New magnetic field instability and magnetar bursts
aa r X i v : . [ a s t r o - ph . H E ] D ec New magnetic field instability and magnetar bursts
Maxim Dvornikov a,b ∗ a Pushkov Institute of Terrestrial Magnetism, Ionosphereand Radiowave Propagation (IZMIRAN),108840 Troitsk, Moscow, Russia; b Physics Faculty, National Research Tomsk State University,36 Lenin Avenue, 634050 Tomsk, Russia
Abstract
We consider the system of massive electrons, possessing nonzero anomalous magneticmoments, which electroweakly interact with background neutrons under the influence ofan external magnetic field. The Dirac equation for such electrons is exactly solved. Basingon the obtained solution, we find that a nonzero electric current of these electrons canflow along the magnetic field. Accounting for the new current in the Maxwell equations,we demonstrate that a magnetic field in this system appears to be unstable. Then weconsider a particular case of a degenerate electron gas, which may well exist in a neutronstar, and show that a seed magnetic field can be amplified by more than one order ofmagnitude. Finally we discuss the application of our results for the explanation of theelectromagnetic radiation emitted by magnetars.
The problem of the magnetic field instability is important, e.g., in the context of the existenceof strong astrophysical magnetic fields [1]. Besides the magnetohydrodynamics mechanismsfor the generation of astrophysical magnetic fields, recently the approaches based on theelementary particle physics were proposed. These approaches mainly rely on the chiral mag-netic effect (CME) [2], which consists in the generation of the anomalous current of masslesscharged particles along the magnetic field J = α em ( µ R − µ L ) B /π , where α em ≈ /
137 is thefine structure constant and µ R , L are the chemical potentials of right and left chiral fermions.If J is accounted for in the Maxwell equations, the magnetic field appears to be unstableand can experience a significant enhancement. The model for the generation of strong mag-netic fields in the dense matter of a neutron star (NS) driven by CME under the influence ofthe electroweak interaction between electrons and neutrons was developed in a series of ourworks [3–7]. Other applications of CME for the generation of astrophysical and cosmologicalmagnetic fields are reviewed by Kharzeev [8].However, the existence of CME in astrophysical media is questionable. Vilenkin [9] andDvornikov [10] found that J can be non-vanishing only if the mass of charged particles,forming the current, is exactly equal to zero, i.e. the chiral symmetry is restored. For the caseof electrons the restoration of the chiral symmetry is unlikely at reasonable densities which ∗ [email protected] J ∼ B for massive particles,which can lead to the magnetic field instability, is quite important for the explanation ofastrophysical magnetic fields. One of the examples of such a current in electroweak mat-ter was proposed by Semikoz & Sokoloff [15]. However, the model developed by Semikoz &Sokoloff [15] implies the inhomogeneity of background matter. This fact imposes the restric-tion on the scale of the magnetic field generated.In the present work, we discuss another scenario for the magnetic field instability. Itinvolves the consideration of the electroweak interaction of massive fermions with backgroundmatter along with nonzero anomalous magnetic moments of these particles. Note that theelectroweak interaction implies the generic parity violation which can provide the magneticfield instability. Recently, the interpretation of CME in terms of an effective magnetic momentwas considered by Kharzeev et al. [16].This work is organized as follows. In Sec. 2, we discuss the Dirac equation for a massiveelectron with a nonzero anomalous magnetic moment, electroweakly interacting with back-ground matter under the influence of an external magnetic field. Then, we describe the mainsteps in finding the exact solution of this Dirac equation which was previously obtained byBalantsev et al. [17]. Using this solution, in Sec. 3, we calculate the electric current of theseelectrons along the magnetic field direction. This current turns out to be nonzero. Thenwe consider a particular situation of a strongly degenerate electron gas, which can be foundinside NS. Finally, in Sec. 4, we apply our results for the description of the amplification ofthe magnetic field in NS and briefly discuss the implication of our findings for the explanationthe electromagnetic radiation of compact stars. Let us consider an electron with the mass m and the anomalous magnetic moment µ . Thiselectron is taken to interact electroweakly with nonmoving and unpolarized background mat-ter consisting of neutrons and protons under the influence of the external magnetic field alongthe z -axis, B = B e z . Accounting for the forward scattering off background fermions in theFermi approximation, the Dirac equation for the electron has the form, (cid:8) γ µ P µ − m − µB Σ − γ (cid:2) V R (cid:0) γ (cid:1) + V L (cid:0) − γ (cid:1)(cid:3) / (cid:9) ψ = 0 , (1)where γ µ = (cid:0) γ , γ (cid:1) , γ = i γ γ γ γ , and Σ = γ γ γ are the Dirac matrices, P µ = i ∂ µ + eA µ , A µ = (0 , , Bx,
0) is the vector potential, and e > V R , L have the form [3], V R = − G F √ n n − n p (1 − ξ )] 2 ξ, V L = − G F √ n n − n p (1 − ξ )] (2 ξ − , (2)where n n,p are the number densities of neutrons and protons, G F = 1 . × − GeV − is theFermi constant, and ξ = sin θ W ≈ .
