New magnetic field instability and magnetar bursts
We consider the system of massive electrons, possessing nonzero anomalous magnetic moments, which electroweakly interact with background neutrons under the influence of an external magnetic field. The Dirac equation for such electrons is exactly solved. Basing on the obtained solution, we find that a nonzero electric current of these electrons can flow 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 we consider a particular case of a degenerate electron gas, which may well exist in a neutron star, and show that a seed magnetic field can be amplified by more than one order of magnitude. Finally we discuss the application of our results for the explanation of the electromagnetic radiation emitted by magnetars.
The problem of the magnetic field instability is important, e.g., in the context of the existence of strong astrophysical magnetic fields . Besides the magnetohydrodynamics mechanisms for the generation of astrophysical magnetic fields, recently the approaches based on the elementary particle physics were proposed. These approaches mainly rely on the chiral magnetic effect (CME) , which consists in the generation of the anomalous current of massless charged particles along the magnetic field , where is the fine structure constant and are the chemical potentials of right and left chiral fermions. If is accounted for in the Maxwell equations, the magnetic field appears to be unstable and can experience a significant enhancement. The model for the generation of strong magnetic fields in the dense matter of a neutron star (NS) driven by CME under the influence of the electroweak interaction between electrons and neutrons was developed in a series of our works [3, 4, 5, 6, 7]. Other applications of CME for the generation of astrophysical and cosmological magnetic fields are reviewed by Kharzeev .
However, the existence of CME in astrophysical media is questionable. Vilenkin  and Dvornikov  found that 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 case of electrons the restoration of the chiral symmetry is unlikely at reasonable densities which can be found in astrophysics . The chiral symmetry can be unbroken in quark matter owing to the strong interaction effects . The magnetic fields generation in quark matter, which can exist in some compact stars, was discussed by Dvornikov [13, 14]. Nevertheless this kind of situation looks quite exotic.
Therefore the issue of the existence of an electric current for massive particles, which can lead to the magnetic field instability, is quite important for the explanation of astrophysical magnetic fields. One of the examples of such a current in electroweak matter was proposed by Semikoz & Sokoloff . However, the model developed by Semikoz & Sokoloff  implies the inhomogeneity of background matter. This fact imposes the restriction on the scale of the magnetic field generated.
In the present work, we discuss another scenario for the magnetic field instability. It involves the consideration of the electroweak interaction of massive fermions with background matter along with nonzero anomalous magnetic moments of these particles. Note that the electroweak interaction implies the generic parity violation which can provide the magnetic field instability. Recently, the interpretation of CME in terms of an effective magnetic moment was considered by Kharzeev et al. .
This work is organized as follows. In Sec. 2, we discuss the Dirac equation for a massive electron with a nonzero anomalous magnetic moment, electroweakly interacting with background matter under the influence of an external magnetic field. Then, we describe the main steps in finding the exact solution of this Dirac equation which was previously obtained by Balantsev et al. . Using this solution, in Sec. 3, we calculate the electric current of these electrons along the magnetic field direction. This current turns out to be nonzero. Then we consider a particular situation of a strongly degenerate electron gas, which can be found inside NS. Finally, in Sec. 4, we apply our results for the description of the amplification of the magnetic field in NS and briefly discuss the implication of our findings for the explanation the electromagnetic radiation of compact stars.
2 Solution of the Dirac equation
Let us consider an electron with the mass and the anomalous magnetic moment . This electron is taken to interact electroweakly with nonmoving and unpolarized background matter consisting of neutrons and protons under the influence of the external magnetic field along the -axis, . Accounting for the forward scattering off background fermions in the Fermi approximation, the Dirac equation for the electron has the form,
where , , and are the Dirac matrices, , is the vector potential, and is the absolute value of the elementary charge. The effective potentials of the electroweak interaction have the form ,
where are the number densities of neutrons and protons, is the Fermi constant, and is the Weinberg parameter.
are the Hermite functions, are the Hermite polynomials, , are the spin coefficients, , and .
where is the unit matrix and are the Pauli matrices.
where is the discrete spin quantum number.
