Galvano-rotational effect induced by electroweak interactions in pulsars

We study electroweakly interacting particles in rotating matter. The existence of the electric current along the axis of the matter rotation is predicted in this system. This new galvano-rotational effect is caused by the parity violating interaction between massless charged particles in the rotating matter. We start with the exact solution of the Dirac equation for a fermion involved in the electroweak interaction in the rotating frame. This equation includes the noninertial effects. Then, using the obtained solution, we derive the induced electric current which turns out to flow along the rotation axis. We study the possibility of the appearance of the galvano-rotational effect in dense matter of compact astrophysical objects. The particular example of neutron and hypothetical quark stars is discussed. It is shown that, using this effect, one can expect the generation of toroidal magnetic fields comparable with poloidal ones in old millisecond pulsars. We also briefly discuss the generation of the magnetic helicity in these stars. Finally we analyze the possibility to apply the galvano-rotational effect for the description of the asymmetric neutrino emission from a neutron star to explain pulsars kicks.


Introduction
The importance of noninertial effects for various areas in modern physics cannot be underestimated. Some of the examples of these effects are mentioned in ref. [1]. One of the most common manifestations of noninertial effects is the description of physical processes in a rotating frame. More interesting phenomena can take place if there is a parity violating interaction between rotating particles. In the present work we will show that, in such a system, one can expect the appearance of the electric current flowing along the rotation axis. We shall call this phenomenon as the new galvano-rotational effect (GRE).
Previously the generation of an electric current due to nontrivial topological effects was studied mainly in connection to the chiral magnetic effect [2]. In this case a nonzero current can be induced in the system of massless particles embedded in an external magnetic field [3], provided there is an imbalance between left and right particles. Recently, in ref. [4], we showed that the electric current can be generated even at zero chiral imbalance if charged particles are involved in the parity violating interaction.
When applied to the astrophysical medium inside a compact star, GRE appears to generate a toroidal magnetic field (TMF) there. Although stellar TMFs cannot be observed directly, they are an internal ingredient of various astrophysical media. For example, the most plausible explanation of the 22 yr solar cycle is the oscillation between poloidal and toroidal components of the solar magnetic field [5]. Moreover, purely poloidal or toroidal stellar magnetic fields were shown in refs. [6,7] to be unstable. Thus TMF is inherent to a magnetized star. In this work we show how TMF can be generated in an old millisecond pulsar basing on GRE.
This paper is organized in the following way. First, in section 2, we briefly describe the Standard Model interaction between leptons and quarks in flat space-time. Then, in section 3, we derive the Dirac equation for a fermion which interacts electroweakly with a rotating background matter. For this purpose we write down this Dirac equation in the corotating frame, using the method of an effective curved space-time. The exact solution of the Dirac equation for an ultrarelativistic fermion, accounting for the noninertial effects, is obtained in section 3. In section 4, we establish GRE, which, in this situation, consists in the appearance of the electric current along the rotation axis. We calculate this current in section 4 using the exact solution of the Dirac equation obtained in section 3. Finally, in section 5, we apply GRE for the generation of TMF and the magnetic helicity in compact rapidly rotating stars. In section 6, we summarize our results and compare them with the findings of other authors.

Electroweak interaction of fermions in flat space-time
In this section we shall briefly remind the description of the electroweak interaction between leptons and quarks in flat space-time.
Let us consider a medium consisting of electrons as well as u and d quarks. We shall assume that quarks are both in confined states, forming nucleons, and hypothetical free particles. The effective Lagrangian for the electroweak interaction in this system in the Fermi approximation has form [8], where G F ≈ 1.17 × 10 −5 GeV −2 is the Fermi constant, J µ is the charged current, and K µ is the neutral current. In the considered system of elementary particles, the charged current has the form, where ψ u,d are the wave functions of u and d quarks, γ L µ = γ µ 1 − γ 5 /2, γ µ = γ 0 , γ are the Dirac matrices, γ 5 = iγ 0 γ 1 γ 2 γ 3 , and V ud ≈ 0.97 is the element of the Cabbibo-Kobayashi-Maskawa matrix. The neutral current can be expressed as where ψ e is the wave function of an electron, γ R µ = γ µ 1 + γ 5 /2, and Here ξ = sin 2 θ W ≈ 0.23 and θ W is the Weinberg angle.
