Effective description of non-equilibrium currents in cold magnetized plasma
EEffective description of non-equilibrium currents in cold magnetized plasma
Nabil Iqbal ∗ Centre for Particle Theory, Department of Mathematical Sciences,Durham University, South Road, Durham DH1 3LE, UK
The dynamics of cold strongly magnetized plasma – traditionally the domain of force-free electro-dynamics – has recently been reformulated in terms of symmetries and effective field theory, wherethe degrees of freedom are the momentum and magnetic flux carried by a fluid of cold strings. Inphysical applications where the electron mass can be neglected one might expect the presence of ex-tra light charged modes – electrons in the lowest Landau level – propagating parallel to the magneticfield lines. We construct an effective description of such electric charges, describing their interactionwith plasma degrees of freedom in terms of a new collective mode that can be thought of as abosonization of the electric charge density along each field line. In this framework QED phenomenasuch 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 associatedwith magnetic flux conservation. We study some simple applications of our effective theory, showingthat the scattering of magnetosonic modes generically creates particles and that the rotating Michelmonopole is now surrounded by a cloud of electric charge.
I. INTRODUCTION
Diverse sets of physical phenomena across vastly differ-ent length scales are controlled by the dynamics of mag-netic fields in plasma. The description of such plasmas interms of coarse-grained hydrodynamic degrees of freedomhas a long history [1]. A particular regime of a stronglymagnetized plasma is obtained when one considers a sit-uation where electric charges are sufficiently plentiful asto screen the electric field to zero, but sufficiently dif-fuse in that one can ignore their collective stress-energy.Such a regime is conventionally described by the equa-tions of force-free electrodynamics (FFE), which describesthe non-linear dynamics of magnetic field lines at zerotemperature [2–4]. Importantly, this theory has no pre-ferred rest frame. The many applications of this theoryinclude the study of the magnetospheres of compact as-trophysical objects [5, 6].Nevertheless, FFE as conventionally formulated maybe considered incomplete. As a coarse-grained theorywith no intrinsic scales, it is insufficient to describe by it-self astrophysical phenomena such as (e.g.) coherent ra-diation and the production of particle winds [7–9]. The-oretically, though FFE is clearly an approximation, it isnot immediately clear what the small parameter is, norhow exactly it could be improved to better approximatereality.Thus motivated, and with an eye towards astrophysicalapplications, [10] (building on a framework constructedin [11]) reformulated force-free electrodynamics as an ef-fective field theory . The authors identified a set of sym-metry principles and wrote down the most general low-energy action respecting those principles, resulting ina realization of the force-free plasma as a fluid of coldstrings. The novel symmetry principle making this pos-sible was that of generalized global symmetries [12], and ∗ [email protected] the utility of such symmetries in the description of mag-netized plasma was first articulated in [11]. To leadingorder in derivatives, the theory of [10] is precisely force-free electrodynamics, where the “strings” are magneticfield lines. There is however an important conceptualdifference in that one no longer supposes that the inertiaof electric charges is neglected; rather these charges havebeen “integrated out” in that they no longer appear inthe low-energy description. A (slightly) different actionprinciple for an EFT for FFE appears in [13].Importantly, this effective field theory formalism nowallows for the systematic inclusion of higher-derivativecorrections to FFE. [11] showed that these correctionscan result in qualitatively new physical effects, such asthe generation of nontrivial electric fields parallel to themagnetic field, i.e. E · B (cid:54) = 0. A. Light charged modes
However, there is reason to believe that the EFT de-scription is still incomplete when applied to some actualphysical settings. Imagine the interaction of our FFEplasma with some extra electric charges that are movingultra-relativistically, perhaps due to their initial condi-tions, or perhaps because the magnetic field strength ismuch stronger than the electron rest mass squared, asin magnetars. Under such conditions, it may be a goodapproximation to consider the electron to be massless.The massless electrons and positrons will then spinrapidly around the magnetic field lines; quantum me-chanically, they will sink into the lowest Landau level.For massless electrons, the energy of the lowest Lan-dau level is exactly zero . Thus the motion of the elec- This occurs through a cancellation between the (negative) en-ergy associated with the Zeeman coupling of the electron spin a r X i v : . [ h e p - t h ] S e p tron transverse to the field lines is gapped, but its mo-tion along the field lines is gapless . Furthermore, if wework on scales longer than the magnetic cyclotron radius (cid:96) B ∼ ( eB ) − , particles moving along different field linesshould be essentially uncorrelated.These gapless modes are expected to be present at lowenergies but are clearly not included in the effective de-scription of [10]. One way to understand this is thatthey are associated with an almost-conserved axial cur-rent, which made no appearance in that discussion.In this work we will construct an effective description ofsuch light electric charges and their interaction with theplasma. We are motivated primarily by practical consid-erations, and so take the viewpoint that these are simply“extra” degrees of freedom that are not in equilibriumwith FFE plasma: thus we will sometimes refer to themas “non-equilibrium” charges. As they move freely onlyalong the two-dimensional world-sheet swept out by thefield line in spacetime, a great deal of intuition for thisproblem can be obtained from the bosonization of two-dimensional fermions, where a bosonic collective modecaptures the dynamics of the fermionic charge density.We similarly introduce a new effective 4d bosonic fieldΘ( x ) that is essentially a bosonized version of the electriccharge current j el along each field line. j σ el = − n µν (cid:15) σρµν ∂ ρ Θ( x ) (1)where here n µν is a unit-norm tensor introduced in [10]that is proportional to the electromagnetic field strength F µν . We will couple this field in a universal manner tothe FFE degrees of freedom. A key technical point is theidentification of a new symmetry principle that ties Θ tothe low-energy degrees of freedom of [10] in a way thatconfines the charges to move along field lines.Though our implementation in terms of EFT is novel,similar ideas have appeared before. In particular, build-ing on work in [15, 16], [17] recently performed a mi-croscopic construction of a similar collective field by di-rectly bosonizing the Landau levels of a 4d Dirac fermionalong a homogenous magnetic field. However, that workwas a perturbative computation, and the existence of aforce-free limit and the validity of various approxima-tions is not clear to us. Our hydrodynamic approach isan attempt to directly arrive at an effective low-energydescription. to the magnetic field and the (positive) energy of the zero-pointcyclotron motion about the field line. This cancellation is re-quired by index theorems that govern the realization of the 4daxial anomaly. At zero EM coupling this statement is obvious. At weak EMcoupling, the situation is somewhat subtle: a single one of thesezero modes – the “center of mass”, i.e. the appropriately defineduniform sum over states in the lowest Landau level – acquires amass by a coupling to the zero mode of the photon (see e.g. [14])but the majority of them remain massless.