23 is the Weinberg parameter.The solution of Eq. 1 has the form [17], ψ T = exp ( − i Et + i p y y + i p z z ) ( C u n − , i C u n , C u n − , i C u n ) (3)2here u n ( η ) = (cid:18) eBπ (cid:19) / exp (cid:18) − η (cid:19) H n ( η ) √ n n! , n = 0 , , . . . , (4)are the Hermite functions, H n ( η ) are the Hermite polynomials, η = √ eBx + p y / √ eB , C i arethe spin coefficients, i = 1 , . . . ,
4, and −∞ < p y,z < + ∞ .Using Eqs. 1 and 3, we get that the spin coefficients C i obey the system of equations, (cid:0) E − ¯ V − p z + V (cid:1) C − √ eB n C + ( m + µB ) C =0 , √ eB n C − (cid:0) E − ¯ V + p z + V (cid:1) C − ( m − µB ) C =0 , ( m + µB ) C + (cid:0) E − ¯ V + p z − V (cid:1) C + √ eB n C =0 , ( m − µB ) C + √ eB n C + (cid:0) E − ¯ V − p z − V (cid:1) C =0 , (5)where ¯ V = ( V L + V R ) /
2, and V = ( V L − V R ) /
2. To derive Eq. 5 we choose the Dirac matricesin the chiral representation [18], γ µ = (cid:18) − σ µ − ¯ σ µ (cid:19) , σ µ = ( σ , − σ ) , ¯ σ µ = ( σ , σ ) , (6)where σ is the unit 2 × σ are the Pauli matrices.Equating the determinant of the system in Eq. 5 to zero, we get the energy levels E forn > E = ¯ V + E , E = q p z + m + 2 eB n + ( µB ) + V + 2 sR ,R = r ( p z V − µBm ) + 2 eB n h V + ( µB ) i , (7)where s = ± C i at n > S = V ˆ S long − µB ˆ S tr p V + ( µB ) , ˆ S long = ( ΣP ) m , ˆ S tr = Σ − i m ( γ × P ) (8)commutes with the Hamiltonian of Eq. 1. Basing on the fact that the wave function in Eq. 3is the eigenfunction of the operator ˆ S , it is convenient to represent C i in terms of the newauxiliary coefficients A and B as (cid:18) C C (cid:19) = 1 √ r − sR ( p z V − µBm ) (cid:18) Z − µB/ZµB/Z Z (cid:19) (cid:18) AB (cid:19) , (cid:18) C C (cid:19) = s √ r sR ( p z V − µBm ) (cid:18) Z µB/Z − µB/Z Z (cid:19) (cid:18) AB (cid:19) , (9)where Z = q V + p V + ( µB ) .Inserting Eq. 9 to Eq. 5, we get that A and B are completely defined by the followingrelation: A = ( − sR + ( µB ) + V E p V + ( µB ) ) C , B = ( sR + ( µB ) + V E p V + ( µB ) ) C , AB = − mV + µBp z E p V + ( µB ) C . (10)3he coefficient C can be found if we normalize the wave function ψ as Z d xψ † p y p z n ψ p ′ y p ′ z n ′ = δ (cid:0) p y − p ′ y (cid:1) δ (cid:0) p z − p ′ z (cid:1) δ nn ′ . (11)In this situation, the spin coefficients obey the relation, X i =1 | C i | = 1(2 π ) , (12)at any n ≥
0. Finally, we get that C = 14(2 π ) p V + ( µB ) . (13)We can see that Eqs. 9, 10, and 13 completely define the spin coefficients C i at n > C = C = 0.Thus, the energy spectrum reads E = ¯ V + E , E = q ( p z + V ) + ( m − µB ) . (14)Using Eq. 12, we obtain that the nonzero spin coefficients C , have the form, | C | = 12(2 π ) E ( m − µB ) ( E + p z + V ) , | C | = E + p z + V π ) E . (15)Note that, while solving Eq. 1, we take into account only electron rather than positronsdegrees of freedom. Using the exact solution of the Dirac equation, which is found above, we can calculate theelectric current of electrons in this matter. This current has the form [9], J = − e ∞ X n=0 X s Z + ∞−∞ d p y d p z ¯ ψ γ ψf ( E − χ ) , (16)where f ( E ) = [exp( βE ) + 1] − is the Fermi-Dirac distribution function, β = 1 /T is thereciprocal temperature, and χ is the chemical potential. First, we notice that J x,y ∼ ¯ ψγ , ψ =0 because of the orthogonality of Hermite functions with different indexes. Hence, only J z ∼ ¯ ψγ ψ should be considered. Then, using Eq. 9, we can derive the identity Z + ∞−∞ d p y ψ † γ γ ψ = eB (cid:0) | C | + | C | − | C | − | C | (cid:1) = − eB (cid:20) µB AB + s V R ( p z V − µBm ) (cid:0) A − B (cid:1)(cid:21) , (17)which is valid in the chiral representation of Dirac matrices in Eq. 6.4ow let us consider the contribution of the energy levels, with n >
