Collective Neutrino Flavor Instability Requires a Crossing
  Neutrinos in supernovae, neutron stars, and in the early Universe may change
flavor collectively and unstably, due to neutrino-neutrino forward-scattering.
We prove that for collective instability to occur, the difference of momentum
distributions of two flavors must change sign, i.e., there is a zero crossing.
This necessary criterion, which unifies slow and fast instabilities, is valid
for Hamiltonian flavor-evolution of ultra-relativistic Standard Model neutrino
occupation matrices, including damping due to collisions in the relaxation
approximation. It provides a simple but rigorous condition for collective
flavor transformations that are believed to be important for stellar dynamics,
nucleosynthesis, and neutrino phenomenology.