B. Summary
We now summarize the remainder of the paper. In Sec-tion II we review some useful background material. InSection III we present our effective description of electriccharges. In Section IV we specialize to a particular the-ory (fixing various thermodynamic functions that appearin the general framework) and in Section V we presentsome simple applications, i.e. we demonstrate that thescattering of magnetosonic modes results in particle cre-ation and that the Michel monopole is now surroundedby a halo of non-equilibrated charges. In Section VI weconclude with some directions for future research.
II. BACKGROUND
Here we review some background material. The readerwho is completely familiar both with conventional 2dbosonization and with the EFT construction of FFE in[10] should skip to Section III, where the addition of gap-less electric charges is presented.
A. Review of 2d bosonization
We briefly discuss 2d bosonization. This is textbookmaterial, see e.g. [18]. Recall the Schwinger model, i.e.a massless 2d Dirac fermion coupled to 2d electromag-netism: S d = (cid:90) d x (cid:18) ¯ ψ (cid:0) i/∂ − i /A (cid:1) ψ − e F (cid:19) (2)This theory has a vector and axial current, written as j µV = ¯ ψγ µ ψ j µA = ¯ ψγ µ γ ψ (3)The vector current is coupled to the gauge field A µ , and iswhat we conventionally call “electric charge”. The axialcurrent is not actually conserved; a one-loop computationshows that it is is afflicted by the well-known 2d axialanomaly: ∂ µ j µA = 12 π (cid:15) µν F µν . (4)We can precisely reformulate this fermionic system as atheory of a single boson φ : S d = (cid:90) d x (cid:18) − π ( ∂φ ) − e F + 12 π A µ (cid:15) µν ∂ ν φ (cid:19) (5)In this formulation, the vector and axial currents are j µV = 12 π (cid:15) µν ∂ ν φ j µA = 14 π ∂ µ φ . (6)Note that the bosonic field φ provides an alternative de-scription of the dynamics of the charge sector, which maybe more convenient for certain purposes. For example,the axial anomaly (4) is now visible at the classical level,as it is equivalent to the equation of motion of the field φ . Note also that the vector electric current is identically conserved, unlike in the fermionic description. Many fea-tures of this story will reappear in our construction be-low. B. Review of EFT of FFE
We now turn to FFE, which is often presented as atheory of the Maxwell field strength tensor F µν , supple-mented with the condition that bare electric charges arepresent in sufficient quantities to screen the electric field,but in insufficient quantities for their energy-momentumexchange with the electromagnetic fields to matter (seee.g. [4]).In [10], a different effective theory viewpoint on FFEwas presented. We review this briefly here, assuming thatthe reader is already somewhat familiar with that work.As usual in EFT, we begin by identifying conserved quan-tities. One of them is the usual stress tensor T µν , whoseconservation follows from general covariance in the pres-ence of a background metric g µν . More interestingly, fora theory including dynamical magnetic fields, the mag-netic flux J µν ≡ (cid:15) µνρσ F ρσ is also a conserved quantity,as the Bianchi identity guarantees that we have ∇ µ J µν = 0 (7)We note that the symmetry associated with this con-served quantity may be somewhat unfamiliar and iscalled a generalized global symmetry [12]. Such symme-tries are present whenever one has a conserved density ofextended objects (such as magnetic field lines); they wereinitially understood in the context of non-Abelian gaugetheory, and have recently begun to be used to constrainhydrodynamic theories [11, 13, 19–27].The main idea of [10] is to realize this symmetry not onthe microscopic photon and electrons, but rather on a dif-ferent set of low-energy collective fields, which are takento be two scalar fields Φ , and a vector field a µ called theworldsheet magnetic photon. The simultaneous level setsof Φ , determine the magnetic field worldsheets, and da measures the magnetic flux on each worldsheet.It is very useful to couple J µν to an external fixed source field b µν . The symmetry associated with J is thenimplemented by demanding that the theory is invariantunder the following simultaneous transformation of dy-namical field a and source b : b → b + dλ a → a + λ (8)where λ is an arbitrary 1-form . The nonlinear transformation of a is similar to that of a 1-form It will be very important for our later purposes thatthis external source b may be interpreted as an externalelectric current density via j σ el = − (cid:15) σρµν ∂ ρ b µν . (9)This relation is explained in detail in [11], and arisesphysically from the idea that the natural source for atheory of dynamical electromagnetism is a fixed exter-nal charge density. Note that the current is identicallyconserved.Given the external sources b and g , we compute theconserved quantities from the action as T µν ≡ √− g δSδg µν J µν ≡ √− g δSδb µν . (10)In addition to this microscopic symmetry, we also de-mand that the theory is invariant under the following emergent symmetries of the description: • Field sheet relabelings, i.e reparametriztionsΦ I → Φ (cid:48) I (Φ I ) (11) • String-dependent 1-form shifts: a → a + ω (Φ ,
2) (12)Here ω is an arbitrary function of Φ , , and so varies fromworldsheet to worldsheet. These symmetries should bethought of as characterizing the low-energy phase that isforce-free electrodynamics.We now write down the most general effective actionthat is invariant under these symmetries. It is convenientto introduce the following 2-form and its magnitude. S µν ≡ ∇ [ µ Φ ∇ ν ] Φ s = (cid:114) S µν S µν n and the volume form ε on the foliation n µν = S µν s ε µν = 12 (cid:15) µνρσ n ρσ (14)which satisfy n µν n µν = − ε µν ε µν = 2 , n ∧ n = ε ∧ ε = 0 (15) S µν is invariant only under volume-preservingreparametrizations of the φ I , but both n and ε areinvariant (up to a sign) under all reparametrizations. Goldstone mode [28, 29] but the 1-form shift (12) means thatthe symmetry is not spontaneously broken in this phase: thisis a generalization of a similar “chemical-shift” symmetry whichplays the same role in effective actions for conventional hydro-dynamics [30].
