Magnetic field instability driven by anomalous magnetic moments of massive fermions and electroweak interaction with background matter
aa r X i v : . [ h e p - ph ] D ec Magnetic field instability driven byanomalous magnetic moments of massivefermions and electroweak interaction withbackground matter
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
It is shown that the electric current of massive fermions along the external mag-netic field can be excited in the case when particles possess anomalous magneticmoments and electroweakly interact with background matter. This current is calcu-lated on the basis of the exact solution of the Dirac equation in the external fields. Itis shown that the magnetic field becomes unstable if this current is taken into accountin the Maxwell equations. Considering a particular case of a degenerate electron gas,which can be found in a neutron star, it is revealed that the seed magnetic fieldcan be significantly enhanced. The application of the results to astrophysics is alsodiscussed.
The problem of the magnetic field instability is important, e.g., in the context of theexistence of strong astrophysical magnetic fields [1]. Besides the conventional magneto-hydrodynamics mechanisms for the generation of astrophysical magnetic fields, recentlythe approaches based on the elementary particle physics were proposed. These approachesmainly rely on the chiral magnetic effect (CME) [2], which consists in the appearance ofthe anomalous current of massless charged particles along the external magnetic field B , J CME = e π ( µ R − µ L ) B , (1) ∗ [email protected] e is the particles charge and µ R , L are the chemical potentials of right and left chiralfermions.If J CME is accounted for in the Maxwell equations, a seed magnetic field appears to beunstable and can experience a significant enhancement. The applications of CME for thegeneration of astrophysical and cosmological magnetic fields are reviewed in Ref. [3].However, the existence of CME in astrophysical media is questionable. As found inRefs. [2, 4], J CME can be non-vanishing only if the mass of charged particles, forming thecurrent, is exactly equal to zero, i.e. the chiral symmetry is restored. For the case ofelectrons the restoration of the chiral symmetry is unlikely at reasonable densities whichcan be found in present universe [5]. The chiral symmetry can be unbroken in quark matterowing to the strong interaction effects [6]. The magnetic fields generation in quark matter,which can exist in some compact stars, was discussed in Refs. [7, 8]. Nevertheless this kindof situation looks quite exotic.Therefore the issue of the existence of an electric current J ∼ B for massive particles,which can lead to the magnetic field instability, is quite important for the development ofastrophysical magnetic fields models. One of the example of such a current in electroweakmatter was proposed in Ref. [9]. However, the model developed in Ref. [9] implies theinhomogeneity of background matter. This fact imposes the restriction on the scale of themagnetic 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 back-ground matter along with nonzero anomalous magnetic moments of these fermions. Notethat the electroweak interaction implies the generic parity violation which can provide themagnetic field instability.This work is organized as follows. First, we discuss the Dirac equation for a mas-sive electron with a nonzero anomalous magnetic moment, electroweakly interacting withbackground matter under the influence of an external magnetic field. Using the previouslyobtained solution of this Dirac equation, we calculate the electric current of these electronsalong the magnetic field direction. This current turns out to be nonzero. Then we considera particular situation of a strongly degenerate electron gas, which can be found inside aneutron star (NS). Finally we apply our results for the description of the amplification ofthe magnetic field in NS and briefly discuss the implication of our findings to explain theelectromagnetic radiation of compact stars.Let us consider a fermion (an electron) with the mass m and the anomalous magneticmoment µ . This electron is taken to interact electroweakly with nonmoving and unpo-larized background matter, consisting of neutrons and protons, under the influence of theexternal magnetic field along the z -axis, B = B e z . Accounting for the forward scatteringoff background fermions in the Fermi approximation, the Dirac equation for the electronhas the form, (cid:2) γ µ P µ − m − µB Σ − γ ( V R P R + V L P L ) (cid:3) ψ = 0 , (2)where P µ = i ∂ µ + eA µ , A µ = (0 , , Bx,
0) is the vector potential, e > P R , L = (1 ± γ ) / γ µ = ( γ , γ ),2 = i γ γ γ γ , and Σ = γ γ γ are the Dirac matrices. The effective potentials of theelectroweak interaction V R , L have the form [10], V R = − G F √ n n − n p (1 − ξ )] 2 ξ, V L = − G F √ n n − n p (1 − ξ )] (2 ξ − , (3)where n n,p are the number densities of neutrons and protons, G F = 1 . × − GeV − isthe Fermi constant, and ξ = sin θ W ≈ .
