By K. Bhattacharya
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Additional info for Elementary Particle Interactions in Bkgd Magnetic Field
To calculate the rates of these processes one has to calculate the effective electromagnetic vertex of neutrinos in a background magnetic field and in a magnetized medium. This section is primarily concerned about the electromagnetic vertex of the neutrinos in a magnetized medium. For simplicity we consider the background temperature and neutrino momenta to be small compared to the masses of the W and Z bosons. We can, therefore, neglect the momentum dependence in the W and Z propagators, which is justified if we are performing a calculation to the leading order in the Fermi constant, GF .
Where we have defined the direction of the incoming neutrino by the angle θ, with qz = Ω cos θ . 71) In Eq. 70), we have denoted the cross-section by σn′ because the electron ends up in a specific Landau level n′ . , n′max σ= σn′ n′ =0 eBG2β = 2π n′max gn′ (G2V + 3G2A ) + 2GA S(GV + GA ) cos θ n′ =0 D+Ω +δn′ ,0 (G2V − G2A ) cos θ + 2GA S(GV − GA ) (D + Ω)2 − m2 − 2n′ eB . 72) The possible allowed Landau level has a maximum, n′max , which is given by the fact that the quantity under the square root sign in the denominator of Eq.
We already observed that σ is direction dependent. Therefore, the value of nn on the “neutrino sphere” depends on the direction as well, and the surface is no longer a sphere. Different values of nn will correspond to different temperatures. Thus, neutrinos will be emitted with different momenta in different directions. This can result in a kick to the star. To estimate the magnitude of the kick, let us abbreviate Eq. 72) as σ = eBG2β (a + b cos θ) . 87) neglecting corrections of order b/a. The neutron gas in a typical proto-neutron star can be considered to be non-relativistic and degenerate.