Generation of strong magnetic fields in dense quark matter driven by the electroweak interaction of quarks

We study the generation of strong large scale magnetic fields in dense quark matter. The magnetic field growth is owing to the magnetic field instability driven by the electroweak interaction of quarks. We discuss the situation when the chiral symmetry is unbroken in the degenerate quark matter. In this case we predict the amplification of the seed magnetic field $10^{12}\,\text{G}$ to the strengths $(10^{14}-10^{15})\,\text{G}$. In our analysis we use the typical parameters of the quark matter in the core of a hybrid star or in a quark star. We also discuss the application of the obtained results to describe the magnetic fields generation in magnetars.


Introduction
The origin of strong magnetic fields B ∼ 10 15 G in some compact stars, called magnetars [1], remains an open problem of modern astrophysics. Despite the popularity of some models describing the generation of such magnetic fields, which are based on magnetohydrodynamics of stellar plasmas, none of them can satisfactory describe the observational data. These models are reviewed in Ref. [1].
Recently, the methods of elementary particle physics, mainly the chiral magnetic effect (CME) [2], were applied in Ref. [3] to generate toroidal magnetic fields in a neutron star (NS), and in particular to solve the problem of magnetars [4]. The major motivation to apply CME to produce magnetic fields in NS is that the nonzero chiral imbalance of electrons µ 5 = (µ R − µ L ) /2 is created in nonequilibrium Urca processes, which are parity violating. It happens since ultrarelativistic left electrons are washed out from the system producing µ 5 > 0. The nonzero µ 5 generates the electric current of ultrarelativistic electrons along the magnetic field. This current, in its turn, leads to the magnetic field instability resulting in the growth of a magnetic field.
Another possibility to utilize the electroweak interaction for the production of the magnetic field instability was proposed in Refs. [5,6]. It consists in the fact that the induced anomalous electric current along the magnetic field gets the contribution proportional to the difference of the effective potentials of the effective electroweak interaction of left and right electrons with background matter. Thus the electroweak interaction becomes a constant driver of the magnetic field instability. Then in Refs. [7][8][9][10] this idea was applied to generate strong large scale magnetic fields in NS due to the electroweak electron-nucleon interaction.
The key issue in the application of CME to generate a stellar magnetic field is the presence of left and right charged fermions in a star. Strictly speaking, the possibility to separate a fermionic field into left and right chiral projections is only possible if this particle is massless, i.e. when the chiral symmetry in unbroken. Despite the typical energy of an electron in the NS matter is much greater than its mass, one cannot claim these electrons are chiral particles there. Thus we can expect that CME for electrons is unlikely to appear in NS. This claim is also true with respect to the model in Refs. [7][8][9][10]. Note that, for the first time the fact that a nonzero particle mass destroys CME was noticed in Ref. [11]. Recently, in Ref. [12], this result of Ref. [11] was confirmed in the presence of the electroweak interaction.
Despite of the above disappointing observation, we can still expect the existence of astrophysical media where the chiral symmetry is unbroken. It is the quark matter in the core of a hybrid star (HS) or in a hypothetical quark star (QS). HS is a NS having the quark core. QS is based on the strange matter hypothesis. The properties of these compact stars are reviewed in Ref. [13]. Note that, despite of the sporadic claims of the observations of HS/QS (see, e.g., Ref. [14]), there is a certain skepticism on the existence of these compact stars.
The present work is devoted to the application of the methods of Refs. [7][8][9][10] to describe the magnetic field instability, leading to its growth, in quark matter in HS/QS. In Sec. 2, we derive the kinetic equations describing the evolution of the magnetic field and chiral imbalances in quark matter. We also formulate the initial conditions corresponding to a typical astrophysical medium. Then, in Sec. 3, we present the results of the numeric solutions of these kinetic equations. Finally, in Sec. 4, we discuss the obtained results and their applicability for modeling magnetic fields in magnetars. The computation of the helicity flip rates of quarks in their mutual collisions is provided in Appendix A.