0, to J z . Basing onEqs. 10, 13, and 17, we obtain that it has the form, J (n > z = − e B (2 π ) ∞ X n=1 X s = ± Z + ∞−∞ d p z E (cid:20) p z (cid:18) s V R (cid:19) − s µBmV R (cid:21) f ( E − χ ) . (18)To find the first nonzero term in Eq. 18 we decompose J (n > z in a series in µB and V . Finally,we get that J z = µmV B e π ∞ X n=1 Z + ∞ d p E (cid:20)(cid:18) − p E (cid:19) (cid:18) f ′ − f E eff (cid:19) + p E eff f ′′ (cid:21) , (19)where E eff = q p + m and m eff = √ m + 2 eB n. The argument of the distribution functionin Eq. 19 is E eff + ¯ V − χ .As an example, we shall consider a strongly degenerate electron gas. In this situation, f = θ ( χ − ¯ V − E eff ), where θ ( z ) is the Heaviside step function. We can also disregard thepositrons contribution to J z . The direct calculation of the current in Eq. 19 gives J z = − µmV B e π ˜ χ ∞ X n=1 q ˜ χ − m θ ( ˜ χ − m eff ) , (20)where ˜ χ = χ − ¯ V . One can see that J z in Eq. 20 is nonzero if B < ˜ B , where ˜ B = (cid:0) ˜ χ − m (cid:1) / e .If the magnetic field is relatively strong and is close to ˜ B , then only the first energy level withn = 1 contributes to J z , giving one J z = − α em π µBmV ˜ χ B p ˜ χ − m − eB → , (21)where α em = e / π . In the opposite situation, when B ≪ ˜ B , one gets that J z = − α em πe µBmV ˜ χ (cid:0) ˜ χ − m − eB (cid:1) / ≈ − α em πe µmV B, (22)i.e. the current is proportional to the magnetic field strength.Finally, let us consider the contribution of the lowest energy level n = 0 to J z . UsingEqs. 14, 15, and 17, we rewrite J (n=0) z in Eq. 16 as J (n=0) z = e B Z + ∞−∞ d p z (cid:0) | C | − | C | (cid:1) f ( E − χ )= − e B (2 π ) Z + ∞−∞ d p z p z + V q ( p z + V ) + ( m − µB ) f ( E − χ ) = 0 , (23)where we take into account the expression for E in Eq. 14. Eq. 23 means that the lowest energylevel with n = 0 does not contribute to the electric current along the magnetic field. Thisresult extends our recent finding [10] to the situation when the anomalous magnetic momentis accounted for. Analogously to Eq. 23, one can show that positrons do not contribute to thecurrent either. Note that the result is Eq. 23 is valid for arbitrary characteristics of plasma,external fields, as well as the mass and the magnetic moment of an electron.5t is interesting to compare the appearance of the new current along the magnetic field inEq. 19 with CME [2, 8]. Vilenkin [9] showed that only massless electrons at the zero Landaulevel in an external magnetic field contribute to the generation of the anomalous current alongthe magnetic field. This feature remains valid in the presence of the background electroweakmatter [3, 4]. The current of such massless particles is exited since, at the zero Landau level,left electrons move along the magnetic field, whereas right particles move in the oppositedirection [3, 4]. Electrons at higher Landau levels can move arbitrarily with respect to themagnetic field. Therefore, if one has a different population of left and right electrons at thelowest Landau level, there is a nonzero current is the system J ∼ ( µ R − µ L ) B , which is themanifestation of CME.In the situation described in the present work, i.e. when massive electrons with nonzeroanomalous magnetic moment move in the electroweak matter, the particles at the lowestenergy level can move in any direction with respect to the magnetic field, i.e. −∞ < p z < + ∞ .Moreover, there is no asymmetry for electrons with p z > p z <