23 is the Weinberg parameter.The solution of Eq. (2) has the form [11], ψ = 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 ) T ,u n ( η ) = (cid:18) eBπ (cid:19) / exp (cid:18) − η (cid:19) H n ( η ) √ n n! , n = 0 , , . . . , (4)where −∞ < p y,z < + ∞ , H n ( η ) are the Hermite polynomials, η = √ eBx + p y / √ eB , and C i , with i = 1 , . . . ,
4, are the spin coefficients. For the definiteness, we will use belowthe chiral representation for the Dirac matrices. It is convenient to normalize the wavefunction 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 ( p z − p ′ z ) δ nn ′ , (5)at any moment of time.The energy levels E for n > E = ¯ V + E , E = q p z + m + 2 eB n + ( µB ) + V + 2 sR ,R = q ( p z V − µBm ) + 2 eB n (cid:2) V + ( µB ) (cid:3) , (6)where s = ± V = ( V L + V R ) /
2, and V = ( V L − V R ) / E = ¯ V + q ( p z + V ) + ( m − µB ) . (7)It should be noted that, at lowest energy level, the electron spin has only one directionsince C = C = 0. In Eqs. (6) and (7), we present the solution only for particles (electrons)rather than for antiparticles (positrons).Using the exact solution of the Dirac equation, we can calculate the electric current ofelectrons in this matter. This current has the form [2], J = − e ∞ X n=0 X s Z + ∞−∞ d p y d p z ¯ ψ γ ψf ( E − χ ) , (8)where f ( E ) = [exp( βE ) + 1] − is the Fermi-Dirac distribution function, β = 1 /T is thereciprocal temperature, and χ is the chemical potential.3irst, we notice that the transverse components of the electric current J x,y ∼ ¯ ψγ , ψ are vanishing because of the orthogonality of Hermite functions with different indexes. Thecontribution of the lowest energy level with n = 0 to the electric current along the magneticfield J z ∼ ¯ ψγ ψ is also vanishing: J (n=0) z = 0. This result is valid for arbitrary parameters m , µ , V , and χ , and B .The contributions of the higher energy levels with n > J z can be obtained usingthe expressions for the spin coefficients C i also found in Ref. [11], 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 ) . (9)The first nonzero term in Eq. (9) is proportional to µB and V , 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) , (10)where E eff = p p + m and m eff = √ m + 2 eB n. The argument of the distributionfunction in Eq. (10) is E eff + ¯ V − χ .Let us consider the case of 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. (10) gives J z = − µmV B e π ˜ χ ∞ X n=1 q ˜ χ − m θ ( ˜ χ − m eff ) , (11)where ˜ χ = χ − ¯ V .One can see that J z in Eq. (11) is nonzero if B < ˜ B , where ˜ B = ( ˜ χ − m ) / e . If themagnetic field is strong enough and is close to ˜ B , then only the first energy level with n = 1contributes to J z , giving one J z = − µmV B α em p ˜ χ − m − eB/π ˜ χ →
0, where α em = e / π ≈ . × − is the fine structure constant. In the opposite situation, when B ≪ ˜ B ,one gets that J z = − α em µmV B ( ˜ χ − m − eB ) / / πe ˜ χ ≈ − α em µmV B/ πe , i.e.the current is proportional to the magnetic field strength.To study the evolution of the magnetic field in the presence of the current in Eq. (11)we return there to the vector notations, J = Π B , Π = − µmV B α em π ˜ χ N X n=1 q ˜ χ − m , (12)where N is maximal integer, for which ˜ χ − m − eBN ≥
0, and take into account thecurrent in Eq. (12) in the Maxwell equations along with the usual ohmic current J = σ cond E ,where σ cond is the matter conductivity and E is the electric field.4onsidering the magnetohydrodynamic approximation, which reads σ cond ≫ ω , where ω is the typical frequency of the electromagnetic fields variation, we derive the modifiedFaraday equation for the magnetic field evolution, ∂ B ∂t = 1 σ cond ∇ × (Π B ) + 1 σ cond ∇ B , (13)where we neglect the coordinate dependence of σ cond .Let us consider the evolution of the magnetic field given by the Chern-Simons wave, cor-responding to the maximal negative magnetic helicity, B ( z, t ) = B ( t ) ( e x cos kz + e y sin kz ),where k = 1 /L is the wave number determining the length scale of the magnetic field L and B ( t ) is the wave amplitude which can depend on time. In this situation we can neglectthe coordinate dependence of Π in Eq. (13) and the equation for B takes the form,˙ B = − kσ cond ( k + Π) B. (14)Since Π in Eq. (12) is negative, the magnetic field described by Eq. (14) can be unstablesince ˙ B > n n = 1 . × cm − and n p ≪ n n .Using Eq. (3), one gets that V = G F n n / √ n e = 9 × cm − ,which gives one χ = (3 π n e ) / = 125 MeV [12]. Thus electrons are ultrarelativistic andwe can take that ˜ χ ≈ χ . We shall study the magnetic field evolution in NS in the timeinterval t < t < t max , where t ∼ yr and t max ∼ yr. In this time interval, NScools down from T ∼ K mainly by the neutrino emission [13]. In this situation, thematter conductivity in Eq. (14) becomes time dependent σ cond ( t ) = σ ( t/t ) / [12], where σ = 2 . × GeV.We shall discuss the amplification of the seed magnetic field B = 10 G, which istypical for a young pulsar. In such strong magnetic fields, the anomalous magnetic momentof an electron was found in Refs. [14, 15] to depend on the magnetic field strength. We canapproximate µ as µ = e m α em π (cid:18) − BB c (cid:19) , (15)where B c = m /e = 4 . × G. Note that Eq. (15) accounts for the change of the signof µ at B ≈ B c predicted in Ref. [14].