Basic equations for the magnetic field evolution in quark matter
In this section we shall derive the equations for the evolution of the spectra of the magnetic helicity density and the magnetic energy density as well chiral imbalances in degenerate matter containing u and d quarks interacting by the parity violating electroweak forces. Let us consider a dense quark matter consisting of u and d quarks. The density of this matter is supposed to be high enough for the chiral symmetry to be restored. In this case we can take that the quarks are effectively massless. Recently, in Ref. [15], it was shown with help of lattice simulations that the chiral symmetry has a tendency to restore in a quark matter at high density. Therefore we can decompose the quark wave functions into left and right chiral components, which evolve independently, and attribute different chemical potentials µ qL,R , where q = u, d, for each chiral component.
Generalizing the results of Refs. [7,8], we get that, in the external magnetic field B, there is the induced electric current where e u = 2e/3 and e d = −e/3 are the electric charges of quarks, e > 0 is the elementary charge, µ 5q = (µ qR − µ qL ) /2 is the chiral imbalance, V 5q = (V qL − V qR ) /2, and V qL,R are the effective potentials of the electroweak interaction of left and right quarks with background fermions. The potentials V qL,R were found in Ref. [16] on the basis of the effective Lagrangian for the ud electroweak interaction, where Here γ L,R 0 = γ 0 1 ∓ γ 5 /2, γ 5 = iγ 0 γ 1 γ 2 γ 3 , γ µ = γ 0 , γ are the Dirac matrices, G F = 1.17 × 10 −5 GeV −2 is the Fermi constant, ξ = sin 2 θ W = 0.23 is the Weinberg parameter, n u,d are the number densities of u and d quarks, and V ud = 0.97 is the element of the Cabbibo-Kobayashi-Maskawa matrix. The matter of the star is supposed to be electrically neutral. Thus we should have n u = n 0 /3 and n d = 2n 0 /3, where n 0 = n u + n d is the total number density of quarks in the star. Using Eq. (3) one obtains that Assuming that n 0 = 1.8 × 10 38 cm −3 , we get that V 5u = 4.5 eV and V 5d = 2.9 eV. Note that in Eqs. (1) and (2) we do not account for the uu and dd interactions. However as shown in Ref. [17], basing on the direct calculation of the two loops contribution to the photon polarization operator, that such contributions to the induced current in Eq. (1) are vanishing.
Using Eq. (1) and the results of Refs. [8], we can obtain the system of kinetic equations for the spectra of the density of the magnetic helicity h(k, t) and of the magnetic energy density ρ B (k, t), as well as the chiral imbalances µ 5u (t) and µ 5d (t), in the form, where Γ u,d are the rates for the helicity flip in ud plasma, α em = e 2 /4π = 7.3 × 10 −3 is the QED fine structure constant, σ cond is the electric conductivity of ud quark matter, and µ u,d = 3π 2 n u,d 1/3 are the mean chemical potentials of u and d quarks. In the electroneutral ud plasma, we get that The functions h(k, t) and ρ B (k, t) in Eq. (5) are related to the total magnetic helicity H(t) and the magnetic field strength by where V is the normalization volume. The integration in Eq. (6) is over all the range of the wave number k variation. Note that we assume the isotropic spectra in Eq. (6). In our model for the magnetic field generation in magnetars, we suggest that background fermions are degenerate. Nevertheless there is a nonzero temperature T of the quark matter, which is much less than the chemical potentials: T ≪ µ q . The conductivity of the degenerate quark matter was estimated in Ref. [18] as where α s is the QCD fine structure constant, T 0 = (10 8 − 10 9 ) K is the initial temperature corresponding to the time t 0 ∼ 10 2 yr, when the star is already in a thermal equilibrium. Using Eq. (7), we obtain that where we assume that α s ∼ 0.1. Note that