In this section we shall consider the case when background fermions are at rest and unpolarized. While averaging the currents in eqs. (2.2) and (2.3) over the Fermi-Dirac distributions . . . , in this case we get that only ψ † f ψ f = n f = 0, where n f is the invariant number density of these fermions. The quantities ψ f γψ f = 0 and ψ f γ µ γ 5 ψ f = 0 are vanishing since they are proportional to the macroscopic velocity and the polarization.

Test particle
Background particles Table 1. The values of the effective potentials V L,R in eq. (2.5) for various channels of the scattering of a test fermion off background particles. Here n n,p are the densities of neutrons and protons, and n u,d are the densities of u and d quarks.
After averaging over the ensemble of background particles, we can rewrite eq. (2.1) in the form, where ψ is the wave function of a test fermion which undergoes a scattering off background particles and the effective potentials V L,R are given in table 1 for any scattering channels. When we consider the electron scattering off nucleons, u and d quarks are confined inside neutrons or protons. In this case only the neutral current contributes to eq. (2.5). If we study the scattering u quarks off d quarks, and vice versa, both the charged and the neutral currents give the contributions to eq. (2.5). To derive the expression for V L for (ud) and (du) interactions on the basis of eq. (2.2), we use the Fierz transformation.

Dirac equation for a fermion interacting with rotating matter
In this section we shall find the exact solution of the Dirac equation for a fermion interacting with a rotating matter by means of the electroweak forces. The obtained solution includes noninertial effects. Let us discuss the interaction of a fermion with matter rotating with the constant angular velocity ω. In this case we cannot directly apply the results of section 2 taking ψ f γψ f ∼ v f = 0, where v f = (ω×r) is the fermions velocity, while averaging over the ensemble of background particles. In the situation, when matter moves with an acceleration, one should account for possible noninertial effects. Nevertheless we can still choose a noninertial reference frame where matter is at rest. Assuming that matter is unpolarized, we get that only ψ † f ψ f = 0 in this reference frame. Thus, formally we can use the effective potentials derived in section 2. For the first time this approach was put forward in ref. [9], where the neutrino interaction with a rotating matter was studied.
It is known that the description of a particle in an accelerated frame is analogous to the motion of this particle in the curved space-time or the interaction with an effective gravitational field. For example, when we study the motion in the rotating frame, the interval takes the form [10], where g µν is the metric tensor of the effective gravitational field. Here we use the cylindrical coordinates x µ = (t, r, φ, z). Using eq. (2.5), we get that the Dirac equation for a test fermion, with the mass m, involved in the parity violating interaction and moving in a curved space-time, has the form (see also ref. [9]), where γ µ (x) are the coordinate dependent Dirac matrices, and V µ L,R are the effective potentials. Note that, since we choose a corotating frame, then V 0 L,R ≡ V L,R = 0 and V i L,R = 0, where V L,R are given in table 1. Analogous Dirac equation was also discussed in ref. [11].
One can check that, using the following vierbein vectors: Let us introduce the constant Dirac matrices in a locally Minkowskian frame by γā = e a µ γ µ (x). As shown in ref. [9], γ 5 (x) = iγ0γ1γ2γ3 = γ5 does not depend on coordinates. The spin connection in the Dirac eq. (3.2) has the form [12], where V V,A = (V L ± V R ) /2 are the vector and axial parts of the effective potentials. Note that the operator D in eq. (3.6) is analogous to that recently studied in ref. [13]. Since eq. (3.6) does not explicitly depend on t, φ, and z, we shall look for its solution in the form, where ψ r = ψ r (r) is the wave function depending on the radial coordinate. The values of J z in eq. (3.7) were found in ref. [14] to be ±1/2, ±3/2, . . . . It is convenient to rewrite eq. (3.6) as is the potential of the effective electromagnetic field, and V = V A γ0 − ωrγ2 γ5.
Let us look for the solution of eq. (3.8) in the form, The solution of eq. (3.10) can be found for ultrarelativistic particles. In the limit m → 0, we can represent Φ = vϕ in eq. (3.10), where ϕ = ϕ(r) is a scalar function and v is a constant spinor satisfying Σ 3 v = σv and γ5v = χv, with σ = ±1 and χ = ±1, since both Σ 3 and γ5 now commute with the operator of eq. (3.10).
