Collective modes in neutrino "beam" electron-positron plasma interactions

We derive semiclassical neutrino-electron transport equations in the collisionless (Vlasov) limit from the coupled Dirac equations, incorporating the charged and neutral weak current-current as well as electromagnetic interactions. A corresponding linear response theory is derived. In particular, we calculate the response functions for a variety of beam-plasma geometries, which are of interest in a supernova scenario. We apply this to the study or plasmons and to a new class of collective pharon resonance modes, which are characterized by omega <q. We find that the growth rates of the unstable modes correspond to a strongly temperature (proportional to(TnuTe3)-T-2) and linearly momentum dependent e-folding length of about 10(10) km under typical conditions for type II supernovas. This appears to rule out such long-wavelength collective modes as an efficient means of depositing neutrino energy into the plasma sphere.