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Chern-simon type photon mass from fermion electric dipole moments at finite temperature in 3+1 dimensions

We study the low energy effective field theory of fermions with electric and magnetic dipole moments at finite temperature. We find that at one loop there is an interaction term of the Chern-Simon form ${\cal L_I}=m_μ\>A_ν{\tilde F}^{μν}$. The four vector $m_μ\simeq d_i μ_i m_i^2 ~{\partial_μ}\>(ln T)$ is interpreted as a Chern- Simon type mass of photons, which is determined by the electric (magnetic) dipole moments $d_i$ ($μ_i$) of the fermions in the vacuum polarisation loop diagram. The physical consequence of such a photon mass is that, photons of opposite circular polarisations, propagating through a hot medium, have different group velocities. We estimate that the time lag between the arrival times of the left and right circularly polarised light signals from pulsars. If the light propagates through a hot plasma (where the temperature in some regions is $T \sim 100 MeV$) then the time lag between the two circularly polarised signals of frequency $ω$ will be $Δt(ω) \simeq 10^{-6} /ω$. It may be possible to observe this effect in pulsar signals which propagate through nebula at high temperatures.