0. It results from Eq. 14if we replace p z → p z − V there. On the contrary, higher energy levels with n > p z → − p z . The reflectional symmetrycannot be restored by any replacement of p z . Therefore electrons having p z > p z < v z = p z / E . The electric current alongthe magnetic field B = B e z is proportional to h v z i . Thus such a current should be nonzero,with only higher energy levels contributing to it. It is interesting to mention that the termin Eq. 7 which violates the reflectional symmetry p z → − p z is proportional to µBmV . It isthis factor which J z in Eq. 19 is proportional to. Returning to the vector notations we get the current in Eq. 20 takes the form, J = Π B , Π = − µmV B α em π ˜ χ N X n=1 q ˜ χ − m , (24)where N is maximal integer, for which ˜ χ − m − eBN ≥
0. To study the magnetic fieldevolution in the presence of the additional current in Eq. 24 we take this current into accountin the Maxwell equations along with the usual ohmic current J = σ cond E , where σ cond isthe matter conductivity and E is the electric field. Considering the magnetohydrodynamicapproximation, which reads σ cond ≫ ω , where ω is the typical frequency of the electromagneticfields variation, we derive the modified Faraday equation for the magnetic field evolution, ∂ B ∂t = Π σ cond ( ∇ × B ) + 1 σ cond ∇ B + 1 σ cond B dΠd B (cid:2) B ( ∇ × B ) − B ( B · ∇ × B ) − ( B × ( B ∇ ) B ) (cid:3) , (25)where we neglect the coordinate dependence of σ cond .Let us consider the evolution of the magnetic field given by the Chern-Simons wave, withthe amplitude A ( t ), corresponding to the maximal negative helicity, A ( z, t ) = A ( t ) (cid:0) e x cos kz + e y sin kz (cid:1) or B ( z, t ) = B ( t ) (cid:0) e x cos kz + e y sin kz (cid:1) , where k = 1 /L is the wave number deter-mining the length scale of the magnetic field L and B ( t ) = − kA ( t ). In this situation, Eq. 256 − t , yr × B , G × (a) t − t , yr × B , G × (b) Figure 1: Magnetic field evolution obtained by the numerical solution of Eq. 26 for differentlength scales. (a) L = 10 cm, and (b) L = 10 cm.can be simplified. The equation for the amplitude of the magnetic field B takes the form,˙ B = − kσ cond ( k + Π) B. (26)Since Π in Eq. 24 is negative, the magnetic field, described by Eq. 26, can be unstable.We shall apply Eq. 24 to describe the magnetic field amplification in a dense degeneratematter which can be found in NS. In this situation, n n = 1 . × cm − and n p ≪ n n .Using Eq. 2 for this number density of neutrons, one gets that V = G F n n / √ n e = 9 × cm − , which gives one χ = (3 π n e ) / = 125 MeV [4]. Thus electronsare ultrarelativistic and we can take that ˜ χ ≈ χ . We shall study the magnetic field evolutionin NS in the time interval t < t < t max , where t ∼ yr and t max ∼ yr. In thistime interval, NS cools down from T ∼ K mainly by the neutrino emission [19]. In thissituation, the matter conductivity in Eq. 26 becomes time dependent σ cond ( t ) = σ ( t/t ) / [4],where σ = 2 . × GeV. Here we use the chosen electron density.We shall discuss the amplification of the seed magnetic field B = 10 G, which is typicalfor a young pulsar. In such strong magnetic fields, the anomalous magnetic moment of anelectron was found by Ternov et al. [20] to depend on the magnetic field strength. We canapproximate µ as µ = e m α em π (cid:18) − BB c (cid:19) , (27)where B c = m /e = 4 . × G. Note that Eq. 27 accounts for the