Collective Neutrino Flavor Transformation In Supernovae

We examine coherent active-active channel neutrino flavor evolution in environments where neutrino-neutrino forward scattering can engender large-scale collective flavor transformation. We point out a key quantity, the "total effective energy", which is conserved in several important regimes. Using this concept, we analyze collective neutrino and antineutrino flavor oscillation in the "synchronized" mode and what we term the "bi-polar" mode. We thereby are able to explain why large collective flavor mixing can develop on short timescales even when vacuum mixing angles are small in, e.g., a dense gas of initially pure $\nu_e$ and $\bar\nu_e$ with an inverted neutrino mass hierarchy (an example of bi-polar oscillation). In the context of the spin precession analogy, we find that the co-rotating frame provides insights into more general systems, where either the synchronized or bi-polar modes could arise. For example, we use the co-rotating frame to demonstrate how large flavor mixing in the bi-polar mode can occur in the presence of a large and dominant matter background. We use the adiabatic condition to derive a simple criterion for determining whether the synchronized or bi-polar mode will occur. Based on this criterion we predict that neutrinos and antineutrinos emitted from a proto-neutron star in a core-collapse supernova event can experience synchronized and bi-polar flavor transformations in sequence before conventional Mikhyev-Smirnov-Wolfenstein (MSW) flavor evolution takes over. This certainly will affect the analyses of future supernova neutrino signals, and might affect the treatment of shock re-heating rates and nucleosynthesis depending on the depth at which collective transformation arises.