Spontaneous magnetization in high-density quark matter

It is shown that the spontaneous magnetization occurs due to the anomalous magnetic moments of quarks in the high-density quark matter under the tensor-type four-point interaction. The spin polarized condensate for each flavor of quark appears at high baryon density, which leads to the spontaneous magnetization due to the anomalous magnetic moments of quarks. The implications to the strong magnetic field in the compact stars is discussed.


§1. Introduction
One of the recent interests about the physics governed by quantum chromodynamics (QCD) may be to clarify the phase structure of QCD with respect to the temperature, baryon chemical potential or baryon number density, external magnetic field and so on. 1) In the region with high temperature and low density, the lattice QCD simulation gives many insights about the QCD world. However, as is well known, in the region of large quark chemical potential, there are some problems to be solved in order to calculate the physical quantities definitely. But, it is expected that there are still many interesting phenomena in the region of low temperature and large chemical potential. Especially, it is expected that there are many exotic phases such as the two-flavor color superconducting phase, the colorflavor locked phase, 2), 3), 4) the quarkyonic phase, 5) the phase with the inhomogeneous condensate 6) and so forth.
In the heavy-ion collision experiments such as relativistic heavy-ion collider (RHIC) experiments at Brookhaven National Laboratory, it is believed that the quark-gluon phase is realized apart from the hadronic phase. Thus, the quark matter may be created and the extreme states of QCD with finite temperature and density may be realized in the heavy-ion collision experiments. On the other hand, the high-density hadronic phase or quark phase may be realized in the inner core of the compact star objects such as the neutron stars, magnetars and quark stars, if they exist. In the core of these compact stars, it is expected that the hadron or quark phase exists at low temperature and high baryon density. For example, the quark matter is considered to exist in the core of pulsars with two solar masses in compact stars, and in the case of the heavy ion collisions. 7) Therefore, the investigation of quark matter in the region of the low temperature and the large quark chemical typeset using PTPT E X.cls Ver.0.9 potential is one of very interesting and important subjects in order to understand the whole world governed by QCD.
It is further known that neutron stars, especially the so-called magnetars, 8), 9) have a strong magnetic field. However, the origin of the strong magnetic field is not so clear. It has been pointed out that the strong magnetic field may be created if the quark liquid exists in the core of the compact stars. 10) Further, the possibility of the quark spin polarization in the high-density quark matter has been investigated when the pseudovector-type interaction between quarks exists. 11) Under the pseudovector-type interaction, it was shown that the spin polarized condensate appears, which leads to a ferromagnetism in a quark matter. 12) However, the spin polarized condensate only appears in a narrow region of the quark chemical potential.
As for the spin polarized condensate, it has been shown that the tensor-type interaction between quarks leads to the spin polarized phase in high-density quark matter. 13), 14), 15) The present authors have shown the possibility of the quark spin polarized phase in the quark matter at high baryon density against the two-flavor color superconducting phase 16) and the color-flavor locked phase 17) under both the quark pairing interaction and tensor-type four-point interaction in the Nambu-Jona-Lasinio (NJL)-type model. 18), 19) However, the magnetic feature is not considered in the tensor-type interaction until now .
In this paper, the magnetic features are investigated under the existence of the spin polarized condensate in the NJL model with the tensor-type four-point interactions. It will be shown that the anomalous magnetic moments of quarks 20) play an essential role, the existence of which leads to the spontaneous magnetization of quark matter. Also, the implication to the magnetic field of compact stars such as neutron stars with quark matter, namely hybrid stars, will be discussed. This paper is organized as follows: In the next two sections, §2 and §3, the thermodynamic potential under the external magnetic field is given with and without the spin polarized condensate. In section §4, an approximate expression of the thermodynamic potential with small magnetic field is derived. In section §5, it is shown that the spontaneous magnetization does not appear in the case of no anomalous magnetic moments of quarks. In section §6, the anomalous magnetic moments of quarks are introduced within the mean field approximation. As a result, spontaneous magnetization occurs in quark matter in the region of high baryon density due to both the spontaneous spin polarization and the anomalous magnetic moments of quarks. In section §7, the implication to the hybrid compact stars is discussed briefly and it is shown that the strong magnetic field in the surface of the compact stars can be revealed. The last section is devoted to a summary and concluding remarks. §2. Thermodynamic potential under external magnetic field We consider the two-flavor case. Let us start from the following Lagrangian density with chiral symmetry: where τ represents the flavor su(2)-generator.
