Spin Polarization versus Color-Flavor-Locking in High Density Quark Matter

It is shown that spin polarization with respect to each flavor in three-flavor quark matter occurs instead of the color-flavor locking at high baryon density by using the Nambu-Jona-Lasinio model with four-point tensor-type interaction. Also, it is indicated that the order of phase transition between the color-flavor locked phase and the spin polarized phase is the first order by means of the second order perturbation theory.


§1. Introduction
One of recent interests about the physics governed by the quantum chromodynamics (QCD) may be to understand the structure of phase diagram on a plane with respect to, for example, temperature and baryon chemical potential or external magnetic field, isospin chemical potential and so forth. 1) Especially, under extreme conditions such as high baryon density, it is interesting what phase is favorable and is realized. In the region with high baryon density and low temperature in the quark matter, it is believed that there exists the two-flavor color superconducting (2SC) phase or the color-flavor locked (CFL) phase. 2) In the preceding study, it was indicated that the spin polarized phase may appear at high baryon density due to the pseudovector-type interaction between quarks. 3) However, in the limit of the quark mass being zero, it has been shown that the spin polarized phase disappears. 4) In our preceding paper, 5) it was shown that the quark spin polarized (SP) phase in two-flavor case is realized in the region of high baryon density by the use of the Nambu-Jona-Lasinio (NJL) model 6) devised by four-point tensor-type interaction with chiral symmetry. 7) Further, since 2SC phase may exist in two-flavor QCD at high baryon density, it is also investigated whether the quark spin polarized phase is realized or not against the 2SC phase. As a result, it was shown that the quark spin polarized phase is actually realized in the same two-flavor NJL model adding to the quark-pair interaction. 8) In this paper, a possibility of quark spin polarization for each flavor is investigated in the three-flavor case by using the NJL model with four-point tensor-type interaction. In three-flavor case, CFL phase may be realized at high baryon density.
typeset using PTPT E X.cls Ver.0.9 Thus, the quark-pairing interaction is introduced and it is investigated which phase, namely CFL phase or SP phase, is favorable energetically and is realized at high baryon density. Also, if both phases exist, it is necessary to discuss the order of phase transition between CFL and SP phase.
This paper is organized as follows: In the next section, the NJL-type model Hamiltonian is explained. In §3 with Appendix A, under the above-derived Hamiltonian, the CFL phase without the condensate of the quark spin polarization and/or the SP phase without color superconducting gap are discussed. In §4, numerical results are given and the realized phase in certain density regions is considered. In §5, based on the CFL phase and dealing with tensor-type interaction as a perturbation term, the order of phase transition from CFL to SP phase is discussed. Some expressions needed in the calculation of second order perturbation are given in Appendix B. The last section is devoted to a summary and concluding remarks. §2. Hamiltonian showing three-flavor color superconductivity and spin polarization based on the NJL model Let us start with the following NJL type Lagrangian density: where ψ C = Cψ T with C = iγ 2 γ 0 being the charge conjugation operator. Also, λ f k and λ c a are the flavor and color su(3) Gell-Mann matrices, respectively. Here, the NJL Lagrangian density contains other four point interaction parts which are not explicitly shown such as G 0 (ψψ)(ψψ) and is invariant under chiral transformation. However in this paper, some terms are omitted because we investigate only the condensates with respect to color-flavor-locking and the each quark-spin polarization in high density quark matter in the mean field approximation. For example, at high baryon density, the chiral condensate ψ ψ is equal to zero. Further, in threeflavor case, it is well known that the Kobayashi-Maskawa-'t Hooft (KMT) term 9) appears, which describes the U A (1)-anomaly and is represented by the six-point interaction with determinant-type form in the NJL model. However, hereafter, since we adopt the mean field approximation, the KMT term only gives contributions as ψ u ψ u ψ d ψ d ψ s ψ s , ψ d ψ d ψ s ψ s (ψ u ψ u ) and so on, where ψ f represents the quark field with flavor f and · · · represents the condensate. Thus, at high baryon density under investigation in this paper, there is no contribution of the KMT term because the chiral condensate ψ f ψ f is zero. As for the quark masses, although the strange quark mass with 0.1 GeV is certainly non-zero compared with the up and down quark masses, we may safely ignore the strange quark mass at high baryon density with the quark chemical potential being 0.4 -0.5 GeV under consideration in this paper. Thus, we ignore the quark mass term in (2.1).
Within the mean field approximation, the above Lagrangian density is expressed as where h.c. represents the Hermitian conjugate term of the preceding one. Here, we used the Dirac representation for the Dirac gamma matrices and σ 3 represents the third component of the 2 × 2 Pauli spin matrices. The symbol · · · represents the expectation value with respect to a vacuum state. The expectation values F 3 and F 8 correspond to the order parameter of the spin alignment which may lead to quark spin polarization. The expectation value ∆ ak corresponds to the quark-pair condensate which means the existence of the color superconducting phase if ∆ ak = 0. The mean field Hamiltonian density with quark chemical potential µ is easily obtained as with N = ψ † ψ. In the Dirac representation for the Dirac gamma matrices, the Hamiltonian matrix of the spin polarization part in where α i = γ 0 γ i and β = γ 0 . Here, we define F τ as For good helicity states, this Hamiltonian matrix is easily diagonalized in the case F τ = 0. For simplicity, we rotate around p 3 axis and we set p 2 = 0 without loss of generality. In this case, we derive κ = U −1 h SP M F U as follows: (2.6) Finally, in the original basis rotated around p 3 -axis, |p 1 | is replaced to p 2 1 + p 2 2 . Thus, the many-body Hamiltonian can be expressed by means of the quark creation and annihilation operators as is seen later.
As for the Hamiltonian matrix of the color superconducting part, H M F c = d 3 xH M F c , we can derive another expression by using the quark-creation and annihilation operators with respect to good helicity states, similarly to Ref. 8) As a result, in the basis of good helicity states, the relevant combination of the mean field Hamiltonian H M F = d 3 xH M F and the quark number N = d 3 xN is given by where V represents the volume in the box normalization. Here, c † pητ α andc † pητ α represent the quark and antiquark creation operators with momentum p, helicity η = ±, flavor index τ and color α. It should be noted that α (= 1, 2, 3) and τ (= u, d, s) represent color and flavor, respectively, in which especially we understand τ 1 = u, τ 2 = d and τ 3 = s. Hereafter, we will use p ≡ (p, η) andp ≡ (−p, η) as abbreviated notations. Also, ǫ τ τ ′ τ ′′ and ǫ αα ′ α ′′ represent the complete antisymmetric tensor for the flavor and color indices. We define p = p 2 1 + p 2 2 + p 2 3 , that is, the magnitude of momentum. * ) In (2.7), the color and flavor are locked due to V CFL . Namely, the combination cp 1u c p2d − cp 1d c p2u , cp 2d c p3s − cp 2s c p3d and cp 3s c p1u − cp 3u c p1s appear in the CFL condensate ∆, in which we define (2.8) The symmetry su(3) CFL remains because the symmetry breaking pattern is su Color-flavor locked phase without spin polarization and spin polarized phase without color-flavor-locking

