Proc. Neutrino Astrophysics - MPP Theory Group

Anisotropic Neutrino Propagation in a
Magnetized Plasma
Per Elmfors
Stockholm University, Fysikum, Box 6730, S-113 85 Stockholm, Sweden
Introduction
The idea that a large-scale magnetic field can be responsible for a significant neutrino emission
asymmetry in a supernova explosion goes back to Chugai [1] in 1984. Since then there
have been a number of suggestions of how to implement this idea in more detail. Lately, in
particular after the Ringberg workshop, several papers appeared dealing with how to calculate
neutrino cross sections in a magnetic field more accurately and with more realistic neutron
star parameters [2, 3]. The various approaches are either to use the exact Landau levels for
the charged particles which is necessary for very strong fields, or to include the field effects
only through the polarization of the medium. The disadvantage with the full Landau level
approach is that the number of Landau levels that have to be included grows quadratically
with the Fermi energy. If the field is B ∼ 1 (MeV) 2 ≃ 1.6×10 14 Gauss and Fermi energy
EF = � µ 2 e − m2e ∼ 100 MeV approximately E2 F /2eB ≃ 15000 Landau levels are filled, and
with all the transition matrix elements that have to be calculated the problem becomes a
numerical challenge. For weak fields, where the strength should be compared with E2 F rather
than m2 e, it seems logical to rely on a weak-field approximation.
To explain the observed peculiar velocities of neutron stars by asymmetric neutrino emission
an asymmetry of a few percent is needed. It is found here, as in other recent papers
[3], that the asymmetry in the damping rate is itself much too small for fields of the order
of 1014 –1015 Gauss. On the other hand, it is certain that the final asymmetry cannot be
esitmated simply by the asymmetry in the damping rate. One step towards a more complete
analysis was taken in [4] where a cumulative parity violation was suggested to enhance the
asymmetry. For a reliable estimate it would be desirable to run a full magneto-hydrodynamics
simulation.
In this presentation I shall discuss the asymmetry from the electron polarization in
neutrino–electron scattering and URCA processes. In addition one will have to add the asymmetry
from proton and neutron polarization in URCA processes which may be as important
as the electron polarization. The reason is that polarization of the degenerate electron gas is
suppressed by the large chemical potential even though the electron magnetic moment is much
larger than the nuclear magnetic moment. Furthermore, I am here using free propagators for
the nucleons which may have to be improved to account for rapid spin-flip processes [5].
Neutrino Damping and the Imaginary Part of the Self-Energy
The simplest formalism for computing damping coefficients in thermal field theory is, in my
opinion, from the imaginary part of the self-energy [6, 7]. The relation to be used for a
neutrino with momentum Q is
Γ(Q) = − 〈ν|ImΣ(Q)|ν〉
〈ν|ν〉
79
. (1)