It will often be convenient to work with the projectorsparallel and perpendicular to the foliation: h µν = − ε µρ ε νρ , h ⊥ µν = n µρ n νρ . (16)The only invariant scalar to lowest order in derivativesis µ = 12 ε µν ( b µν − ∂ µ a ν + ∂ ν a µ ) . (17)We note that the microscopic symmetry requires that a always appear in the combination b − da , and the string-dependent shift further requires that this be projectedagainst ε .We may now use this scalar to write down an invariantaction; for example, the leading order action is S [Φ , a ; g, b ] = (cid:90) d x √− g p ( µ ) . (18)where p is an arbitrary function of µ . Borrowing ter-minology from hydrodynamics, we call this the “ideal”action: from here we can use (10) to construct the idealstress tensor and magnetic flux as J µν = ρε µν T µν = pg µν − µρh µν , (19)where ρ = dpdµ . The equations of motion for this theoryare simply the conservation equations for T and J , whichare: ∇ µ T µν = 12 ( db ) νρσ J ρσ , ∇ µ J µν = 0 , (20)If we further make the choice p ( µ ) = µ , this the-ory is precisely equivalent to usual FFE. Recall that E · B = (cid:15) µνρσ J µν J ρσ ; if we take J to be given by itsideal expression (19), we see that E · B = 0, as requiredfor ideal FFE.Of course, the value of the EFT formalism is that itis now possible to systematically include corrections toFFE, simply by adding higher derivative terms to theaction (18). We emphasize one particular class of correc-tions, e.g. consider the following invariant second orderterm: S R [Φ , a ; g, b ] = 12 (cid:90) d x √− gR ( µ ) ∇ α (cid:15) βγ ( db ) αβγ (21)This results in the following correction J R to the idealflux (19): J = J + J R J µνR = − ∇ σ (cid:16) R ( µ ) ∇ [ σ (cid:15) µν ] (cid:17) (22)In the presence of this correction, we generically havethat E · B (cid:54) = 0; thus it creates an accelerating electricfield. However, this correction to J is identically con-served, meaning that within this theory it merely altersthe relationship between the flux J and the dynamicalfields without changing the equations of motion. As onemight expect, this will change once we add light electriccharges for this electric field to pull on in the next section.From now on, we will refer to the theory reviewed hereas the “FFE sector”, with total action (including possiblehigher-derivative corrections) S FFE [Φ , a ; g, b ]. III. EFFECTIVE THEORY OFNON-EQUILIBRIUM CURRENTSA. Electric charge and symmetries
We now finally turn to the addition of non-equilibriummassless charges; as motivated above, we would like tocouple the FFE EFT described above to a density ofmassless electric charges which are confined to movealong field lines. Note that the key object needed to per-form this coupling already exists within the FFE EFT; inparticular, through (9), we know that the external source b already has the interpretation as an electric charge den-sity.We thus introduce a new dynamical field Θ( x ), andcouple it to the FFE degrees of freedom through b . Wewrite b (Θ) = ¯ b + Θ( x ) n (23)where ¯ b is now the fixed external source, and where n is the binormal defined in (14). To get some intuitionfor this choice, take n to be constant (corresponding to aconstant magnetic field): from (9) we see that the electriccurrent arising from Θ is j σ el = − n µν (cid:15) σρµν ∂ ρ Θ( x ) (24)As desired, this is exactly a current propagating alongthe magnetic field directions. Indeed comparing it to theexpression for the 2d vector current (6), we see that Θplays a role very similar to the field φ appearing in thebosonized description of the 2d Dirac fermion.It is however not enough to demand that the currentflow mostly along the worldsheet. As motivated ear-lier, the current on each field-sheet should be also essen-tially uncorrelated, as we are studying infrared physics onscales much longer than the magnetic cyclotron length.To ensure that derivatives perpendicular to the world-sheet do not enter the theory, we further demand invari-ance under the following symmetry: • String-dependent scalar shift: Θ( x ) → Θ( x ) + s ( x ) f (Φ , Φ ) (25)Here f is an arbitrary function of Φ , , whereas s ( x ) wasdefined in (13). As we will discuss, this symmetry isclosely related to axial charge conservation. We also dis-cuss the rationale for the factor of s ( x ) below.We now write the full action of the system as S = S Θ [Φ , a, Θ; g, b (Θ)] + S FFE [Φ , a ; g, b (Θ)] (26)where the notation b (Θ) serves to stress that everywhere b is written in terms of Θ as in (23). Here S Θ is a newterm that describes the dynamics of Θ itself, whereas S FFE is the FFE theory constructed previously. Notethat S FFE depends on Θ only through b (Θ). This is akind of “minimal coupling”, indicating that the chargedegrees of freedom affect the FFE sector only throughtheir electric charge density (9).We now discuss the realization of the symmetries. Inparticular, it is not at all clear that the FFE sector to-gether with the coupling (23) is itself invariant under theshift symmetry (25). To see that it is, note that under(25), b shifts as b → b + f (Φ , Φ ) d Φ ∧ d Φ (27)where we have used that sn = d Φ ∧ d Φ .Now the magnetic shift symmetry (8) guarantees that b can enter the action only as db , or together with themagnetic worldsheet photon a in the combination ( b − da ). db is manifestly invariant under (27). Furthermore,the 1-form string-dependent shift of a (12) means that b − da must be projected down onto the worldsheet (e.g. asin (17)). However this projection is also invariant under(27), as d Φ , is perpendicular to the worldsheet.The upshot is that any FFE theory where b enjoys thesymmetries recorded in the previous section can be cou-pled to a Θ field as in (23) and is automatically invariantunder (25). The factor of s present in that expressionis crucial for this invariance. One way to understandthis is that f (Φ , Φ ) is not a scalar in the space of Φ , but rather a 2-form, and s transforms in the appropriatemanner to allow us to add it to a true scalar Θ in (25). B. Charge dynamics