σ cond in quark matter is several orders of magnitude less than the conductivity of electrons in the nuclear matter in NS [19]. The volume density of the internal energy of degenerate background quarks is ε T = ε 0 + δε T [9], where ε 0 ∼ µ 4 q is the temperature independent part and δε T = µ 2 u + µ 2 d T 2 /2 is the temperature correction. In Ref. [9] we suggested that the growth of the magnetic field is powered by the transmission of δε T to the magnetic energy density ρ B = B 2 /2. The energy conservation law in the magnetized ud plasma reads d (δε T + ρ B ) /dt = 0 [10]. Integrating this expression with the appropriate initial condition one gets where we assume that initially the thermal energy is greater than the magnetic energy, which is the case for a young pulsar. Indeed, if one starts with a seed field B 0 = 10 12 G, one gets that ρ B (t 0 ) = 1.9 × 10 −4 MeV 4 and δε T (t 0 ) = 5.5 MeV 4 . It means that where the equipartition magnetic field can be found from the following expression [9]: Note that Eq. (10) describes the magnetic cooling, i.e. the temperature decreasing because of the magnetic field enhancement. As we will see later, other channels of the star cooling, such as the neutrino emission [20], are negligible on the time scale of the magnetic field growth in our model. The dependence of the temperature on the magnetic field is analogous to the quenching of the parameter Π in Eq. (1) introduced in Ref. [9] (see also Ref. [21]). Although we suppose that the chiral symmetry is restored in the star and quarks are effectively massless, there are induced quark masses due to the interaction with dense matter. The effective masses of u and d quarks were computed in Ref. [22], Note that the effective quark masses in Eq. (12) should be accounted for only in quarks collisions (see Appendix A). It implies the transitions between left and right particles in their mutual collisions. The helicity flip rates Γ u,d for each quark types are computed in Appendix A, where we use Eq. (28). Let us introduce the following dimensionless functions: where we assume k min < k < k max , k min = 1/R = 2 × 10 −11 eV, R = 10 km is the star radius, k max = 1/Λ (min) B , and Λ (min) B is the minimal scale of the magnetic field, which is a free parameter. Using the dimensionless parameters, as well as Eqs. (8), (10), and (13), we can rewrite Eq. (5) in the form, where κ max = k max /k min , B 2 and B 2 eq are given in Eqs. (6) and (11). While solving of Eq. (16) numerically, we use the initial Kolmogorov spectrum of the magnetic energy density, ρ B (k, t 0 ) = Ck −5/3 , where the constant C can be obtained by equating the initial magnetic energy density, computed on the basis of Eq. (6), to B 2 0 /2 (see Ref. [8]). The initial spectrum of the magnetic helicity density is h(k, t 0 ) = 2rρ B (k, t 0 )/k, where the parameter 0 ≤ r ≤ 1, corresponds to initially nonhelical, r = 0, and maximally helical, r = 1, fields.
In Ref. [7] we found that the evolution of the magnetic field is almost independent on the initial values of the chiral imbalances µ 5(u,d) (t 0 ) because of the huge helicity flip rates Γ u,d . Therefore we can take almost arbitrary values of µ 5(u,d) (t 0 ) only requiring that µ 5(u,d) (t 0 ) ≪ µ u,d . In our simulations we shall take that µ 5u (t 0 ) = µ 5d (t 0 ) = 1 MeV.

Results of the numeric solution of the kinetic equations
In this section we present the results of the numerical solution of Eq. (16) with the initial conditions corresponding to a quark matter in a compact star.
In Fig. 1 we show the amplification of the initial magnetic field B 0 = 10 12 G by two or three orders of magnitude. This result is obtained by numerically solving Eq. (16) with the initial conditions discussed in Sec. 2. These initial conditions are quite possible in a dense quark matter in a HS/QS.