To determine at we notice that the spin operator 
commutes with the Hamiltonian of Eq. 1. Basing on the fact that the wave function in Eq. 3 is the eigenfunction of the operator , it is convenient to represent in terms of the new auxiliary coefficients and as
The coefficient can be found if we normalize the wave function as
In this situation, the spin coefficients obey the relation,
at any . Finally, we get that
At , both the energy spectrum and the spin coefficients can be found directly from Eq. 2 since, in this case, the electron spin has only one direction and hence . Thus, the energy spectrum reads
Using Eq. 12, we obtain that the nonzero spin coefficients have the form,
Note that, while solving Eq. 1, we take into account only electron rather than positrons degrees of freedom.
3 Calculation of the electric current
Using the exact solution of the Dirac equation, which is found above, we can calculate the electric current of electrons in this matter. This current has the form ,
where is the Fermi-Dirac distribution function, is the reciprocal temperature, and is the chemical potential. First, we notice that because of the orthogonality of Hermite functions with different indexes. Hence, only should be considered. Then, using Eq. 2, we can derive the identity
which is valid in the chiral representation of Dirac matrices in Eq. 6.
To find the first nonzero term in Eq. 18 we decompose in a series in and . Finally, we get that
where and . The argument of the distribution function in Eq. 19 is .
As an example, we shall consider a strongly degenerate electron gas. In this situation, , where is the Heaviside step function. We can also disregard the positrons contribution to . The direct calculation of the current in Eq. 19 gives
where . One can see that in Eq. 20 is nonzero if , where . If the magnetic field is relatively strong and is close to , then only the first energy level with contributes to , giving one
where . In the opposite situation, when , one gets that
i.e. the current is proportional to the magnetic field strength.
where we take into account the expression for in Eq. 14. Eq. 3 means that the lowest energy level with does not contribute to the electric current along the magnetic field. This result extends our recent finding  to the situation when the anomalous magnetic moment is accounted for. Analogously to Eq. 3, one can show that positrons do not contribute to the current either. Note that the result is Eq. 3 is valid for arbitrary characteristics of plasma, external fields, as well as the mass and the magnetic moment of an electron.
It is interesting to compare the appearance of the new current along the magnetic field in Eq. 19 with CME [2, 8]. Vilenkin  showed that only massless electrons at the zero Landau level in an external magnetic field contribute to the generation of the anomalous current along the magnetic field. This feature remains valid in the presence of the background electroweak matter [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 opposite direction [3, 4]. Electrons at higher Landau levels can move arbitrarily with respect to the magnetic field. Therefore, if one has a different population of left and right electrons at the lowest Landau level, there is a nonzero current is the system , which is the manifestation of CME.
In the situation described in the present work, i.e. when massive electrons with nonzero anomalous magnetic moment move in the electroweak matter, the particles at the lowest energy level can move in any direction with respect to the magnetic field, i.e. . Moreover, there is no asymmetry for electrons with and . It results from Eq. 14 if we replace there. On the contrary, higher energy levels with in Eq. 2 are not symmetric with respect to the transformation . The reflectional symmetry cannot be restored by any replacement of . Therefore electrons having and will have different energies and hence different velocities . The electric current along the magnetic field is proportional to . Thus such a current should be nonzero, with only higher energy levels contributing to it. It is interesting to mention that the term in Eq. 2 which violates the reflectional symmetry is proportional to . It is this factor which in Eq. 19 is proportional to.
4 Instability of the magnetic field in NS
Returning to the vector notations we get the current in Eq. 20 takes the form,
where is maximal integer, for which . To study the magnetic field evolution in the presence of the additional current in Eq. 24 we take this current into account in the Maxwell equations along with the usual ohmic current , where is the matter conductivity and is the electric field. Considering the magnetohydrodynamic approximation, which reads , where is the typical frequency of the electromagnetic fields variation, we derive the modified Faraday equation for the magnetic field evolution,
where we neglect the coordinate dependence of .