Having coupled Θ to the magnetic field, we now turnto the dynamics of Θ itself. We would like to construct akinetic term for Θ. Arbitrary derivatives of Θ are clearlyforbidden by the shift symmetry (25). The 3-form d (Θ n )is however invariant under all symmetries, and we canuse it to construct a kinetic term. At leading order inderivatives, the only candidate is S Θ = − (cid:90) d x √− gQ ( µ ) ∇ [ µ (cid:0) n ρσ ] Θ (cid:1) ∇ [ µ (cid:16) n ρσ ] Θ (cid:17) (28)Here Q ( µ ) is an arbitrary function of µ . Again to obtainintuition consider the case where n is constant, in whichcase we find S Θ = − (cid:90) d x √− gQ ( µ ) h µν ∇ µ Θ ∇ ν Θ (29)where from (16) h µν is a projector parallel to the world-sheet; thus in equilibrium Θ becomes effectively a collec-tion of two-dimensional fields, each with dynamics onlyon the worldsheet.We note that in situations where n is not constant,then both derivatives and electric current off the world-sheet will appear; however the precise manner in whichthis happens is dictated by the symmetry principlesabove, and is thus a prediction of our EFT. We now discuss the equations of motion, starting withthat for Θ. The variation of the total action with respectto Θ takes the form δ Θ S = δ Θ S Θ + δSδ ¯ b µν n µν (30)where the second term arises from the implicit depen-dence of b on Θ in (23). Putting in the explicit formof S Θ and using the definition of J in (35), we find thefollowing equation of motion: ∇ α (cid:104) ∇ [ α (cid:16) n βσ ] Θ (cid:17) Q ( µ ) (cid:105) n βσ + J µν n µν = 0 (31)This is a wave equation Θ on the field-sheets, sourcedby a term that depends on the magnetic field. To un-derstand the source term, note that within our construc-tion, magnetic domination means that J always pointsmostly in the direction of ε – indeed at ideal order, wesee from (19) that it is precisely proportional to ε . As n = (cid:63)ε , the source term J µν n µν is essentially propor-tional to J ∧ J ∼ E · B , i.e. to the presence of unscreenedaccelerating electric fields. As we describe, under somecircumstances this sourced wave equation can be inter-preted as pair creation. C. Axial anomaly and pair creation
This equation of motion is closely related to theshift symmetry (25). Recall that this is actually in-finitely many symmetries, parametrized by a free func-tion f (Φ , Φ ); thus this leads to infinitely many Noethercharges, one on each worldsheet. As we will argue, thiscan be thought of as an independent axial current on eachfield line.To understand this, it is instructive to consider thecontribution to this Noether current not from the fullaction, but only from S Θ ; denoting this by j µf , we find j αf = ∇ [ α (cid:16) n βσ ] Θ (cid:17) Q ( µ ) n βσ f (Φ , Φ ) s ( x ) (32)As we have neglected the contribution from the FFE sec-tor, this is not quite conserved, and instead we have ∇ α j αf = − J µν n µν f (Φ , Φ ) s ( x ) (33)In the case f = 1, this is equivalent after some manip-ulation to the usual Θ equation of motion (31), thoughwe can only write the left-hand side of that equation asa divergence if we re-introduce the field s ( x ).This non-conservation equation may be understood asa hydrodynamic manifestation of the following micro-scopic Adler-Bell-Jackiw anomaly equation arising in thetheory of fermions coupled to QED: ∇ µ j µA = − π (cid:15) µνρσ F µν F ρσ . (34)where j µA is the axial current density ¯ ψγ µ γ ψ .We see that the role of j µA is played approximately by ∂ µ Θ, weighted by Q ( µ ) and projected in an appropriatemanner onto the worldsheet. Thus Θ is something like aworldsheet axion [31].It is instructive to examine how this equation relatesto pair creation in a magnetic field; our discussion hereis closely related to the chiral magnetic effect [32] (see[33] for a review). Consider pair creation of a massless e ± pair by a nonzero E · B . Energetically, the electronand positron will want to appear in their lowest Landaulevel. The spin in this lowest Landau level is correlatedwith the magnetic field: for the electron it is aligned andfor the positron it is anti-aligned. Under the influenceof the electric field, the electron and positron will moveoff in opposite directions along the magnetic field; thusfor both particle and anti-particle the spin is correlatedwith the motion. They both have the same helicity, andthe whole process thus results in the creation of net axial charge. Microscopically, this process is governed by theABJ anomaly (34), which directly relates E · B to axialcharge creation.In our hydrodynamic setup, we are describing the sameprocess, except at scales much longer than the magneticcyclotron length. Thus each field line is independent, andno charge can move from one field line to the next, so weobtain the stronger relation (33), corresponding to an in-dependent (non) conservation law on each field line. It isextremely interesting to see the physics of the anomalyemerging naturally from our purely hydrodynamic con-struction. We note also that the resulting expression isvery similar to the bosonization of the 2d axial anomalydiscussed around (6), except that the structure inducedby Φ , provides the information required to sew differentfield sheets together.Finally, while the qualitative physics of the anomalydoes appear from our construction, the right-hand sideof the equation is not precisely F ∧ F , but is only pro-portional to it, where the constant of proportionality is“dynamical” in that it depends on details of the thermo-dynamic function p ( µ ). There does not appear to be aprotected anomaly coefficient. This is related to the factthat the right hand side of the anomaly equation (34) isa dynamical operator and not an external source (as it isin usual examples of anomalous hydrodynamics [34, 35]),and thus in a theory of dynamical electromagnetism axialcurrent is simply genuinely non-conserved. Similar issuesrelated to non-universality have been studied in [36]. To be more precise: in standard conventions, the particle andanti-particle both have the same helicity, but for an anti-particlethe chirality is defined to be the opposite of the helicity [32]. Theleft-hand side of the (integrated) anomaly equation may be un-derstood as (the sum of the numbers of particles and anti-particlewith right-handed helicity) minus (the sum of the numbers ofparticles and anti-particles with left-handed helicity).