change of the sign of µ at B ≈ B c predicted by Ternov et al. [20].The evolution of the magnetic field for the chosen initial conditions is shown in Fig. 1 fordifferent length scales. One can see that, if the magnetic field is enhanced from B = 10 G,it reaches the saturated strength B sat ≈ . × G. Thus, both quenching factors in Eqs. 24and 27 are important. One can see in Fig. 1 that a larger scale magnetic field grows slower.The further enhancement of the magnetic field scale compared to L = 10 cm correspondingto Fig. 1(b) is inexpedient since the growths time would significantly exceed 10 yr. At suchlong evolution times, NS cools down by the photon emission from the stellar surface ratherthan by the neutrino emission [19]. 7he energy source, powering the magnetic field growth shown in Fig. 1, can be the kineticenergy of the stellar rotation. To describe the energy transmission from the rotational motionof matter to the magnetic field one should take into account the advection term ∇ ( v × B ) inthe right hand side of Eq. 25. Here v is the matter velocity. Moreover one should assume thedifferential rotation of NS [21]. For this purpose we should take that NS is not in a superfluidstate. This case is not excluded by the observational data [22]. We have estimated the spindown of NS with the radius R ∼
10 km and the initial rotation period P ∼ − s basing onthe fact that the total energy, I Ω / B V /
2, is constant. Here I is the moment of inertiaof NS, Ω is the angular velocity, and V is the NS volume. For B sat ≈ . × G shown inFig. 1, the relative change of the period is ( P − P ) /P ∼ − . Hence only a small fractionof the initial rotational energy is transmitted to the energy of a growing magnetic field.The obtained results can be used for the explanation of electromagnetic flashes emitted bymagnetars [23]. Beloborodov & Levin [24] suggested that magnetar bursts, happening in thestellar magnetosphere, are triggered by plastic deformations of the magnetar crust driven bya thermoplastic wave (TPW). TPW can be excited by a fluctuation of the internal magneticfield with the length scale of about several meters [25] having the strength B & G [26].As one can see in Fig. 1, these conditions are fulfilled in our case. Therefore the instabilityof the magnetic field predicted in our model can excite TPW which then causes a magnetarburst.
In conclusion we mention that, in the present work, we have considered the generation ofthe electric current of charged fermions, e.g., electrons, flowing along the external magneticfield. This current is nonzero if electrons electroweakly interact with background matter aswell as if the nonzero mass and the nonzero anomalous magnetic moment are accounted for.Unlike the situation of massless fermions, when, owing to CME, J ∼ B is created by thepolarization effects at the lowest energy level [3, 4], in our case, only higher energy levelswith n > ≥ B = 10 G by more than one order of magnitude. The generated magnetic field hasa relatively small scale L ∼ (10 − ) cm. The time for the field growth is (10 − ) yrdepending on the length scale. Finally we have considered the implication of our results forthe explanation of magnetar bursts. Acknowledgments
I am thankful to the organizers of the 52 nd Rencontres de Moriond for the invitation anda financial support, as well as to the Tomsk State University Competitiveness ImprovementProgram and RFBR (research project No. 15-02-00293).8 eferences [1] H.C. Spruit,
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