Spectral split in a prompt supernova neutrino burst: Analytic three-flavor treatment

The prompt nu(e) burst from a core-collapse supernova is subject to both matter-induced flavor conversions and strong neutrino-neutrino refractive effects. For the lowest-mass progenitors, leading to O-Ne-Mg core supernovae, the matter density profile can be so steep that the usual Mikheyev-Smirnov-Wolfenstein matter effects occur within the dense-neutrino region close to the neutrino sphere. In this case a split occurs in the emerging spectrum, i.e., the nu(e) flavor survival probability shows a steplike feature. We explain this feature analytically as a spectral split prepared by the Mikheyev-Smirnov-Wolfenstein effect. In a three-flavor treatment, the steplike feature actually consists of two narrowly spaced splits. They are determined by two combinations of flavor-lepton numbers that are conserved under collective oscillations.