Neutrino Transport in Strongly Magnetized Proto-Neutron Stars and the Origin of Pulsar Kicks: I. The Effect of Parity Violation in Weak Interactions

In proto-neutron stars with strong magnetic fields, the neutrino-nucleon scattering/absorption cross sections depend asymmetrically on the direction of neutrino momentum with respect to the magnetic field axis, a manifestation of parity violation in weak interactions. We develop the moment formalism of neutrino transport in the presence of such asymmetric neutrino opacities. For a given neutrino species, there is a drift flux of neutrinos along the magnetic field in addition to the usual diffusive flux. Although the drift flux of $\nu_{\mu(\tau)}$ and that of $\bar\nu_{\mu(\tau)}$ completely cancel due to a crossing symmetry in the scattering matrix element, there is a net drift flux associated with $\nu_e$ and $\bar\nu_e$ since a proto-neutron star contains more $\nu_e$ than $\bar\nu_e$. This net drift flux induces globally asymmetric temperature and composition profiles in the star, and leads to asymmetric neutrino emission from the star. We demonstrate that significant asymmetry in neutrino emission can be obtained due to multiple neutrino-nucleon scatterings. We also show that in the bulk interior of the neutron star, the asymmetry associated with neutrino absorption is cancelled by the asymmetry associated with neutrino emission. We numerically evolve the asymmetric profiles of temperature and lepton number fractions of a newly-formed, magnetized neutron star through the deleptonization and thermal cooling phases. For an ordered magnetic field threading the neutron star interior, the fractional asymmetry in the total neutrino emission is about $0.006(B/10^{14}G)$, corresponding to a pulsar kick velocity of about $200(B/10^{14}G)$ km/s for a total radiated neutrino energy of $3\times 10^{53}$ erg.