One can see in Fig. 1 that the magnetic field reaches the saturated strength B sat . This result is analogous to the findings of Refs. [9,10]. For T 0 = 10 8 K in Figs. 1(a) and 1(b), B sat ≈ 1.1 × 10 14 G; and for T 0 = 10 9 K in Figs. 1(c) and 1(d), B sat ≈ 1.1 × 10 15 G. However, unlike Refs. [9,10], B sat in Fig. 1 is defined entirely by T 0 . The obtained B sat is close to the magnetic field strength predicted in magnetars [1], especially for T 0 = 10 9 K.
The time of the magnetic field growth to B sat is several orders of magnitude shorter than in Refs. [9,10]. This fact is due to the smaller value of the electric conductivity σ cond in quark matter in Eq. (7) compared to σ cond for electrons in nuclear matter which we used in Refs. [9,10]. This fact can be understood with help of the Faraday equation, which is equivalent to the first two lines in Eq. (5). Using Eq. (17) one gets that the saturation time t sat ∼ σ cond /ΠΛ B , where Λ B is the magnetic field scale. It means that the smaller σ cond is, the faster the magnetic field reaches B sat . Moreover, we can see that short scale magnetic field should reach B sat faster. The later fact, which was also established in Refs. [8][9][10], is confirmed by the comparison of Figs. 1(a) and 1(b) as well as Figs. 1(c) and 1(d).
In our model of the magnetic field generation, the thermal energy of background fermions is converted to the magnetic energy. One can say that a star cools down magnetically. The typical values of t sat are 10 h in Figs. 1(a) and 1(b) and 10 2 min in Figs. 1(c) and 1(d). At such short time scales, other cooling channels, such as that due to the neutrino emission [20], do not contribute to the temperature evolution significantly. Therefore, unlike Refs. [8][9][10], we omit them in our present simulations.
In Fig. 1 we can see that, although the initial magnetic helicity can be different (see solid and dashed lines there), the subsequent evolution of such magnetic fields is almost indistinguishable, especially at t ∼ t sat . It means that, besides the generation of strong  magnetic field, we also generate the magnetic helicity in quark matter. This result is in the agreement with Refs. [8][9][10].

Discussion
In the present work we have applied the mechanism for the magnetic field generation, proposed in Refs. [7][8][9], to create strong large scale magnetic fields in dense quark matter. This mechanism is based on the magnetic field instability driven a parity violating electroweak interaction between particles in the system. We have established the system of kinetic equations for the spectra of the magnetic helicity density and the magnetic energy density, as well as for the chiral imbalances, and have solved it numerically.
Although there is a one-to-one correspondence between the mechanisms for the magnetic field generation in Refs. [7][8][9][10] and in the present work, the scenario described here is likely to be more realistic. As mentioned in Ref. [12] the generation of the anomalous current in Eq. (1) is impossible for massive particles. Electrons in NS are ultrarelativistic but have a nonzero mass. As found in Ref. [23], the chiral symmetry can be restored at densities n ∼ M 3 W ∼ 10 46 cm −3 , that is much higher than one can expect in NS. Therefore the chiral magnetic effect for electrons as well as the results of Refs. [7][8][9][10] are unlikely to be applied in NS. Recently this fact was also mentioned in Ref. [12].
On the contrary, the chiral symmetry was found in Ref. [24] to be restored for lightest u and d quarks even at densities corresponding to a core of HS or in QS. Accounting for the existence of the electroweak parity violating interaction between u and d quarks, we can conclude that the application of the methods of Refs. [7][8][9][10] to the quark matter in a compact star is quite plausible.
We have obtained that, in quark matter, the seed magnetic field B 0 = 10 12 G, which is typical in a young pulsar, is amplified up to B sat ∼ 10 14 − 10 15 G, depending on the initial temperature. Such magnetic fields are predicted in magnetars [1]. Therefore HS/QS can become a magnetar. The obtained growth time of the magnetic field to B sat is much less than that in electron-nucleon case studied in Refs. [7][8][9][10]. It means that, in our model, strong magnetic fields are generated quite rapidly with t sat ∼ several hours after a star is in a thermal equilibrium.
