Fast Pairwise Conversion of Supernova Neutrinos: A Dispersion Relation Approach

Collective pair conversion ₄{}₄ₗ{}ₗ by forward scattering, where x= or, may be generic for supernova neutrino transport. Depending on the local angular intensity of the electron lepton number carried by neutrinos, the conversion rate can be ``fast, '' i. e. , of the order of 2G₅ (n₄-n{₄}) m₀ₓ₌^2/2E. We present a novel approach to understand these phenomena: a dispersion relation for the frequency and wave number (, K) of disturbances in the mean field of ₄ₗ flavor coherence. Runaway solutions occur in ``dispersion gaps, '' i. e. , in ``forbidden'' intervals of and/or K where propagating plane waves do not exist. We stress that the actual solutions also depend on the initial and/or boundary conditions, which need to be further investigated.