In previous study, we have found that a spin polarized condensate F 3 = −G ψ Σ 3 τ 3 ψ may be realized at high baryon density, where Σ 3 = −iγ 1 γ 2 is the spin operator. Thus, the Lagrangian density under the mean field approximation is obtained as , denote the eigenvalues of τ 3 . Here, σ 3 is the third component of the Pauli spin matrices.
Hereafter, let us consider the system under the external magnetic field B along the z-axis. The Lagrangian density is recast into where D µ represents the covariant derivative: Here, for up (down) quark, Q = 2e/3 (−e/3) where e is the elementary charge. Because we investigate the quark matter at finite density system, the Hamiltonian density with quark chemical potential µ is obtained as 5) where N represents the quark-number operator. Here, h can be expressed as where, by using the Dirac representation of the Dirac gamma matrices, Hereafter, let us consider two cases, namely F > 0 and F = 0, respectively.

F > 0 case
The Dirac equation is written by i∂ψ/∂t = hψ = Eψ, namely where φ and ϕ are two-component spinors. Eliminating ϕ and noting [P x ,P y ] = iQB, we obtain the following equation: where 1 is the identity matrix in the isospin space. Let us consider Q > 0 case, that is, the case of up quark. We introduce new operators instead ofP x andP y as (2.14) Here,p z and a † a should be replaced by their eigenvalues p z and ν (= 0, 1, 2, · · · ). Further, since the case Q > 0 is treated, namely the case of up quark, so F 3 = F . Then from the coefficient of φ 1 being 0, we can get the eigenvalue of the Dirac equation, E, which is expressed as ǫ flavor pz,ν , as with Q = 2e/3. It should be here noted that it is necessary to pay a special attention for the case ν = 0 because aφ(ν = 0) = a|ν = 0 = 0 satisfies. For ν = 0, from for positive energy solution. Thus, for ν = 0, the solution only appears once, and the energy corresponding to ν = 0 is not degenerate.
In the same way, we can get the eigenvalue E in the case of down quark where Q < 0. Instead of (2.10), we define new operators as where [ a , a † ] = 1 is satisfied. Then, we obtain the equation instead of (2.11) as Eliminating φ 1 which is the upper component of the two-component spinor φ, we obtain the following equation: Thus, we obtain the eigenvalue for down quark, ǫ down pz,ν by replacingp z and a † a into their eigenvalues p z and ν (= 0, 1, 2, · · · ): Thus, the single-particle energy of quarks with flavor f = u or d for up and down quarks can be expressed as The thermodynamic potential Φ can be expressed in terms of the vacuum expectation values of the HamiltonianĤ and the particle number operatorN , in which the quarks occupy the energy levels from that with the lowest energy to that with the Fermi energy. Thus, we obtain Φ as where V represents the volume under consideration and the factor 3 represents the color degree of freedom. Here, it should be noted that the sum with respect to p z and ν can be regarded as the following * ) : Here, p 2 x + p 2 y can be regarded as 2|Q|Bν (= p 2 ⊥ ). Thus, the following correspondence may be understood: Thus, the thermodynamic potential can be obtained finally as Next, let us consider the F = 0 case. For Q > 0 and Q < 0, Eqs.(2.11) and (2.19) are valid with F = 0. Thus, the following equation for the two-component spinor φ is obtained: where the upper (lower) sign in front of σ z corresponds to the case Q > 0 (Q < 0). Replacingp z and a † a by their eigenvalues and φ = t (φ 1 , φ 2 ), the above equation is written as Thus, the single-particle energy E is obtained as .

(2.29)
This result is included in Eq.(2.23). Thus, the thermodynamic potential Φ 0 can be calculated as where p 0 and ν f M should be determined later. §3. The thermodynamic potential in three cases: F > µ, 0 < F < µ and F = 0 First, let us integrate out with respect to p z in the thermodynamic potential (2.26) and (2.30). The condition F + η 2|Q f |Bν 2 + p 2 z ≤ µ 2 gives the range of integration with respect to p z . Namely, By using the following integration formula, we can carry out the integration with respect to p z in Eq.(2.26), which leads to f,M are determined by the condition which guarantees that p z is real. Here, g η (x) is defined by For η = 1, the condition (3.4) is not satisfied for any F . On the other hand, Thus, we summarize the condition for ν as follows: where [· · · ] represents the Gauss symbol. Thus, the thermodynamic potential can be expressed as with (3.6).

