7 g cm^-3 that typically causes significant modifications of the overall ₄ and {}₄ survival probabilities. This effect is surprisingly sensitive to deviations from maximal 23-mixing, being different for each octant.