D. Effect of collective mode on FFE
Having exhaustively discussed the equation of motionof Θ, we return finally to the FFE sector. The stresstensor and magnetic flux of the system are now obtainedby differentiation of the total action with respect to g and ¯ b respectively: T µν ≡ √− g δSδg µν J µν ≡ √− g δSδ ¯ b µν . (35)Note that as µ depends on ¯ b , both J and T receive con-tributions from S Θ .As before, the the equations of motion are simply theconservation equations for the currents: ∇ µ T µν = 12 ( d ¯ b ) νρσ J ρσ , ∇ µ J µν = 0 , (36)with the modification that generically both T and J willnow receive extra contributions from the Θ sector. Wewill compute these contributions for a particular choiceof theory below. Note that it is the external source ¯ b thatappears on the right hand side of the non-conservationequation for T ; in most situations it is set to 0.Finally, we discuss a general feature of the equationsof motion. Let us imagine that the FFE sector has nohigher derivative corrections and is given by (18). In thatcase J ∝ (cid:15) , and thus the source term in (31) is zero. Itis then consistent to set Θ to 0, and thus all solutions toFFE remain solutions of the coupled theory. Linearizedfluctuations of Θ about any FFE solution will decouple.On the other hand, in the presence of higher derivativecorrections such as (21), E · B is no longer zero, and nowthe source term in (31) will turn on Θ. As expected, ac-celerating electric fields can have a dramatic effect on freeelectric charges, including the hydrodynamic manifesta-tion of the pair creation process discussed previously. Wewill study some aspects of this below. E. Formal aspects
We now note some formal aspects of the above con-struction. In particular, the sufficiently universally-minded reader may be somewhat puzzled that after go-ing through all the effort of constructing an effectivesymmetry-based description of FFE in [10], we then per-form a brutal operation – i.e. couple in “extra” mass-less charges – that is motivated not by global symmetrystructure but rather mostly by phenomenological consid-erations.We do not really feel that we have a completely sat-isfactory response to such a reader, but we note that ata formal level this construction corresponds to gauging aparticular part of the 1-form symmetry associated withmagnetic flux conservation.To be more precise, in the construction of [10], an ex-ternal b field couples to the 2-form current J . By makingpart of this b field dynamical as in (23), b = ¯ b + Θ( x ) n (37)and providing it with a kinetic term, we are gauging “thepart of the 2-form current perpendicular to the magneticfield lines.” This means that excitations of those gaugedcomponents of J – i.e. precisely those that create E · B – are now parts of a gauge current and not a global one,and Θ is a new sort of gauge field for these componentsof J .As usual in gauge theory, exciting gauge charges costsenergy in terms of the gradients of Θ, i.e. through theequation of motion (31), which can be thought of asGauss’s law for the new gauge symmetry. (One can com-pare this to the solution in conventional weak-couplingelectrodynamics, where excitations of the “gauged” elec-tric charge cost energy in terms of the gradients of thevector potential).We find this somewhat suggestive but still incomplete,as the identification of which part of the 1-form symmetrywe gauge is made through the binormal n , which is itselfstill a low-energy construct. Thus we do not see a purelyuniversal way to characterize this gauging procedure. Itwould be extremely interesting if one could be found andrelated to the structure of the axial anomaly discussed inthe previous subsection. IV. SPECIFIC THEORY
We briefly summarize. Given an EFT construction ofFFE governed by an effective action S FFE , there is a wayto “minimally” couple it to free electric charges confinedto move along string worldsheets, where the dynamics ofthese charges is given by S Θ , as in (26): S = S Θ [Φ , a, Θ; g, b (Θ)] + S FFE [Φ , a ; g, b (Θ)] . (38)For concreteness, in the remainder of this paper wewill work with the specific theory given by (38), wherethe FFE sector is given by the choice S FFE = S + S R (39)with S and S R given by (18) and (21) respectively. S and S Θ are the unique choices at leading order in deriva-tives. The choice of S R (and no other higher derivativecorrections) is arbitrary, and was made purely for conve-nience to illustrate the physics of pair creation; we do notexpect the qualitative physics to change if generic higherderivative corrections are added.We now discuss length scales. This theory containsthree arbitrary functions of µ : p ( µ ) , Q ( µ ), and R ( µ ). Us-ing the fact that both µ and Θ have mass dimension 2(for the latter, see (23)) the leading order expansion ofeach of these quantities in powers of µ is: p ( µ ) = 12 µ + · · · Q ( µ ) = Q | µ | + · · · R ( µ ) = µ Λ + · · · (40) where Λ is some UV mass scale. In all concrete compu-tations from here on, we will restrict to just the leadingterm in each of these expressions.Notably, both p and Q are scale-free to leading or-der. The non-analytic behavior of Q ( µ ) as a function of µ may seem surprising. To understand this, note thatin the FFE limit the magnitude of the magnetic field B = µ . Microscopically, the physics of Landau levels isindeed non-analytic as a function of the magnetic field;for example, the density of states of Landau zero modesis controlled by e | B | , and this is the ultimate origin ofthe non-analyticity in µ .At the level of the EFT, Q is a free parameter control-ling the strength of interactions of Θ with the FFE de-grees of freedom. In principle it can be determined from aUV description; in Appendix A we discuss a preliminaryattempt at matching with the microscopic treatment of[17], where the UV completion is provided by Dirac elec-trons coupled to QED with electromagnetic coupling e .In that case we find Q = 2 π e (41)This identification comes with caveats, and we refer theinterested reader to the Appendix for a full discussion.We will keep Q arbitrary in what follows.Thus this minimal theory has a single length scale Λ − ;it controls the magnitude of possible E · B , and thus canbe interpreted as defining the scale over which the perfectscreening of E in the plasma breaks down. An unscreened E will pair-create charges (parametrized by Θ) throughthe source term in (31). These charges will then moveabout, interacting nonlinearly with the plasma in a waythat is captured by the evolution equations below.We will provide an extremely preliminary discussion ofthe possible physics arising from this in some applicationsto simple geometries (wave scattering about a homoge-nous background, and the rotating Michel monopole) be-low. A. Detailed expressions for stress tensor and flux
For completeness, we write down the stress tensor andflux. We find J µν = ( ρ + ρ Θ ) ε µν − ∇ σ (cid:16) R ( µ ) ∇ [ σ ε µν ] (cid:17) (42)where as before ρ = dpdµ , but ρ Θ is a scalar correctionat O (Θ ) to the effective magnetic flux arising from the µ -dependence of the Θ kinetic term: ρ Θ ≡ − dQdµ ∇ [ µ (cid:0) n ρσ ] Θ (cid:1) ∇ [ µ (cid:16) n ρσ ] Θ (cid:17) (43)As the expression for T µν is somewhat lengthier, webreak it into several pieces: T µν = T µν + T µνR + T µν Θ (44)where T µν was given in (19) and where T R and T Θ arisefrom differentiating (21) and (28) with respect to g re-spectively, and are: T Rµν = 12 (cid:18) − R ( µ ) g µν + µ dRdµ h µν (cid:19) ∇ α (cid:15) βγ ( db ) αβγ +3 R ( µ ) ∇ [ µ (cid:15) ρσ ] ( db ) ναβ g ρα g σβ (45)where we remind the reader that b = ¯ b + Θ n with ¯ b thefixed external source, and T Θ µν = − (cid:20) (cid:18) − Q ( µ ) g µν + µ dQdµ h µν (cid:19) ( ∇ n Θ) Q ( µ ) ∇ [ µ (cid:0) n ρσ ] Θ (cid:1) ∇ [ ν (cid:0) n αβ ] Θ (cid:1) g ρα g σβ (cid:21) (46)where the notation ( ∇ n Θ) ≡ ∇ [ µ (cid:0) n ρσ ] Θ (cid:1) ∇ [ µ (cid:0) n ρσ ] Θ (cid:1) .To simplify the above computations, it was helpful tonote that the variation of n with respect to g is: δ g n µν = − n µν h ⊥ αβ δg αβ (47)This means that the metric variation of the combinationΘ n is proportional to n , and thus takes the same formas a variation of Θ n with respect to Θ. Such variationsvanish on-shell, i.e. when the Θ equations of motion (31)are imposed. In this situation it is then consistent toset all metric variations of (Θ n ) to zero when computingthe stress tensor. We have done so in the above expres-sions, which are thus valid only on-shell. (Genericallythey contain other terms proportional to the Θ equationof motion multiplying h ⊥ ).To recap: the degrees of freedom can be taken to be theFFE variables (cid:15), µ together with the charge mode Θ. Theequations of motion are the conservation equations (48),which for convenience we reproduce in the case where¯ b = 0: ∇ µ T µν = 0 , ∇ µ J µν = 0 , (48)together with the Θ equation of motion, which we writeout explicitly for arbitrary R ( µ ). ∇ α (cid:104) ∇ [ α (cid:16) n βσ ] Θ (cid:17) Q ( µ ) (cid:105) n βσ − ∇ σ (cid:16) R ( µ ) ∇ [ σ ε µν ] (cid:17) n µν = 0(49) V. APPLICATIONS
In this section we present some simple applications ofthis formalism.
