The neutrino-neutrino interaction effects in supernovae: the point of view from the 'matter' basis

We consider the Hamiltonian for neutrino oscillations in matter in the case of arbitrary potentials including off-diagonal complex terms. We derive the 'matter basis' Hamiltonian in terms of quantities one can derive from the flavor basis Hamiltonian and its derivative, for an arbitrary number of neutrino flavors. We make our expressions explicit for the two-neutrino flavor case and apply our results to the situation where the Hamiltonian includes both coupling to matter and to neutrinos, which describes neutrino propagation in core-collapse supernovae. We show that the neutrino flavour evolution depends on the mixing matrix derivatives involving not only the derivative of the matter mixing angles but also of the phases. In particular, we point out the important role of the phase derivatives, that appear due to the neutrino-neutrino interaction, and show how it can cause an oscillating degeneracy between the diagonal elements of the matter basis Hamiltonian. Our results also reveal, that the end of the synchronization regime is due to a rapid increase of the phase derivative, and identify the condition to be fulfilled for the onset of bipolar oscillations involving both the off-diagonal neutrino-neutrino interaction contributions and the vacuum terms.