Analysis of collective neutrino flavor transformation in supernovae

We study the flavor evolution of a dense gas initially consisting of pure monoenergetic ${\nu }_e$ and ${\overline{\nu }}_e$. Using adiabatic invariants and the special symmetry in such a system we are able to calculate the flavor evolution of the neutrino gas for the cases with slowly decreasing neutrino number densities. These calculations give new insights into the results of recent large-scale numerical simulations of neutrino flavor transformation in supernovae. For example, our calculations reveal the existence of what we term the “collective precession mode.” Our analyses suggest that neutrinos which travel on intersecting trajectories subject to destructive quantum interference nevertheless can be in this mode. This mode can result in sharp transitions in the final energy-dependent neutrino survival probabilities across all trajectories, a feature seen in the numerical simulations. Moreover, this transition is qualitatively different for the normal and inverted neutrino mass hierarchies. Exploiting this difference, the neutrino signals from a future galactic supernova can potentially be used to determine the actual neutrino mass hierarchy.

Figure 1

The equivalence of a pendulum and a symmetric ($n_{\nu ,1}=n_{\nu ,2}=n_{\nu }$) bipolar system initially consisting of pure ${\nu }_e$ and ${\overline{\nu }}_e$. In the limit $n_{\nu }/n_{\nu }^0\gg 1$ two NFIS blocks ${\mathbf{S}}_1=n_{\nu ,1}{\mathbf{s}}_1$ (for neutrino) and ${\mathbf{S}}_2=n_{\nu ,2}{\mathbf{s}}_2$ (for antineutrino) are always roughly antialigned and ${\mathbf{S}}_1$ points roughly in the same direction as the pendulum in flavor space.

Figure 2

Flavor oscillations of the simple symmetric bipolar system in a nearly adiabatic process [Eq. (34) with $\gamma =10$ and $n_{\nu }(0)/n_{\nu }^0=100$] for ${\tilde{\theta }}_{\mathrm{v}}=0.01$. The dashed line is the value of $-\cos {\vartheta }_1$ as a function $n_{\nu }$ computed from Eq. (3) with the initial conditions in Eq. (4). The solid line is $-\cos {\vartheta }_{\max }$ computed from Eq. (30).

Figure 3

Values of $\cos {\vartheta }_1$ (solid lines) and $\cos {\vartheta }_2$ (dashed lines), where ${\vartheta }_1$ and ${\vartheta }_2$ are the polar angles of the NFIS’s ${\mathbf{s}}_1$ and ${\mathbf{s}}_2$ with respect to ${\mathbf{H}}_{\mathrm{V}}$, as functions of neutrino number density $n_{\nu }$ for simple asymmetric bipolar systems. Panels (a), (b), and (d) have mixing angle ${\tilde{\theta }}_{\mathrm{v}}=0.01$ and panel (c) has ${\theta }_{\mathrm{v}}=0.6$. The asymmetry parameter is $\alpha =0.8$ for (a) and (c), and $\alpha =0.2$ for (b) and (d). The thin lines are computed from the original e.o.m. of NFIS’s ${\mathbf{s}}_1$ and ${\mathbf{s}}_2$ [Eq. (3) with the initial conditions in Eq. (4)] assuming a nearly adiabatic process [Eq. (34) with $n_{\nu }(0)/n_{\nu }^{\mathrm{x}}=2$]. The adiabatic parameter is $\gamma =40$ for panels (a–c) and is 200 for panel (d). The thick lines are computed from Eq. (66) assuming that bipolar systems stay in the pure precession mode. The vertical dashed-dotted lines in panels (a), (b), and (d) correspond to $n_{\nu }=n_{\nu }^{\mathrm{c}}$.

Figure 4

Scaled precession frequency $\Omega /{\mu }_{\mathrm{V}}$ of the flavor pendulum as a function of neutrino number density $n_{\nu }$. The pendulum is assumed to be in the pure precession mode. The vacuum mixing angle is ${\tilde{\theta }}_{\mathrm{v}}=0.01$ for the solid and dashed lines, and ${\theta }_{\mathrm{v}}=0.6$ for the dotted line. The asymmetry parameter is $\alpha =0.8$ for the solid and dotted lines and is 0.2 for the dashed line. The vertical dashed-dotted line and dashed-dot-dotted line correspond to $n_{\nu }=n_{\nu }^{\mathrm{c}}$ for $\alpha =0.8$ and 0.2, respectively.

Figure 5

Ratio of the potential energy $E_{\mathrm{pot}}$ of the flavor pendulum to its kinetic energy $E_{\mathrm{kin}}$ [Eq. (49)] as functions of neutrino number density $n_{\nu }$. The parameters and the meaning of the lines are the same as those in Fig. 4.

Figure 6

The evolution of $x$ components of the average NFIS’s of supernova neutrinos in the flavor basis obtained from the multiangle simulations as presented in Fig. 4 of Ref. 22. The left panels are for neutrinos emitted as ${\nu }_e$, and the right panels are for antineutrinos emitted as ${\overline{\nu }}_e$. The top panels employ the normal mass hierarchy with ${\theta }_{\mathrm{v}}=0.1$, and the bottom panels employ the inverted mass hierarchy with ${\tilde{\theta }}_{\mathrm{v}}=0.1$. The solid, dotted and dashed lines give $⟨s_{\nu x}^{\mathrm{f}}⟩$ along the trajectories with $\cos {\Theta }_0=1$, 0.5 and 0, respectively, where ${\Theta }_0$ is the emission angle defined in Ref. 22. The neutrino mass-squared difference is taken to be $\delta m^2=3\times {10}^{-3}{\mathrm{eV}}^2$.