# Does the Standard Model plasma develop a spontaneous magnetisation at finite temperature?

Reference: arXiv:1204.3604v1 [hep-ph] Long-range magnetic fields in the ground state of the Standard Model plasma. Alexey Boyarsky, Oleg Ruchayskiy, Mikhail Shaposhnikov.

The authors of this paper claim that, at least in a certain regime, the standard model at finite temperature and density is unstable to the development of a long range magnetic field. They also claim that this field can be cosmologically significant but hasn't been accounted for in prior work.

The argument runs as follows:

1. Finite temperature and density breaks Lorentz invariance to 3D rotational invariance.
2. This means that the photon vacuum polarization can develop a term $i \epsilon_{ijk} p^k \Pi_2(p^2)$ involving the 3-momentum $p^k$.
3. This term can be expressed as an effective Chern-Simons term in the Lagrangian, which they connect to the axial anomaly (I'm fuzzy on how this works).
4. Computing $\Pi_2(0)$ in an effective theory they show that for some range of fermion densities and temperatures the vacuum polarization develops a negative eigenvalue. The corresponding mode is unstable and condenses into a long range magnetic field.

There are a number of technical points in the derivation that confuse me (mainly the chemical potential stuff) since I'm only just learning thermal field theory. But I have a few questions of a more general nature:

1. What is the status of this result? Does it hold up? Is there a precedent for this in other models? I've only found 2 citing articles on InspireHEP, though it is a fairly recent article so maybe this is no big issue?

2. How sensitive are the standard cosmological observables to a long range magnetic field in the early universe? Are there limits on the field present during, e.g. BBN?

3. Is this effect genuinely new? How could it be missed for so long?

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