Let us first study left particles, (1 + γ5)ψ = 0, corresponding to χ = +1. The case of right particles with χ = −1 can be studied analogously. Using the new variable ρ = |V L |ωr 2 we can write the equation for ϕ σ as Assuming that ϕ σ → 0 at r → ∞, we get that ϕ + = I N,s and ϕ − = I N −1,s , for V L > 0, as well as ϕ + = I N −1,s and ϕ − = I N,s , for V L < 0. Here N = 0, 1, 2 . . . , s = N − l, and I N,s = I N,s (ρ) is the Laguerre function 1 . The energy spectrum can be found if we take that κ = N in eq. (3.12). We can present E is the form, One can see that the neutrino energy depends on the sign of V L . Note that one should understand m = 0 in eq. (3.13) in the perturbative sense. The total radial wave function ψ r can be found in the explicit form if we choose the In eq. (3.14) we assume that the Dirac matrices are in the chiral representation [15], where σ k are the Pauli matrices. Using eq. (3.14), we get the radial wave functions corresponding to different spin projections, and for V L > 0, and for V L < 0. The signs in eqs. (3.19) and (3.20) are correlated with the sign in eq. (3.13).
It should be noted that the spinors η + and η − in eqs. (3.19) and (3.20) are linearly dependent as it should be for ultrarelativistic particles. Thus, one can use any independent pair of η + and η − . As usual, we shall attribute η − with the upper sign to a particle degree of freedom and η + with the lower sign to antiparticles. This choice of independent spinors is convenient for N > 0. If N = 0, one can better use η + for V L > 0 and η − for V L < 0 as independent degrees of freedom.
Using the normalization condition for the total wave function, which includes the dependence on p z and φ, we get the coefficients C σ in eqs. (3.19) and (3.20) as which is valid for any sign of V L . On the basis of eqs. (3.19) and (3.20) one can notice that, at N = 0, p z is correlated with the particle helicity. Using eq. (3.13) with m = 0 as well as eqs. (3.19) and (3.20), one can find the possible values of p z at N = 0. For the convenience, they are listed in table 2. Note that at N > 0, −∞ < p z < +∞.
It should be noted that p z in table 2 is a formal quantum number. The physical value of p z for antiparticles is opposite to that shown in table 2: p (phys.) z = −p z . Otherwise the electric charge would not be conserved.
Right particles can be treated in the same way as left ones. That is why we just present only the final results. The expression for the energy has the same structure as eq. (3.13) with the replacement V L → V R . In eq. (3.16) one has ψ ± r = (ξ ± , 0) T , where for V R > 0, and for V R < 0. The argument of the Laguerre functions is ρ = |V R |ωr 2 now. The new normalization constant in eq. (3.23) and (3.24) reads .

Induced electric current along the rotation axis
In this section we show that there is a nonzero induced electric current flowing along the rotation axis in the system of electroweakly interacting particles. In our calculation we shall use the exact solution of the Dirac equation obtained in section 3.
In section 3 we already mentioned that there is a correlation between p z and the helicity at N = 0. Thus one expects that there can be macroscopic fluxes of particles in the rotating matter. Let us first examine this issue for left fermions. We shall calculate the mean hydrodynamic currents of particles and antiparticles with respect to the coordinates x µ in the rotating frame. These currents have the form, where ρ L f,f (E) = {exp(β[E ∓ µ L ]) + 1} −1 is the Fermi-Dirac distribution for fermions, with the lower sign staying for antifermions, β = 1/T is the reciprocal temperature of the fermion gas, and µ L is the chemical potential of left particles. The spinor in eq. (4.1) corresponds to the exact solution of the Dirac equation in eqs. (3.19) and (3.20). Note that, for the first time, this method for the calculation of the current was proposed in ref. [3].
We will be interested in the expression for j µ Lf linear in ω. That is why we use E + V L instead of the total particle energy, cf. eq. (3.13), in the distribution function. We should study only j 3 Lf since it is this component of the current that is linear in ω. Using the orthogonality of Laguerre functions, where the upper sign stays for particles and the lower one for antiparticles. On the basis of eq. (4.3) and table 2 we find that only the lowest level N = 0 contributes to the current. Finally we get that Analogously to eq. (4.4) we can obtain the following expression: which is valid for antifermions. The contribution to the hydrodynamic current from right femions is analogous to eqs. (4.4) and (4.5). It is for particles, and for antiparticles. Here ρ R f,f (E) are the distributions of right particles and antiparticles which can be obtained from ρ L f,f (E) by replacing µ L → µ R , where µ R is the chemical potential of right fermions. Now, using eqs. (4.4)-(4.7), we can obtain the expression for the third component of the electric current as Rf , where q f is the electric charge of the fermion f including the sign. For example, q e = −e for an electron, q u = 2e/3 for an u quark, and q d = −e/3 for a d quark. Here e > 0 is the absolute value of the elementary electric charge. In the expression for J 3 , we use the convention that the direction of the electric current coincides with the motion of the positive electric charge. Finally we get for the electric current, where we restore vector notations. We remind that in eq. (4.8) we keep only the terms linear in ω and V L,R . We can attribute the existence of the induced electric current in rotating matter, where the parity violating interaction is present, to the new GRE; cf. section 1.

