Neutrino-nucleon interactions in magnetized neutron-star matter: The effects of parity violation

We study neutrino-nucleon scattering and absorption in a dense, magnetized nuclear medium. These are the most important sources of neutrino opacity governing the cooling of a proto-neutron star in the first tens of seconds after its formation. Because the weak interaction is parity violating, the absorption and scattering cross sections depend asymmetrically on the directions of the neutrino momenta with respect to the magnetic field. We develop the moment formalism of neutrino transport in the presence of such asymmetric opacities and derive explicit expressions for the neutrino flux and other angular moments of the Boltzmann transport equation: For a given neutrino species, there is a drift flux of neutrinos along the magnetic field in addition to the usual diffusive flux. This drift flux depends on the deviation of the neutrino distribution function from thermal equilibrium. Hence, despite the fact that the neutrino cross sections are asymmetric throughout the star, the asymmetric neutrino Aux can be generated only in the outer region of the proto-neutron star where the neutrino distribution deviates significantly from thermal equilibrium. The deviation from equilibrium is similarly altered by the asymmetric scattering and absorption, although its magnitude will still be quite small in the interior of the star. We clarify two reasons why previous studies have led to misleading results. First, inelasticity must be included in the phase space integrals in order to satisfy detail balance. Second, nucleon recoil must be included in order to find the leading order asymmetric cross sections correctly, even though it can be ignored to leading order to get the zero field opacities. In addition to the asymmetric absorption opacity arising from nucleon polarization, we also derive the contribution of the electron (or positron) ground state Landau level. For neutrinos of energy less than a few times the temperature, this is the dominant source of asymmetric opacity. Last, we discuss the implication of our result to the origin of pulsar kicks: in order to generate kick velocity of a few hundred km s(-1) from asymmetric neutrino emission using the parity violation effect, the proto-neutron star must have a dipole magnetic field of at least 10(15)-10(16) G. [S0556-2821(99)02616-8].