Note that, in the present work, instead of the quenching of the parameter Π in Eq. (1) suggested in Ref. [9] to avoid the excessive growth of the magnetic field, we used the conservation of the total energy in Eq. (10) and the dependence of the electric conductivity on the temperature in Eq. (8); cf. Ref. [10]. It results in a more explicit saturation of the magnetic field in Eq. (16); cf. Fig. 1.
Despite the plausibility of the results, several important assumptions were made. Firstly, while calculating the helicity flip rates in Appendix A, we have taken that quarks exchange by plasmons in their scattering. It is, however, known (see, e.g., Ref. [25]) that modified effective interaction potentials can exist in a dense degenerate matter. If one takes into account these interactions it can somehow change the values of Γ u,d . Nevertheless, since the present work is a qualitative study of the magnetic field generation in the degenerate quark matter, we shall restrict ourselves to the the plasmon interaction of quarks.
Secondly, we have considered the simplest case of a compact star consisting of only u and d quarks. However, strange stars, having a certain fraction of s quarks are also actively studied [13, pp. 414-440]. The nonzero fraction of s quarks, which cannot exceed 1/3, is also required by the beta equilibrium. Nonetheless s quarks are unlikely to contribute significantly to the generation of magnetic fields in our model. Firstly, the mass of an s quark m s = 150 MeV is quite great, i.e. the chiral symmetry will remain broken for these particles. Thus, s quarks do not contribute to the induced current in Eq. (1). Secondly, even if s quarks contribute to the helicity flip rates of u and d quarks, it will not change the evolution of the magnetic field. Indeed, Γ u,d computed in Appendix A, is already great enough to wash out the initial chiral imbalances µ 5(u,d) (0). Any bigger contribution to Γ u,d will eliminate µ 5(u,d) (0) faster. However, the growth of the magnetic field is driven by V 5(u,d) , which is constant, rather than by µ 5(u,d) .
Summarizing, we have described the generation of strong large scale magnetic fields in dense quark matter driven by the magnetic field instability caused by the electroweak interaction of quarks. The described phenomenon may well exist in the core of HS or in QS. We suggest that the obtained results can have implication to the problem of magnetars since the generated magnetic fields have strength close to that predicted in these highly magnetized compact stars.

A Helicity flip rates in degenerate quark matter
In this Appendix we shall compute the helicity flip rates of u and d quarks in their collisions in dense matter as well as derive the kinetic equations for the chiral imbalances.
As mentioned in Sec. 2, quarks acquire effective masses in dense matter. Thus the helicity of quarks will change when the particles collide. There are three types of reactions: (a) scattering of identical quarks, with helicities of both particles being changed; (b) scattering of different quark flavors, with helicities of both particles being changed; and (c) scattering of different quark flavors, with helicity of only one particle being changed. We shall successively discuss all the cases. Quarks are supposed to interact by the plasmon exchange.
Scattering of identical quarks There are four reactions in this group: We study in details only the process , where p µ 1,2 = (E 1,2 , p 1,2 ) are the momenta of incoming quarks and p ′µ 1,2 = E ′ 1,2 , p ′ 1,2 are the momenta of outgoing quarks. In this reaction, the number of left particles is decreased by two units and the number of right particles is increased by two units. Other reactions in this group can be studied analogously.