Generation of TMF and magnetic helicity in a pulsar
In this section we apply GRE, for the calculation of TMF inside a pulsar. We also briefly consider the generation of the magnetic helicity in a compact rotating star. If we consider a rapidly rotating compact astrophysical object, like a neutron star (NS) or even a hypothetical quark star (QS), then the mechanism described in section 4 will induce the electric current along the rotation axis of such a star. We shall suppose that this current forms a closed circuit connected somewhere at the stellar surface. Thus, using the Maxwell equation (∇ × B) = J, we get that this current should induce a TMF, B tor ∼ RJ, where R ∼ 10 km is the stellar radius.
It should be noted that a compact star typically has a poloidal magnetic field B pol , which is measured in astronomical observations. For instance, the radiation of a pulsar can be explained by the emission of electromagnetic waves by the rotating magnetic dipole associated with B pol , provided there is a nonzero angle between B pol and ω. However, as shown in ref. [16], using general arguments for the magnetohydrodynamic equilibrium of an axisymmetric NS, a purely poloidal magnetic field configuration turns out to be unstable.
Thus an internal nonzero TMF should exist in a compact star.
Let us first consider the generation of TMF in NS composed of degenerate electrons and nucleons, like neutrons and protons. In this case u and d quarks are confined inside nucleons. The typical electron density in NS is n e ≈ 9 × 10 36 cm −3 , which corresponds to the electron fraction Y e ≈ 0.05. It gives the chemical potential of electrons µ e ≈ 125 MeV ≫ m e . Therefore electrons are ultrarelativistic whereas neutrons and protons are nonrelativistic. Note that the nonzero electron mass was shown in ref. [17] to slightly contribute to the electric current in eq. (4.8). Thus we can assume that electrons are approximately massless in NS and the results of sections 3 and 4 are valid.
Since the chiral symmetry is unbroken, we can consider left and right chiral projections as independent degrees of freedom. For simplicity we shall take that left and right electrons are in equilibrium with µ L ∼ µ R ∼ µ e . The situations, when the chiral imbalance ∆µ = µ R − µ L = 0 is important in NS, are studied, e.g., in refs. [4,17]. Since n p ≪ n n in NS, (V L − V R ) = G F n n / √ 2 ≈ 12 eV for n n ≈ 1.8 × 10 38 cm −3 . Eventually, taking that ω ∼ 10 3 s −1 and R ∼ 10 km as well as using eq. (4.8), we get that B tor ≈ 2.5 × 10 8 G can be generated in a rotating NS.
The obtained value of B tor is comparable with B pol (10 8 − 10 9 ) G in weakly magnetized old millisecond pulsars [18]. It should be noted that the stability of magnetic fields in NS can be reached if 0 ≤ E tor /E mag < 0.07 [16], where E tor ∼ B 2 tor is the energy of TMF and E mag ∼ (B 2 tor + B 2 pol ) is the total magnetic energy. Our estimate for B tor satisfies this criterion.
Let us discuss the creation of TMF in a hypothetical QS. Although QSs have not been observed yet, their properties are actively studied theoretically [19]. Various models of QS predict that it consists of free u and d quarks with some admixture of s quarks. After the analysis of various equations of state of QS matter, the strangeness fraction was found in ref. [20] not to exceed ∼ 0.3. Thus we can approximately omit the contribution of s quarks in the calculation of the induced current.
Finally, using, in eq. (4.8), the adopted values of densities and chemical potentials of quarks, the values of V L,R in table 1 as well as for ω ∼ 10 3 s −1 and R ∼ 10 km, we get that B tor ≈ 7.2 × 10 8 G can exist inside a rotating QS. The obtained value of TMF is slightly greater than B tor for NS. Note that the derived strength of TMF is also in agreement with stability criterion obtained in ref. [16] for old weakly magnetized millisecond pulsars [18]. It should be noted that, in our estimate of TMF in QS, we account for only (ud) and (du) contributions to the electric current. Using the analogy of the rotating electroweak matter with the presence of an effective magnetic field [22] (see eq. (3.8) in section 3) and the results of ref. [23], we get that (uu) and (dd) interactions do not contribute to the current in eq. (4.8).