A. Michel monopole
We first discuss how the above theory behaves on theMichel monopole background, which is an exact solution to conventional FFE [37]. This is a rotating magneticmonopole, and is a toy model for the magnetosphere out-side of a rotating uniformly magnetized star. One shouldimagine that the core of the monopole is shielded by thestar radius at r = r (cid:63) .In terms of the degrees of freedom (cid:15) and µ , the solutioncan be written as: ε = d ( t − r ) ∧ ( dr − r Ω sin θdφ ) µ = qr . (50)which corresponds to J = qr d ( t − r ) ∧ ( dr − r Ω sin θdφ ) (51)Though it is not necessary for our purposes, a valid choiceof the foliation degrees of freedom isΦ = θ, Φ = φ − Ω( t − r ) , a = qr dt (52)This is a solution to pure FFE. Does it remain a so-lution to the theory of free charges described above? Asexplained in [10], in the presence of the higher derivativecoupling (21), a non-trivial E · B is created. This has noeffect in the original FFE theory, but in the model withlight charges it sources the Θ field, and we can no longerset it to zero. Instead we must solve the wave equation(49): Q ∇ α (cid:20) ∇ [ α | µ | (cid:16) n βσ ] Θ (cid:17)(cid:21) n βσ = 3Λ ∇ σ (cid:16) µ ∇ [ σ (cid:15) µν ] (cid:17) n µν (53)where we have inserted the form of Q ( µ ) and R ( µ ).In general, this is now a nonlinear problem that canonly be solved numerically. In this work we tackle it bytaking the scale Λ − to be much smaller than any otherscales in the problem, giving us a small parameter inwhich we can perform a perturbative expansion.
1. Magnetic worldsheets and dS We first examine the kinetic term appearing in (53).It is convenient to define outgoing time u = t − r and arescaled field Θ( t, r, θ, φ ) ≡ r ˜Θ( r, t, θ, φ ) (54)For future convenience, we define the wave operator onthe left-hand side of (53) as (cid:3) : Q ∇ α (cid:34) ∇ [ α | µ | (cid:32) n βσ ] ˜Θ r (cid:33)(cid:35) n βσ ≡ (cid:3) ˜Θ (55)We then find: (cid:3) ˜Θ = 2 Q q (cid:2) (1 − r Ω sin θ ) ∂ r − ∂ u + Ω ∂ φ + r Ω sin θ ) ∂ r (cid:3) ˜Θ (56)Note that there are no θ derivatives at all, and so eachlatitude completely decouples from the rest. However therotation does introduce a φ derivative Ω ∂ φ . Indeed thefull dependence on ( u, φ ) is now through the combination( ∂ u + Ω ∂ φ )Curiously, the operator within square brackets is pre-cisely the wave operator of two-dimensional de-Sitterspace written in outgoing Eddington-Finkelstein coordi-nates: ds = − (cid:18) − r L (cid:19) du − dudr (57)provided we make the substitution( ∂ u ) dS → ( ∂ u + Ω ∂ φ ) Michel (58)and where the de Sitter radius L is L = 1Ω sin θ (59)In particular, the location of the ( θ -dependent) de Sit-ter horizon is r = (Ω sin θ ) − , i.e. at the light cylinder.Heuristically, the light cylinder is the point where an ob-server rotating with the star would have to move fasterthan light. It acts as a horizon for Alfvenic perturbationsof the background, and apparently also for the chargefluctuations developed in this paper.The relevance of dS space to the Michel monopole wasanticipated in [4], where it was pointed out that the in-trinsic geometry of a Michel field sheet is indeed dS . Itis thus unsurprising that charges that are confined to thisworld-sheet obey the de Sitter wave equation, albeit withan interestingly shifted notion of dS time (58). Studyingthe propagation of waves on this effective curved geome-try will now require a treatment of boundary conditionsat the de Sitter horizon. We would find it extremely sat-isfying if the physics of de Sitter horizons were to playan unexpected role in an understanding of the dynamicsof pulsars, and hope to return to this point in the future.Finally, as we will require it for the next section, wenote that in the case that Θ depends only on r , the waveoperator above reduces to (cid:3) ˜Θ ≡ Q q (cid:2)(cid:0) − r Ω sin θ (cid:1) r ∂ r − r Ω sin θ∂ r (cid:3) ˜Θ(60)The two independent solutions to this equation areΘ( r ) = 1 r ( A + B arctanh( r Ω sin θ )) (61)where we have expressed them in terms of the originalfield Θ. Note that the term in B is singular at the lightcylinder.
