Particle Dispersion in External Active Media
This chapter is devoted to an analysis of the dispersion properties of photons and neutrinos in external active media: magnetic field, plasma, and magnetized plasma. Possible astrophysical manifestations of particle processes influenced by external active
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Particle Dispersion in External Active Media
This chapter is devoted to an analysis of the dispersion properties of photons and neutrinos in external active media: magnetic field, plasma, and magnetized plasma. Possible astrophysical manifestations of particle processes influenced by external active media are also considered.
4.1 Dispersion in Media: Main Definitions Dispersion effects in the medium significantly affect the propagation of particles with small masses (photons, neutrinos), while other particles remain almost insensitive to the influence of the environment (e.g., axions and other Nambu–Goldstone bosons). The direct way to investigate the dispersion relations of photons and neutrinos is to analyze the link between forward scattering and refractive index. In accordance with the general concepts of quantum field theory, particles are quantized excitations of the corresponding fields: the electromagnetic field produces photons, the electron-positron field produces the electrons, and so on. Usually, it is convenient to describe these fields by means of plane waves, characterized by a frequency ω and wavevector k. Then the excitations of these modes have the time and spatial dependence, which is described by a factor exp [−i (ωt − kx)]. With a wave vector given, the frequency is determined by the dispersion relation. Since (ω, k) is a 4-vector, basing on Lorentz invariance we find that in vacuum the value ω 2 −k2 = m2 is the same for all frequencies and m is the particle mass. One consequence of the covariant dispersion relation is that the decay of the form 1 → 2 + 3 is possible only if m1 > m2 + m3 , so that the particle 1 in its rest frame had enough energy for production of the final state. In a medium, dispersion relations are changed, as a rule, by the coherent interaction with the background. In the simplest case, a particle acquires an effective mass caused by the presence of a medium. For example, dispersion relation for photons in a nonrelativistic plasma is of the form ω 2 = ωP2 + k2 , where ωP is the so-called plasma frequency, defined by the expression A. Kuznetsov and N. Mikheev, Electroweak Processes in External Active Media, Springer Tracts in Modern Physics 252, DOI: 10.1007/978-3-642-36226-2_4, © Springer-Verlag Berlin Heidelberg 2013
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4 Particle Dispersion in External Active Media
ωP2 =
4π α Ne , me
(4.1)
where Ne is the electron density. For a process in an environment which induces effective masses of particles, the kinematic condition for the process realization should be considered more carefully. For example, the kinematic condition for the decay 1 → 2 + 3, instead of the simplified vacuum relation m1 > m2 + m3 , is expressed in its original form through the squares of masses: 2 m14 − 2 m12 m22 + m32 + m22 − m32 > 0 .
(4.2)
In this form, the kinematic condition for the possibility of the process is applicable for the case of the negative effective mass squared. The appearance of the photon effective mass in a medium leads to the fact that if ωP2 > 4 mν2 , the
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