We should mention that, in the generation of TMF in a compact star, we discuss a thermally relaxed stage in the evolution of this astrophysical object. This approximation is valid since we consider old millisecond pulsars with ages ∼ (10 8 − 10 10 ) yr [18]. It means that we discard any possible effects related to turbulence which should be treated on the basis of the Navier-Stokes equation. In our analysis we also do not consider a differential rotation either.
The creation of TMF in a compact star is closely related to the problem of the generation of the magnetic helicity defined as where A is the 3D vector potential. If there is configuration of magnetic fields in a star consisting of toroidal and poloidal fields, then H in eq. (5.1) has the form [24], H = 2LΦ tor Φ pol , where Φ tor and Φ pol are the fluxes of toroidal and poloidal fields and |L| = 1 is the linkage number. The magnetic helicity is a conserved quantity in a perfectly conducting medium. It is this fact which provides the stability of a poloidal field in a compact star. Note that another mechanism for the generation of the magnetic helicity in a nonrotating NS, based on the electron-nucleon electroweak interaction, was recently proposed in ref. [4].

Conclusion
In conclusion we note that in the present work we have studied the evolution of particles, involved in the parity violating electroweak interaction, in the rotating matter. In section 3, we have obtained the the new exact solution of the Dirac equation for a test ultrarelativistic particle, which account for the noninertial effects. Then, in section 4, we have computed the induced electric current along the rotation axis on the basis of the exact solution of the Dirac equation. Finally, in section 5, we applied our results for the generation of TMF and the magnetic helicity in compact rotating stars. Several new results have been obtained in this work. Firstly, we mention that the vierbein vectors in eq. (3.3) have never been used previously in the Dirac eq. (3.2), which accounts for the electroweak interaction with background matter in curved space-time. Another veirbein was recently used in ref. [9]. However, the choice of the vierbein in the present work is likely to be more appropriate for ultrarelativistic particles in a rotating frame. In particular, here we have obtained the correct form of the "centrifugal" energy, or the energy of the rotation-angular momentum coupling, E cf = −(J · ω); cf. eq. (3.13). The obtained expression for E cf coincides with the result of ref. [25] derived on the basis of the general analysis. The form of E cf obtained in ref. [9], where another vierbein was used, is slightly different. Therefore, the vierbein adopted in ref. [9] is likely to be more appropriate for the description of nonrelativistic particles in a rotating frame; cf. ref. [26].
Secondly, we have predicted the new GRE. This effect consists in the appearance of the electric current in the rotating matter composed of massless particles involved in the parity violating electroweak interaction. This electric current flows along the rotation axis. The new GRE is analogous to the chiral magnetic effect, known in QED, which consists in the generation of the electric current of massless charged particles along the external magnetic field [2,3]. It should be noted that, for the first time, the analogy between the motion in a rotating electroweak matter and in an external magnetic field was mentioned in ref. [22].
Note that the appearance of the electric tension in a rotating conductor owing to the noninertial effects was also discussed in ref. [27]. The electric tension, predicted in ref. [27], is induced mainly by the Coriolis force acting on charged particles in a rotating conductor. If ω is chosen along the z-axis, this tension is found in ref. [27] to be along the azimuthal direction. In our case, the electric current is owing to both the matter rotation and the presence of the parity violating electroweak interaction. We predict that the induced electric current flows along the rotation axis.
We have used the solution of a Dirac equation in the rotating frame for the calculation of the induced electric current. It means that this current flows inside the rotating matter since the quantum states of charged particles are measured by a corotating observer. For example, if one used the wave functions obtained in ref. [28], although they look similar to those in eqs. (3.19) and (3.20), we would get the electric current with respect the a nonrotating observer, which cannot be applied for the generation of the internal TMF.
Finally, we have used the calculated electric current to generate TMF and the magnetic helicity inside a rotating compact star, like NS or QS. The strength of TMF generated turned out to be moderate, B tor 10 8 G, for both NS and QS. However, such TMF is comparable with a poloidal field in weakly magnetized old millisecond pulsars [18]. Note that the obtained strength of TMF is in agreement with a criterion for the magnetic field stability derived in ref. [16]. It should be noted that our model for the generation of TMF does not require the existence of a significant chiral imbalance between left and right charged particles. Such an imbalance is essential if the chiral magnetic effect is used to create TMF; cf. ref. [17].
We should also note that the results of the present work could be potentially used to describe the asymmetric emission of electroweakly interacting particles, like neutrinos, from compact rotating stars. This effect could be used to explain, e.g., pulsar kicks. However, the estimates made show that the effect is outside the observationally tested region.