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 inEqs. (12) and (15) are important. One can see in Fig. 1 that a larger scale magnetic fieldgrows slower. The further enhancement of the magnetic field scale compared to L = 10 cm,shown in Fig. 1(b), is inexpedient since the growths time would significantly exceed 10 yr.5 − t , yr × B , G × (a) t − t , yr × B , G × (b) Figure 1: Magnetic field evolution obtained by the numerical solution of Eq. (14) fordifferent length scales. (a) L = 10 cm, and (b) L = 10 cm.At such long evolution times, NS cools down by the photon emission from the stellar surfacerather than by the neutrino emission [13].The energy source, powering the magnetic field growth shown in Fig. 1, can be thekinetic energy of the stellar rotation. To describe the energy transmission from the rota-tional motion of matter to the magnetic field one should take into account the advectionterm ∇ ( v × B ) in the right hand side of Eq. (13). Here v is the matter velocity.Moreover one should assume the differential rotation of NS [16]. The NS spin-downbecause of the magnetic field enhancement can be estimated basing on the conservation ofthe total energy of a star: I Ω / B V / I is the moment of inertia ofNS, Ω is the angular velocity, and V is the NS volume.Taking the NS radius R ∼
10 km and the initial rotation period P ∼ − s, we get for B sat ≈ . × G, shown in Fig. 1, that the relative change of the period is ( P − P ) /P ∼ − . Hence only a small fraction of the initial rotational energy is transmitted to theenergy of a growing magnetic field.The obtained results can be used for the explanation of electromagnetic flashes emittedby magnetars within the recently proposed model of a thermoplastic wave [17], which canbe excited by small-scale, with L ∼ (10 − ) cm, fluctuations of the magnetic field havingthe strength B & G [8]. The evolution of fields with such characteristics is shown inFig. 1.In conclusion it is interesting to compare the appearance of the new current along themagnetic field in Eq. (10) with CME [2], which is known to get the contribution onlyfrom massless electrons at the zero Landau level in an external magnetic field. Since leftelectrons move along the magnetic field and right particles move in the opposite direction,the current in Eq. (1) is nonzero until there are different populations of the zero Landaulevel by left and right particles, i.e. µ R = µ L . Electrons at higher Landau levels can movearbitrarily with respect to the magnetic field. Therefore, CME is caused by an asymmetricmotion of massless particles along the external magnetic field.If we consider massive electrons with nonzero anomalous magnetic moments, elec-6roweakly interacting with background matter, then, unlike CME, the motion of suchparticles at the lowest energy level with n = 0 is symmetric with respect to the magneticfield, i.e. −∞ < p z < + ∞ for them (one can see it if we replace p z → p z − V in Eq. (7)).On the contrary, higher energy levels with n > p z → − p z . The reflectional symmetry cannot be restored byany replacement of p z . Therefore electrons having p z > p z < v z = p z / E . Thus J z ∼ h v z i 6 = 0, with only higherenergy levels contributing to it. It is interesting to mention that the term in Eq. (6), whichviolates the reflectional symmetry p z → − p z , is proportional to µBmV . It is this factorwhich J z in Eq. (10) is proportional to. Thus, one can see that the nonzero current J ∼ B results from the asymmetric motion of particles along B . This asymmetry is caused bythe simultaneous presence of three factors: nonzero m and µ , as well as the electroweakinteraction with background matter ∼ V .It should be noted that, in addition to the electroweak interaction between electronsand background fermions, taken into account in Eqs. (2) and (3), and leading to an electriccurrent J ∼ B in Eq. (12), electrons also interact electromagnetically with background pro-tons and neutrons. Considering, for definiteness, the electromagnetic interaction betweenelectrons and a homogeneous gas of non-moving protons with a constant density, we findthat the following additional term appears in the left hand side of Eq. (2): ∼ [ · · · + e γ f ] ψ ,where f ∼ n p /ω p is a quantity proportional to the zero component of the proton currentand ω p is the plasma frequency in the considered matter.The strength of the electromagnetic interaction is much higher than that of the elec-troweak interaction, e f ≫ G F n p , since, in the degenerate matter, one has ω p ∼ α em χ [18]and e f ∼ GeV − × n p for χ ∼ MeV (see above). Nevertheless, the additional con-tribution of the electromagnetic interaction in Eq. (2) can be eliminated by the gaugetransformation ψ → ψ ′ = exp( − ie f t ) ψ in case of matter with constant density, f ∼ n p = const. The contribution of the electroweak interaction in Eq. (2), γ ( V R P R + V L P L ) → γ γ V can not be eliminated by any gauge transformation because of the presence of thematrix γ indicating the parity violation in electroweak interactions. Similarly, it can beshown that the electromagnetic interaction between electrons and neutrons, due to thepresence of the magnetic form factor of neutrons [19], does not give rise to a current J ∼ B in the case of a homogeneous, unpolarized neutron matter with a constant density.The necessity of the presence of the contribution of a parity violating interaction in thegeneration of the current J = Π B in Eq. (12) in the system of massive fermions follows alsofrom the fact that the parameter Π should be a pseudoscalar. Electromagnetic interactionis known to be parity conserving. That is why it does not contribute to Π in the systemin question. Acknowledgements
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