# Effective description of non-equilibrium currents in cold magnetized plasma

### Submission summary

 Authors (as Contributors): Nabil Iqbal
Submission information
Date accepted: 2022-02-04
Date submitted: 2022-01-14 20:07
Submitted by: Iqbal, Nabil
Submitted to: SciPost Physics
Ontological classification
Specialties:
• High-Energy Physics - Theory
Approach: Theoretical

### Abstract

The dynamics of cold strongly magnetized plasma -- traditionally the domain of force-free electrodynamics -- has recently been reformulated in terms of symmetries and effective field theory, where the degrees of freedom are the momentum and magnetic flux carried by a fluid of cold strings. In physical applications where the electron mass can be neglected one might expect the presence of extra light charged modes -- electrons in the lowest Landau level -- propagating parallel to the magnetic field lines. We construct an effective description of such electric charges, describing their interaction with plasma degrees of freedom in terms of a new collective mode that can be thought of as a bosonization of the electric charge density along each field line. In this framework QED phenomena such as charged pair production and the axial anomaly are described at the classical level. Formally, our construction corresponds to gauging a particular part of the higher form symmetry associated with magnetic flux conservation. We study some simple applications of our effective theory, showing that the scattering of magnetosonic modes generically creates particles and that the rotating Michel monopole is now surrounded by a cloud of electric charge.

Published as SciPost Phys. 12, 078 (2022)

I thank the referees for their careful reading and overall positive comments, and I am happy that they both view it as a worthwhile attack on this physical problem. I would like to comment on a few specific weaknesses that the referees point out:

Referee 1: 1- The problem of extending the framework to massive charge carriers is left unresolved.

I entirely agree, and I confess I feel this will be difficult and beyond the scope of the paper, for reasons briefly mentioned in the conclusion.

2- While the paper contains quite a bit of background, the discussion of the EFT of force free electrodynamics is not self-contained (as the author also points out).

Agreed; I did not want to make the paper too lengthy with a more detailed review, but I have expanded this a bit and taken more care to make it accessible, as described further below.

Referee 2: 1) The main weakness, as the author himself recognizes and discusses in section III.E, is the rather ad-hoc way in which the scalar sector is introduced, without fitting into the framework of derivation from symmetry principles. This is fine for a phenomenological model but leaves open the question of universality of the construction.

I agree with this. At the moment the lack of an underlying principle is a deficiency of the model, as I have tried to be completely clear about in Section III.E. I do stress however that the model is actually quite constrained, as the realization of the symmetries around Eq (28) is rather non-trivial, i.e. I cannot think of another way to implement the charge carriers while still confining them along field lines. I thus believe that there is actually an underlying principle here that I have not yet completely understood. I leave for further work along the lines hinted at in Section III.E

2) It is not very clear, or clearly explained, what is the regime of validity of the model. Since the scalar sector describes out-of-equilibrium currents, one is not describing fluctuations around equilibrium as in conventional hydrodynamics, so it is not obvious what is the relevant expansion parameter.

I have made an effort to be more clear about this and where the new degrees of freedom can be neglected, as discussed in more detail below. In general, this theory is valid for long distances, i.e. in a derivative expansion, as usual for hydrodynamics. (Fluctuations are still around equilibrium configurations, but there is now more scope for degrees of freedom to be excited that are not thermalized with the FFE plasma).

The specific changes suggested by the referees are below under "List of changes".

### List of changes

Suggested changes from Referee 1:

1- Present the EFT of force free electrodynamics in a more self-contained manner, especially the meaning of the EFT degrees of freedom (perhaps through a figure).

I have presented more background material in Section IIB; I have tried to focus on which aspects may seem surprising at first glance — notably on page 3 — and given the reader references to the literature where appropriate. The suggestion of a figure is very well taken, and there is now a Figure 1 on page 3 which illustrates the geometry of the fields in a simple setup.

2- Explain explicitly (around eqs. (18)-(22)) what the EFT is an expansion in, i.e. what physical scale makes the derivative expansion dimensionless. (I understand that this issue is discussed later in the paper.)

I have added some more explanatory words about this; below Eq (17) I explain the derivative expansion in more detail, and I have added eq (23) where I both explicitly introduce the new scale, speculate about its meaning, and then anticipate that it will be discussed in more detail later on.

3- Explain why in Sec IV. one higher derivative term is kept. Why isn't it enough to analyze the coupled theory of FFE and \theta?

This is a perfectly fine thing to do, but it misses some interesting effects arising from the physics of pair-creation, which is only captured from the term in S_R (as otherwise E.B — which drives pair creation through the anomaly — is identically zero). I have added some more words explaining this both below Eq (32) where the wave equation for $\Theta$ is discussed and at the beginning of page 8 in Section IV.

Suggested changes from Referee 2:

1) The author should address the issue of the regime of validity of the model, and clarify what is the relevant expansion parameter.

I now further emphasize that we work in an effective field theory framework — where the theory is correct only at long distances and low energies, and the expansion is done in derivatives — below Eq (23), and above Eq (24).

2) It should also be better explained under what conditions the charged matter can be included in the way presented in the model, but still without spoiling the assumptions of force-free electrodynamics.

Further explanation is now given on this point in various places; in particular below Eq (32) where the wave equation for $\Theta$ is discussed, at the bottom of the second column on Page 7. The relevant part there now reads:

“Finally, we discuss a general feature of the equations of motion. Let us imagine that the FFE sector has no higher derivative corrections and is given by (18). In that case J ∝ ε, and thus the source term in (32) is zero. It is then consistent to set Θ to 0, and thus all solutions to FFE remain solutions of the coupled theory. Linearized fluctuations of Θ about any FFE solution will decouple. In this sense, in the extreme infrared the addition of this charged matter does not alter the structure of FFE.

On the other hand, let us now consider moving away from the extreme infrared, i.e. turning on higher derivative corrections such as (21). In this case, E · B is no longer zero, and now the source term in (32) will turn on Θ. As expected, accelerating electric fields can have a dramatic effect on free electric charges, including the hydrodynamic manifestation of the pair creation process discussed previously. We note that though they are pro- duced by a term that is higher-order in derivatives, they can nevertheless have a large effect at long distances due to non-trivial kinematics. We will study some aspects of this below.”

3) There are several terms in the effective action, that are all supposed to be at leading order in derivatives, but have different number of derivatives, can the author explain how the counting works?

The derivative counting is now explained further below Eq (17); indeed this is somewhat unfamiliar, but is actually standard in action formulations of hydrodynamics, as is now explicitly mentioned with a reference to earlier literature given.

4) Can the author elaborate more on why the dependence on the fields $\Phi^{1,2}$ has to come through the normal n_{\mu\nu}:

This happens because n_{\mu\nu} is the lowest order in derivatives object that can be constructed from the $\Phi^{1,2}$ while still being invariant under the worldsheet relabeling symmetry (11); this is now explained explicitly below Eq. (15)

5) The function R(\mu) has an expansion as in eq. (40), however couldn't there be also a constant, -independent term?

Actually this is ruled out as the term in the action R(\mu) would then be odd under orientation-reversing worldsheet reparametrizations; this is now made more explicit on footnote 4 on p4.