Multi-azimuthal-angle effects in self-induced supernova neutrino flavor conversions without axial symmetry

The flavor evolution of neutrinos emitted by a supernova (SN) core is strongly affected by the refractive effects associated with the neutrino-neutrino interactions in the deepest stellar regions. Till now, all numerical studies have assumed the axial symmetry for the "multi-angle effects" associated with the neutrino-neutrino interactions. Recently, it has been pointed out in \cite{Raffelt:2013rqa} that removing this assumption, a new multi-azimuthal-angle (MAA) instability would emerge in the flavor evolution of the dense SN neutrino gas in addition to the one caused by multi-zenith-angle (MZA) effects. Inspired by this result, for the first time we numerically solve the non-linear neutrino propagation equations in SN, introducing the azimuthal angle as angular variable in addition to the usual zenith angle. We consider simple energy spectra with an excess of \nu_e over anti-\nu_e. In these cases, we find that even starting with a complete axial symmetric neutrino emission, the MAA effects would lead to significant flavor conversions in normal mass hierarchy, in cases otherwise stable under the only MZA effects. The final outcome of the flavor conversions, triggered by the MAA instability, depends on the initial asymmetry between \nu_e and anti-\nu_e spectra. If it is sufficiently large, final spectra would show an ordered behavior with spectral swaps and splits. Conversely, for small flavor asymmetries flavor decoherence among angular modes develops, affecting the flavor evolution also in inverted mass hierarchy.