2. Induced charges
We now return to our original problem of determininghow the Θ field behaves in the presence of the Michel background. In particular, we will perform a perturba-tive expansion in powers of Λ − :Θ = Θ (0) + Θ (2) Λ − + · · · J = J (0) + J (2) Λ − + · · · (62)Here J (0) is the force-free bare Michel solution, and itis consistent to set Θ (0) to zero. To find the next orderterm, we insert this form into the wave equation (53).In terms of the wave operator (60), the equation for theterm in Λ − becomes (cid:3) ˜Θ (2) = − q Ω cos θr (63)To solve this equation, we need to specify boundary con-ditions on the field. We will demand that the solutionremain regular at the light cylinder; this amounts to set-ting the coefficient B to zero in (61).At the neutron star surface r = r (cid:63) , the physics is likelyto be rather complicated and non-linear, and we do notat the moment have a good understanding of the cor-rect boundary condition. For simplicity, we impose theNeumann boundary condition ∂ r Θ = 0, noting that aDirichlet condition would imply an explicit breaking ofaxial symmetry. Our result are largely insensitive to thischoice, changing the final expression only by an O (1) fac-tor.With these choices, we find that to leading order inΛ − , the Θ field is:Θ( r ) = 3 q (3 r − r (cid:63) ) Q r r (cid:63) Λ Ω cos θ + · · · (64)Using the identification (24), the electric current is j el = − q ( r − r (cid:63) ) Ω cos θr Q r (cid:63) Λ ( ∂ t + Ω ∂ φ ) (65)This induced current is proportional to the co-rotatingKilling vector of the Michel geometry ∂ t + Ω ∂ φ . It ap-pears that the electric field created has populated spacewith this charge density, which subsequently attempts torotate with the star. This rigid rotation happens fasterand faster as we move out in r until it reaches the speedof light at the light cylinder. The current vector is thustimelike at small r , null at the light cylinder, and space-like outside it.It would be very interesting to understand the conse-quences of this charge density in more physically realisticsituations with less symmetry. This will likely require nu-merical simulation. B. Particle creation by wave scattering
FFE linearized about a homogenous magnetic fieldsupports two types of linearly dispersing wave excitation;the transverse Alfven mode and the longitudinal magne-tosonic (or fast) mode. The addition of the Θ field results0in a new gapless mode that disperses linearly along thefield lines . To leading order in the linearized theory, allof these modes are independent and do not mix. In thissection we demonstrate that at nonlinear order, the scat-tering of two magnetosonic waves will generically resultin particle creation.Consider a homogenous background magnetic field ofmagnitude B pointing in the z direction; this correspondsto (cid:15) = dt ∧ dz n = dx ∧ dy µ = B (66)so that the magnetic flux is simply J = Bdt ∧ dz (67)We first review the dispersion relation of a single mag-netosonic wave (see e.g. [11]). We work with the FFEequation of state p ( µ ) = µ .Let the spacetime dependence of the wave be e ik µ x µ .We define the momentum parallel and perpendicular tothe background field to be: k µ (cid:107) ≡ ( h ) µν k ν k µ ⊥ = ( h ⊥ ) µν k ν , (68)where the projectors h, h ⊥ were defined in (16) and arehere evaluated on the homogenous background field (66).We now consider linearized perturbations around thebackground solution (66), (cid:15) → (cid:15) + δ(cid:15)e ik µ x µ , µ → B + δµe ik µ x µ . For convenience, let a, b run over the directionsparallel to the field t, z and let i, j run over x, y . By solv-ing the linearized equations of motion, the componentsof δ(cid:15) can be written in terms of the scalar perturbation δµ : δ(cid:15) ia = − k ⊥ B k i k b (cid:15) ba δµ (69)(Both δ(cid:15) ij and δ(cid:15) ab identically vanish by the normal-ization and degeneracy constraints (15) on (cid:15) ). For theFFE equation of state all modes propagate at the speedof light, and the dispersion relation can be written in amanifestly Lorentz and rotationally invariant manner as k (cid:107) = − k ⊥ (70)Now we turn to the interactions with Θ. A single lin-earized wave does not turn on the source term in (31);thus, as mentioned earlier, at the linearized level fluctu-ations of Θ and of the FFE modes decouple. We can compare this with the results of [38], which studies amodel of spontaneously broken axial symmetry coupled to freeMaxwell EM with a background magnetic field. They find amixing between an unscreened E (cid:107) and the axial Goldstone, re-sulting in a quadratically dispersing mode. Our results differbecause our electromagnetic sector is governed by FFE, where E (cid:107) is already gapped out and cannot mix with anything. However if we consider the superposition of two mag-netosonic modes with momenta k , k , then after somealgebra we find for the source term J µν n µν ( x ) = e i ( k + k ) · x g ( k , k ) δµ δµ (71)where the “interaction vertex” g ( k , k ) is the followingsymmetric function of the momenta: g ( k , k ) = − B (cid:16) k ⊥ , + k ⊥ , (cid:17) k ⊥ , k ⊥ , n µν k µ, k ν, (cid:15) µν k µ, k ν, (72)Thus we see that non-trivial E · B has been created,and will generically create electric charges through thesourced wave equation (31). This is the main result ofthis section. It would be very interesting to understand ifthis provides a systematic way to understand an energycascade from the magnetic field to electric charges, e.g.during FFE turbulence [39].We briefly sketch how to go further and determine theΘ field itself. We must then solve the sourced wave equa-tion (31), which to lowest order in the perturbations sim-ply reduces to:2 Q B (cid:0) ∂ t − ∂ z (cid:1) Θ = − J µν n µν (73)As usual this can be solved by introducing a Green’sfunction G Θ ( x, y ) for Θ, which is a delta function in thetransverse directions but propagates gaplessly along thefield lines: G Θ ( x, y ) ≡ B Q δ (2) ( x i − y i ) (cid:90) d p (2 π ) e ih ab p a ( x − y ) b h ab p a p b (74)A particular solution to (73) is then simplyΘ( x ) = − (cid:90) d yG Θ ( x, y ) J µν n µν ( y ) (75)Combining this expression with (71), it should be clearthat we are constructing a classical solution by evaluat-ing tree-level Feynman diagrams; in particular, we haveshown that there is an amplitude for two magnetosonicwaves to scatter and create particle charge, and thus theFeynman rules for this theory have a vertex of the formshown in Figure 1.We note that this is a sort of “inverse pion decay”: con-ventionally, the π (which, like Θ, is roughly a Goldstonemode for a spontaneously broken axial symmetry) decaysto two electromagnetic excitations (photons) through achannel mediated by the axial anomaly (34). Here twoelectromagnetic excitations (magnetosonic modes) com-bine through the anomaly-mediated channel (33) to cre-ate an excitation of Θ.It would be very interesting to further develop andunderstand the physical consequences – if any – of thisdiagrammatic expansion.1 k k g ( k , k ) G ⇥ µ µ ⇥ FIG. 1. “Feynman” diagram associated with the evaluation ofthe term computed in (75); solid lines indicate magnetosonicmodes (where in our kinematics the propagators have beenamputated) and dotted line indicates Θ propagator G Θ . VI. CONCLUSIONS
