Dispersion relations of collective flavor excitations in dense neutrino media

We apply the Landau damping theory to collective flavor excitations in dense neutrino media. We demonstrate explicitly that the (meta-)stable modes of the collective flavor excitations in a physical dense neutrino gas are indeed weakly damped. We show that there should in general exist infinitely many collective flavor excitation modes most of which are damped. In a neutrino gas with one or more zero crossings in the distribution of the difference of the (neutrino) lepton numbers (DLNs), the least damped resonant modes and the (originally) metastable modes at certain wave numbers can suffer from inverse Landau damping and thus become unstable. We also introduce the concept of convoluted DLN distributions which can be used to understand the physical origin of the slow and fast flavor instabilities in dense neutrino media in core-collapse supernovae and neutron star mergers.