The matrix element has the form, where t = (p ′ 1 − p 1 ) 2 and u = (p ′ 2 − p 1 ) 2 are the Mandelstam variables. The square of the matrix element in Eq. (18) is where the density matrices are [26, pp. 106-111] Here m u is the effective mass given in Eq. (12) and the polarization vectors are [10] a µ which correspond to left and right particles. Here n 1,2 and n ′ 1,2 are the unit vectors along p 1,2 and p ′ 1,2 . Choosing the center-of-mass frame of colliding quarks and assuming the elastic scattering, one gets that tr ρ ′ 2 γ µ ρ 2 γ ν · tr ρ ′ 1 γ µ ρ 1 γ ν = 16m 4 u sin 4 θ cm 2 , where θ cm is the scattering angle, i.e. the angle between p 1 and p ′ 1 in the center-of-mass frame. In the same frame one has where E cm is the energy of colliding quarks in the center-of-mass frame. In Eq. (23), we also assume that the scattering is elastic. We can express E cm in term of the variables in the laboratory frame, i.e. where the star is at rest, as E 2 cm ≈ m 2 u + E 1 E 2 [1 − (n 1 · n 2 )] /2. Since we study the probability in the lowest order in the effective mass and traces in Eq. (22) are proportional to m 4 u , we neglect m u in Eq. (23) as well as in the following calculations. Finally, Eq. (19) takes the form The total probability of the process has the form [26, pp. 247-252], where f (E) = [exp (βE) + 1] −1 is the Fermi-Dirac distribution of quarks, β = 1/T is the reciprocal temperature, µ L,R are the chemical potentials of left and right quarks, and |M| 2 is given in Eq. (24). Here we assume that quarks are degenerate, i.e. f (E − µ) = Θ(µ − E), where Θ(z) is the Heaviside step function. In Eq. (25) we introduce the additional factor 4 = 2! × 2! in the denominator to take into account identical particles in the initial and final states. The direct calculation of the integrals over the phase space in Eq. (25) accounting for |M| 2 in Eq. (24) gives Analogously we can compute the probabilities of other reactions in this group. The kinetic equations for the evolution of the total number of left and right u quarks Accounting for Eq. (26) and the analogous expression for d quarks, one gets the evolution of the chiral imbalances µ 5(u,d) = µ (u,d)R − µ (u,d)L /2 in the form, In Eq. (28) we take into account the relation between the number densities n L,R = N L,R /V and chemical potentials of left and right quarks where we assume that massless quarks have only one polarization. In particular we get from Eq. (29) thatd(n (u,d)R − n (u,d)L )/dt ≈μ 5(u,d) µ 2 u,d /π 2 .
Scattering of ud quarks: both particles change helicity There are also four reactions: Let us first study the following process: . The matrix element has the form, Instead of using Eqs. (20) and (21) to compute |M| 2 , we can utilize the solution of the Dirac equation, corresponding to left and right particles where w ± (p) are the helicity amplitudes which can be found in Ref. [ where we keep the leading term in the effective quark masses and assume the elastic scattering. Analogously to Eq. (25) one obtains the total probability for the reaction u L d L → u R d R in the form, After the computation of the integrals over the quark momenta in Eq. (33) one has where W 0 ∼ e 2 u e 2 d m 2 (12) and Ref. [22]). It means that the contribution of the reactions in the considered group to the helicity flip rates is negligible.
Scattering of ud quarks: only one particle changes helicity One has eight reactions u L d L,R → u R d L,R , u R d L,R → u L d L,R , d L u L,R → d R u L,R , and d R u L,R → d L u L,R present in this group. Let us first study the process u L (p 1 ) + d L (p 2 ) → u R (p ′ 1 ) + d L (p ′ 2 ). The matrix element for this reaction is The calculation of |M| 2 can be made with help of Eq. (31). Here we present only the final result, We just mention that, to get Eq. (38) we have to avoid the infrared divergence. For this purpose we introduce the plasma frequency in the degenerate ud quark matter [27]. Comparing Eq. (38) with Eq. (26) one can see that W (u L d L → u R d L ) ≪ W (u L u L → u R u R ) since T ≪ ω p in the degenerate matter. That is why the reactions in this group can be omitted as well.
At the end of this Appendix we mention that we do not study the influence of the electroweak interaction between quarks on the helicity flip in quark collisions. The contribution of the electroweak interaction to the scattering probability of electrons off protons was studied in Ref. [10], where it was found that V 5 does not enter to the analog of Eq. (28) for the evolution of the chiral imbalance µ 5 .