In this work, we showed how to couple extra light elec-tric charges – charges that are not in equilibrium withthe plasma – to the FFE EFT of [10]. We conclude withsome directions for future work.One original motivation for this study was astrophys-ical. Ideally, one might hope that this model could pro-vide a useful caricature of the dynamics of compact astro-physical objects, where open questions remain regarding(e.g.) the origins of coherent radiation and the produc-tion of particle winds [7–9].In particular, the truncated model described in Sec-tion IV provides a concrete deformation away from FFE,parametrized by a single scale Λ that controls the scaleat which unscreened electric fields can self-consistentlyappear. Such unscreened fields will subsequently pair-create and accelerate charges, as captured by ripples inthe collective field Θ. It would be very interesting tounderstand the physical consequences of such nonlineardynamics in more realistic geometries with less symme-try than the Michel monopole. This will likely requirenumerical simulation. We note that the viewpoint takenhere – deforming in a controlled manner away from FFE– is the opposite to that taken in usual particle-in-cellsimulations (see e.g. [40, 41]), where one starts micro-scopically with free Maxwell electrodynamics coupled tocharge dynamics and arrives (at long distances) at FFE.One clear deficiency of the system is the fact that itprovides a clean description only in situations when themass of the electron can be neglected. It is somewhatnontrivial to study the electron mass as a perturbation;it seems that any attempt to do so and thus break the ax-ial shift symmetry (25) also ultimately correlates fluctua-tions on different field lines, essentially because a massiveelectron has a finite transverse radius.For any potential application to real-life systems, itis also clearly crucial to understand the magnitude ofscales such as Λ. Such higher-derivative corrections canin principle be precisely matched to a UV description (e.g. QED) using the analogue of hydrodynamic Kuboformulas (see e.g. [42] for a review). We hope to reporton this in the future.
ACKNOWLEDGEMENTS
I am very thankful to S. Gralla for collaboration onthe initial stages of this project and helpful conversa-tions throughout. I am also grateful for conversationson related issues with S. Grozdanov, U. Gursoy, D. Hof-man, A. Lupsasca and N. Poovuttikul. I thank the As-pen Center for Physics (which is supported by NationalScience Foundation grant PHY-1607611) where part ofthis work was performed in the working group “Apply-ing tools from high-energy theory to problems in astro-physics”, and where my trip was supported by a grantfrom the Simons Foundation and an International En-gagement Grant from Durham University. I am sup-ported in part by the STFC under consolidated grantST/L000407/1.
Appendix A: Comparison to microscopic strong-fieldbosonization
Here we compare our theory with the top-down con-struction of [17], which follows earlier foundational workin [15] (see also [16] for a similar computation in a differ-ent context).We briefly review [17]: there QED with electromag-netic coupling e is studied with a single species of massiveDirac fermion on a constant magnetic field background.The fermion is decomposed into Landau levels, and theneach fermion state in the lowest Landau level is bosonizedinto a massless scalar φ i with a kinetic term with deriva-tives only along the magnetic field directions, where i runs over the N = eBA π states in the lowest Landau level.It is then assumed that all of the bosons φ i move in sync,i.e. φ = φ = · · · = φ N = Φ . (A1)An effective action is constructed for Φ and its couplingto the electromagnetic field, and it is then assumed thatthe fields are allowed to vary slowly in the transversedirections to obtain a 4d dynamical theory. Derivativesof Φ in the transverse directions do not appear in thismodel.This sounds structurally similar to our EFT. We nowattempt a comparison: this is possible only for small fluc-tuations around a homogenous background (66). Φ re-sults in an effective electric current of the form j µ el = − e π (cid:15) µνρσ ∂ ν Φ F ρσ (A2)Comparing this to our (24), we see that these take the2same form if we identifyΘ = e π µ Φ (A3)(Here µ is taken to be constant). Comparing now ourkinetic term (29) with that in Eq (3.2) of [17], we seethat they agree if Q ( µ ) = 2 π e | µ | , (A4)thus motivating the value of Q (41) described in themain text.We now discuss the issues with such a comparison.Firstly as our Θ is gapless, the fermion mass m in [17] must be ignored; as mentioned in that work, this appearsto be a necessary condition for a force-free limit in anycase. A further issue is that even if m is set to zero,linearized fluctuations of Φ coupled with the electromag-netic field are still gapped, for essentially the same reasonthat the Schwinger model shown in (2) is gapped. Thisis clearly not the case for our gapless Θ field, and we be-lieve that this occurs because the dynamics of the N − φ i (which remain gapless [14]) have been neglected.We find it plausible that the gapped mode in [17] iseaten by the electromagnetic field and helps to sustainthe FFE architecture in the manner originally proposedin [15], whereas the remaining N − [1] H. Alfv´en, Cosmic radiation as an intra-galacticphenomenon . Almqvist & Wiksells Boktryckeri, 1937.[2] T. Uchida, “Theory of force-free electromagnetic fields.I. General theory,” Phys. Rev. E (Aug., 1997)2181–2197.[3] S. S. Komissarov, “Time-dependent, force-free,degenerate electrodynamics,” MNRAS (Nov., 2002)759–766, astro-ph/0202447 .[4] S. E. Gralla and T. Jacobson, “Spacetime approach toforce-free magnetospheres,” Mon. Not. Roy. Astron.Soc. no. 3, (2014) 2500–2534, arXiv:1401.6159[astro-ph.HE] .[5] P. Goldreich and W. H. Julian, “PulsarElectrodynamics,” ApJ (Aug., 1969) 869.[6] R. D. Blandford and R. L. Znajek, “Electromagneticextraction of energy from Kerr black holes,” MNRAS (May, 1977) 433–456.[7] J. I. Katz, “Fast radio bursts – A brief review: Somequestions, fewer answers,”
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