Let us consider the evolution of the magnetic field given by the Chern-Simons wave, with the amplitude , corresponding to the maximal negative helicity, or , where is the wave number determining the length scale of the magnetic field and . In this situation, Eq. 4 can be simplified. The equation for the amplitude of the magnetic field takes the form,
We shall apply Eq. 24 to describe the magnetic field amplification in a dense degenerate matter which can be found in NS. In this situation, and . Using Eq. 2 for this number density of neutrons, one gets that . The number density of electrons can reach several percent of the nucleon density in NS. We shall take that , which gives one . Thus electrons are ultrarelativistic and we can take that . We shall study the magnetic field evolution in NS in the time interval , where and . In this time interval, NS cools down from mainly by the neutrino emission . In this situation, the matter conductivity in Eq. 26 becomes time dependent , where . Here we use the chosen electron density.
We shall discuss the amplification of the seed magnetic field , which is typical for a young pulsar. In such strong magnetic fields, the anomalous magnetic moment of an electron was found by Ternov et al.  to depend on the magnetic field strength. We can approximate as
The evolution of the magnetic field for the chosen initial conditions is shown in Fig. 1 for different length scales. One can see that, if the magnetic field is enhanced from , it reaches the saturated strength . Thus, both quenching factors in Eqs. 24 and 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 corresponding to Fig. 1 is inexpedient since the growths time would significantly exceed . At such long evolution times, NS cools down by the photon emission from the stellar surface rather than by the neutrino emission .
The energy source, powering the magnetic field growth shown in Fig. 1, can be the kinetic energy of the stellar rotation. To describe the energy transmission from the rotational motion of matter to the magnetic field one should take into account the advection term in the right hand side of Eq. 4. Here is the matter velocity. Moreover one should assume the differential rotation of NS . For this purpose we should take that NS is not in a superfluid state. This case is not excluded by the observational data . We have estimated the spin down of NS with the radius and the initial rotation period basing on the fact that the total energy, , is constant. Here is the moment of inertia of NS, is the angular velocity, and is the NS volume. For shown in Fig. 1, the relative change of the period is . Hence only a small fraction of 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 by magnetars . Beloborodov & Levin  suggested that magnetar bursts, happening in the stellar magnetosphere, are triggered by plastic deformations of the magnetar crust driven by a thermoplastic wave (TPW). TPW can be excited by a fluctuation of the internal magnetic field with the length scale of about several meters  having the strength . As one can see in Fig. 1, these conditions are fulfilled in our case. Therefore the instability of the magnetic field predicted in our model can excite TPW which then causes a magnetar burst.
In conclusion we mention that, in the present work, we have considered the generation of the electric current of charged fermions, e.g., electrons, flowing along the external magnetic field. This current is nonzero if electrons electroweakly interact with background matter as well as if the nonzero mass and the nonzero anomalous magnetic moment are accounted for. Unlike the situation of massless fermions, when, owing to CME, is created by the polarization effects at the lowest energy level [3, 4], in our case, only higher energy levels with contribute to the current. We also mention that the role of a nonzero anomalous magnetic moment is crucial since, as found by Dvornikov , the current of massive charged particles electroweakly interacting with background matter is vanishing at any .
We have revealed that a magnetic field turns out to be unstable if the new current in Eq. 19 is taken into account. As an example of the obtained results, we have discussed the enhancement of the magnetic field in a dense degenerate matter. Using the background matter with characteristics typical to NS, we have obtained the amplification of the seed field by more than one order of magnitude. The generated magnetic field has a relatively small scale . The time for the field growth is depending on the length scale. Finally we have considered the implication of our results for the explanation of magnetar bursts.
I am thankful to the organizers of the 52 Rencontres de Moriond for the invitation and a financial support, as well as to the Tomsk State University Competitiveness Improvement Program and RFBR (research project No. 15-02-00293).
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