Multi-azimuthal-angle instability for different supernova neutrino fluxes

It has been recently discovered that removing the axial symmetry in the “multiangle effects” associated with the neutrino-neutrino interactions for supernova (SN) neutrinos, a new multi-azimuthal-angle (MAA) instability will trigger flavor conversions in addition to the ones caused by the bimodal and multi-zenith-angle (MZA) instabilities. We investigate the dependence of the MAA instability on the original SN neutrino fluxes, performing a stability analysis of the linearized neutrino equations of motion. We compare these results with the numerical evolution of the SN neutrino nonlinear equations, looking at a local solution along a specific line of sight, under the assumption that the transverse variations of the global solution are small. We also assume that self-induced conversions are not suppressed by large matter effects. We show that the pattern of the spectral crossings (energies where $F_{{\nu }_e}=F_{{\nu }_x}$ and $F_{{\overline{\nu }}_e}=F_{{\overline{\nu }}_x}$) is crucial in determining the impact of MAA effects on the flavor evolution. For neutrino spectra with a strong excess of ${\nu }_e$ over ${\overline{\nu }}_e$, presenting only a single crossing, MAA instabilities will trigger new flavor conversions in normal mass hierarchy. In our simplified flavor evolution scheme, these will lead to spectral swaps and splits analogous to what is produced in inverted hierarchy by the bimodal instability. Conversely, in the presence of spectra with a moderate flavor hierarchy, having multiple crossing energies, MZA effects will produce a sizable delay in the onset of the flavor conversions, inhibiting the growth of the MAA instability. In this case, the splitting features for the oscillated spectra in both the mass hierarchies are the ones induced by the only bimodal and MZA effects.

Figure 1

Spectrum $g(\omega )$ for cases $\mathcal{A}$ (upper panel) and $\mathcal{C}$ (lower panel).

Figure 2

Case $\mathcal{A}$. Radial evolution of the $\kappa$ function in NH (left panel) and IH (right panel). Continuous curves correspond to the MZA case, while the dashed ones to the SZA case. The growth of $\kappa$ in the NH is associated to the MAA instability, while in the IH to the bimodal instability.

Figure 3

Case $\mathcal{A}$. Upper panels: Radial evolution of the integrated $z$ component ${\overline{P}}_z$ of the polarization vector for $\overline{\nu }$. Lower panels: Radial evolution of the integrated dipole ${\overline{P}}_{\phi }$ for $\overline{\nu }$. Left panels refer to normal hierarchy, right panels to inverted hierarchy. Dashed curves are for the MAA, SZA evolutions, while continuous curves are for the MAA, MZA cases.

Figure 4

Case $\mathcal{A}$. Upper panels: Initial (thin continuous curves) and final $g(\omega )$ for the MAA evolution in the SZA (dashed curves) and MZA (thick continuous curves) cases for normal hierarchy (left panel) and inverted hierarchy (right panel). Lower panels: Swap function, i.e., the ratio of final with initial spectra in SZA (dashed curves) and MZA (thick continuous curves).

Figure 5

Case $\mathcal{A}$. MZA and MAA flavor evolution for $\nu$’s (left panel) and $\overline{\nu }$’s (right panel) in NH (upper panels) and IH (lower panels). Energy spectra initially for ${\nu }_e$ (black dashed curves) and ${\nu }_x$ (light dashed curves) and after collective oscillations for ${\nu }_e$ (black continuous curves) and ${\nu }_x$ (light continuous curves).

Figure 6

Case $\mathcal{C}$. Radial evolution of the $\kappa$ function in NH (left panel) and IH (right panel). Upper panels correspond to the bimodal instability, while the lower ones to the MAA instability. Continuous curves correspond to the MZA case while dashed ones to the SZA case.

Figure 7

Case $\mathcal{C}$. Upper panels: Radial evolution of the integrated $z$ component ${\overline{P}}_z$ of the polarization vector for $\overline{\nu }$. Lower panels: Radial evolution of the integrated dipole ${\overline{P}}_{\phi }$ for $\overline{\nu }$. Left panels refer to normal hierarchy, right panels to inverted hierarchy. Dashed curves are for the MAA, SZA evolution; while continuous curves are for the MAA, MZA case.

Figure 8

Case $\mathcal{C}$. Upper panels: Initial (thin continuous curves) and final $g(\omega )$ for the MAA evolution in the SZA (dashed curves) and MZA (thick continuous curves) cases for normal hierarchy (left panel) and inverted hierarchy (right panel). Lower panels: Swap function, i.e., the ratio of final with initial spectra in SZA (dashed curves) and MZA (thick continuous curves).

Figure 9

Case $\mathcal{C}$. MZA and MAA flavor evolution for $\nu$’s (left panel) and $\overline{\nu }$’s (right panel) in NH (upper panels) and IH (lower panels). Energy spectra initially for ${\nu }_e$ (black dashed curves) and ${\nu }_x$ (light dashed curves) and after collective oscillations for ${\nu }_e$ (black continuous curves) and ${\nu }_x$ (light continuous curves).