Stepwise spectral swapping with three neutrino flavors

We develop a framework for studying collective three-flavor neutrino oscillations based on the density matrix formalism. We show how techniques proven useful for collective two-flavor neutrino oscillations such as corotating frames can be applied readily to three-flavor mixing. Applying two simple assumptions and the conservation of two “lepton numbers,” we use this framework to demonstrate how the adiabatic/precession solution emerges. We illustrate with a numerical example how two stepwise spectral swaps appear naturally if the flavor evolution of the neutrino gas can be described by such a solution. For the special case where mu and tau flavor neutrinos are equally mixed and are produced with identical energy spectra and total numbers, we find that one of the spectral swaps in the three-flavor scenario agrees with that in the two-flavor scenario when appropriate mixing parameters are used. Using the corotating-frame technique we show how the adiabatic/precession solution can obtain even in the presence of a dominant ordinary matter background. With this solution we can explain why neutrino spectral swapping can be sensitive to deviations from maximal 23-mixing when the “mu-tau” matter term is significant.

Figure 1

Conversion probabilities $P({\nu }_{\alpha }\to {\nu }_i)$ (left panels) and $P({\overline{\nu }}_{\alpha }\to {\overline{\nu }}_i)$ (right panels) as functions of neutrino and antineutrino energies $E_{\nu }$ and $E_{\overline{\nu }}$, respectively, in the bare, hot neutron star example. The top, middle, and bottom panels are for neutrinos that are initially in pure $e$, $\mu$, and $\tau$ flavors, respectively. The solid, dashed, and dot-dashed lines are for neutrinos that end up in the 1st, 2nd, and 3rd vacuum mass eigenstates, respectively, when $n_{\nu }^{\mathrm{tot}}\to 0$.