F < µ case
For η = 1, the condition (3.4) gives the condition 0 gives 0 ≤ 2|Q f |Bν ≤ F . Thus, we summarize the condition for ν as follows: Thus, the thermodynamic potential is (3.3) with (3.8) for η = ±. Namely, Next, let us consider F = 0 case. Thus, the thermodynamic potential Φ 0 can be given in (2.30), which we show again: (3.11) §4. Approximation of the thermodynamic potential by replacing summation by integration with respect to the Landau level In the thermodynamic potential (2.26) and (2.30), the quantum number ν, which labels the Landau level, has to be summed up. However, since it is interesting to consider the spontaneous magnetization, it may be assumed that the external magnetic field B is small and finally B becomes 0. Therefore, let us replace the sum with respect to ν by an integration approximately. 13) In general, let us consider a function f (x). Here, we introduce a small quantity a and let us consider the Tailor expansion around x = aν as follows: Thus, the following relations is obtained : Here, it should be noted that the definition of integral can be used when a is infinitesimally small, namely, and so on. Thus, adding af (aν m ) on both sides of Eq.(4.2), a useful approximate formula is obtained as follows: (4.4) After here, let us approximate the thermodynamic potential in the case of small B.
For F > µ, Eq.(3.7) is approximated by using the formula (4.4) as follows: f,M , respectively. For F < µ, the thermodynamic potential (3.9) is approximated by using (4.2) directly as where ν f,M , respectively. For F = 0, the thermodynamic potential (3.10) is approximated as Here, since B is small, ν may be regarded as a continuum variables. Remembering the condition which determines ν where we used the integration formulae: (4.9) §5. Spontaneous magnetization for the quark matter under the tensor-type interaction between quarks First, let us calculate the spin polarization F with B = 0 as a function of the chemical potential µ. This task has been done in our previous papers 15) or 16). The thermodynamic potential can be expressed as where the factor 6 represents the color and flavor degrees of freedom and p 1 = p x , p 2 = p y and p 3 = p z . Here, (1/V )· p can be replaced to the integration d 3 p/(2π) 3 .
Integrating the three-momentum, we have obtained the thermodynamic potential as for F > µ .   The baryon number density of quark matter divided by the normal nuclear density ρ0 = 0.17 fm −3 is shown as a function of the quark chemical potential. Here, µc ≈ 0.407 GeV is the value at which the spin polarization occurs. Also, in the region of µ ≥ µF ≈ 0.5605 GeV, the spin polarization condensate is greater than the chemical potential, F ≥ µ.
which has been used * ) in our previous papers. 15), 16) About µ = µ c ≈ 0.407 GeV which corresponds to the baryon density being 3.53 ρ 0 , the spin polarization appears against the free quark phase. Of course, if the two-flavor color superconductivity is considered, it has been shown that the spin polarized phase appears about µ = 0.442 GeV (ρ B ≈ 5.85ρ 0 ). 16) We derive the spontaneous magnetization through the thermodynamic relation. The spontaneous magnetization per unit volume, M, is defined by In the right-hand side, B = 0 is adopted. However, from (4.8), the thermodynamic potential does not depend on B linearly. Thus, the spontaneous magnetization is equal to zero under this consideration, namely even if the quark-spin polarization occurs, while the magnetic polarization which is proportional to B may appear. §6. Spontaneous magnetization originated from the anomalous magnetic moments of quarks In the previous section it has been found that no spontaneous magnetization of polarized high density quark matter is predicted by the normal coupling to an external magnetic field. However, the spin polarization F should be observed in some way. We know that the quark has an anomalous magnetic moment. In this section, let us investigate the effect of the anomalous magnetic moment of quarks.
The effect of the anomalous magnetic moment µ A is introduced at the level of the mean field approximation in this paper. Here, it is known that the anomalous magnetic moments reveal as 20) We introduce the effects of the anomalous magnetic moment in the Lagrangian density within the mean field approximation as 2) * ) If the vacuum polarization is taken into account, the parameter G with rather small value gives the same results quantitatively such as G = 11.1 GeV −2 under the standard three-momentum cutoff Λ = 0.631 GeV, although we do not consider the vacuum polarization in this paper.
