Lepton number crossings are insufficient for flavor instabilities

Abstract

In dense neutrino environments, the mean field of flavor coherence can develop instabilities. A necessary condition is that the flavor lepton number changes sign as a function of energy and/or angle. Whether such a crossing is also sufficient has been a longstanding question. We construct an explicit counterexample: a spectral crossing without accompanying flavor instability, with an even number of crossings being key. This failure is physically understood as Cherenkov-like emission of flavor waves. If flipped-lepton-number neutrinos never dominate among those kinematically allowed to decay, the waves cannot grow.

Figures (3)

• Figure 1: DLN distributions for the stable (solid line) and unstable (dashed line) double-crossed setup. The vacuum energy splitting is $\omega_E=10^{-2}\, \mu\,(1\,\mathrm{MeV}/E)$. The DLN is shown in terms of $\omega_E$, so $D(\omega_E)=D(E)\,E/\omega_E$. For a given shifted frequency $\omega$ and shifted wave number $k$, all neutrinos below (above) the threshold $\omega_{E,\rm thr}=\omega-k$ can decay for $k<0$ ($k>0$).
• Figure 2: Dispersion relations for the stable (solid) and unstable (dashed) distributions. Different colors for the different solutions, namely the two longitudinal modes (L1 and L2) from Eq. \ref{['eq:dispersion_longitudinal']} and the transverse mode (T) from Eq. \ref{['eq:dispersion_transverse']}. The light cone $\mathrm{Re}(\omega)=\pm k$ is shown as thin black lines.
• Figure 3: Comparison of the numerical growth rates with the analytical approximations, valid for $|\mathrm{Im}(\omega)|\ll |\mathrm{Re}(\omega)-k|$.