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

The prompt nu_e burst from a core-collapse supernova (SN) 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 SNe, the matter density profile can be so shallow that the usual MSW 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 step-like feature. We explain this feature analytically as a ``MSW prepared spectral split.'' In a three-flavor treatment, the step-like feature actually consists of two narrowly spaced splits. They are determined by two combinations of flavor-lepton numbers that are conserved under collective oscillations.