Asymmetric neutrino emission due to neutrino-nucleon scatterings in supernova magnetic fields

We derive the cross section of neutrino-nucleon scatterings in supernova magnetic fields, including weak-magnetism and recoil corrections. Since the weak interaction violates parity, the scattering cross section depends asymmetrically on the directions of the neutrino momenta with respect to the magnetic field; the origin of pulsar kicks may be explained by the mechanism. An asymmetric neutrino emission (a drift flux) due to neutrino-nucleon scattering is absent at the leading level of $O({\mu }_{B}B/T),$ where ${\mu }_B$ is the nucleon magneton, B is the magnetic field strength, and T is the matter temperature at a neutrino sphere. This is because at this level the drift flux of the neutrinos is exactly canceled by that of the antineutrinos. Hence, the relevant asymmetry in the neutrino emission is suppressed by a much smaller coefficient of $O({\mu }_{B}B/m),$ where m is the nucleon mass; a detailed form of the relevant drift flux is also derived from the scattering cross section, using a simple diffusion approximation. It appears that the asymmetric neutrino emission is too small to induce the observed pulsar kicks. However, we note the fact that the drift flux is proportional to the deviation of the neutrino distribution function from the value of thermal equilibrium at the neutrino sphere. Since the deviation can be large for nonelectron neutrinos, it is expected that there occurs a cancellation between the deviation and the small suppression factor of $O({\mu }_{B}B/m).\mathrm{}$ Using a simple parametrization, we show that the drift flux due to neutrino-nucleon scattering may be comparable to the leading term due to beta processes with nucleons, which has been estimated to give a relevant kick velocity when the magnetic field is as strong as ${10}^{15}–{10}^{16}\mathrm{G}.$