where F µν = ∂ µ A ν − ∂ ν A µ and F 12 ≡ −B z = −B. Here, L is nothing but Eq.(2.3). We only take µ and ν as µ = 1, ν = 2 and µ = 2, ν = 1 because the magnetic field has only z-component. Then, in the mean field approximation, Eq.(2.3) is recast into Here, F 3 τ 3 = F 1 where 1 is the 2 × 2 identity matrix for the isospin space, which are denoted in Eqs.(2.2) and (2.9). Thus, we introduce the flavor-dependent variables F f as Therefore, the Lagrangian density can be expressed as Thus, we learn that the variable F should be replaced to F f = F + µ f B for each flavor except for the last term in (6.5). In this replacement, we can derive the thermodynamic potential in the same way developed in §4. Namely, the results including the effects of the anomalous magnetic moment of quarks should be obtained by replacing F into F f for each flavor in the previous calculation developed in §4, except for the term originated from the mean field approximation, F 2 /2G. The results are summarized as follows: Thus, the spontaneous magnetization per unit volume, M, can be derived thorough Eq.(5.4), that is From (6.6), we can derive the spontaneous magnetization originated form the anomalous magnetic moment and the spin polarization: where the gap equation ∂Φ/∂F = f =u,d ∂Φ/∂ F f + ∂(F 2 /(2G))/∂F = 0 is used from the second line to the third line in (6.8b). It should be noted that if F = 0, the spontaneous magnetization disappears because F = 0 in Eq.(6.8b). In Fig.2, the spontaneous magnetization is depicted as a function of the quark chemical potential. The spontaneous magnetization is shown in SI unit. For µ < µ c ≈ 0.407 GeV, the magnetization does not occur because the spin polarization does not appear, F = 0. At µ = µ c , the spontaneous magnetization suddenly appears. §7. Magnetic field of hybrid compact star As seen in the previous section, for µ > µ c , the spontaneous magnetization in the quark matter appears. Here, the spontaneous magnetization per unit volume M may be regarded as the magnetic dipole moment. Under this identification, we can calculate the strength of the magnetic filed yielded by the spontaneous magnetization. As is well known in the classical electromagnetism, the magnetic field at the position r, namely magnetic flux density B(r), created by the magnetic dipole moment m can be expressed as where µ 0 represents the vacuum permeability. In our case, m = (0, 0, M× V ), where V represents a volume because M is nothing but the magnetization per unit volume.
Here, let us consider the hybrid star with quark matter in the core of neutron star. Let us assume that the hybrid (neutron) star has radius R = 10 km. If there exists quark matter in the inner core of star from the center to r q km, the strength of the magnetic flux density on the surface at the north or south pole of hybrid star is roughly estimated as It should be noted that the SI unit, Tesla, is converted to Gauss, namely, 1 T (Tesla) = 10 4 G (Gauss). In the magnetar, the strength of the magnetic field at the surface of star is near 10 15 G. Thus, in our calculation, if quark matter exists and the spin polarization occurs, a strong magnetic field about 10 13 or 10 14 Gauss may be created. §8. Summary and concluding remarks It has been shown that spontaneous magnetization occurs due to the anomalous magnetic moments of quarks in the high-density quark matter under the tensor-type four-point interaction. In the Nambu-Jona-Lasinio model as an effective model of QCD, the tensor-type four-point interaction has been introduced. As for the tensortype four-point interaction, this interaction term was also introduced to investigate the meson spectroscopy, especially, for vector and axial-vector mesons. 21) Owing to this interaction, the spin polarized condensate appears for each quark flavor in the region of large quark chemical potential. It has been shown that the spin polarized condensate leads to spontaneous magnetization of quark matter due to the anomalous magnetic moments of quarks. Also, it has been pointed out that spontaneous magnetization does not occur if there exists no anomalous magnetic moments of quarks.
In this paper, furthermore, the implication to the strong magnetic field in compact stars such as the hybrid stars has been discussed. If there exists quark matter in the core of neutron stars and then the quark number density is rather high, spontaneous magnetization may occurs. If quark matter occupies a volume with 1% of neutron star, the strength of magnetic field at the surface of neutron star is of order of 10 14 Gauss, which is comparable to the strength of magnetic field in the so-called magnetar.
As indicated in this paper, the spin polarized condensate appears in the region with a large quark chemical potential. Thus, the chiral symmetry is broken in this model because the starting Lagrangian density is constructed with chiral symmetry. Under the magnetic field, the chiral symmetry is broken and it is shown that the chiral condensate grows at most linearly as a function of the magnetic field B. 22) Further, in Ref. 23), the tensor-type four-point interaction was introduced in the NJL model in the case of one quark-flavor. In that paper, both the chiral condensate and spin polarized condensate are equally treated at finite temperature system. Thus, it may be an interesting problem that the coexistence of the chiral condensate and the spin polarized condensate is also considered at finite baryon density system.
As for the implication to compact stars, it has been shown that there is a possibility of the existence of the massive hybrid quark stars with two solar mass under the strong magnetic field. 24) Thus, the investigation of the equation of state of quark matter in the spin polarized phase revealing the spontaneous magnetization may be one of the interesting future problems.