Color-flavor locked phase without spin polarization
Let us consider the case F k = 0, which leads to the color superconductor without spin polarization. 10), 11) The Hamiltonian is expressed as Hereafter, we use an abbreviated notation p = (p, η) andp = (−p, η). The commutation relations are calculated as Thus, we find φp = −φp with p = (p, η) andp = (−p, η). [ where ǫ pu = ǫ pd = ǫ ps = |p|(= ǫ p ). Thus, (1u), (2d) and (3s) become combined with each other. Further, the sets (2u, 1d), (3u, 1s) and (3d, 2s) are mixed each other. First, let us consider the case ε pτ > µ. Thus, for example, we are led to consider new operators such as the following one: Here, we demand that this operator should satisfy the following commutation relation in order to diagonalize the Hamiltonian H eff : Then, X p and Y p are determined from the following equation: As a result, we can derive the following: Similarly, we can introduce new operators and the results are summarized as follows for |p| > µ: Inversely, we can derive the following: As for ǫ p < µ with ǫ p = |p|, we can introduce the new operators as is similar to the case of ǫ p > µ. The new operators,dp ;i , satisfy the diagonalized commutation relations such as where ω 4 = ω 5 = · · · = ω 9 = ω. Then, the new operators can be derived as The following inverse relations are obtained: By using the above operators, we rewrite H eff as is shown in (A . 1) in appendix. Noting X p etc. in (3.7) and (3.10), we finally obtain the following diagonalized many-body Hamiltonian without spin polarization: Next, let us derive and solve the gap equation for CFL phase with F 3 = F 8 = 0 to obtain ∆. The vacuum state is written as |Φ which is a vacuum with respect to the quasi-particle operators d p;a andd p;a : d p;a |Φ =d p;a |Φ = 0 . (3.13) Thus, the thermodynamic potential Φ 0 can be expressed as (3.14) The gap equation, ∂Φ 0 /∂∆ = 0, can be expressed as where Λ represents a three-momentum cutoff. Here, the helicity η = ±1 is considered which gives a factor 2. Namely, the above gap equation with ∆ = 0 is written as where p = |p|. Solving (3.16) with respect to ∆ and substituting its solution into (3.14), the thermodynamic potential (3.14) is obtained: (3.17) The right-hand sides of Eqs.(3.16) and (3.17) can be analytically expressed by using the following formulae: Of course, if ∆ = 0, the thermodynamic potential is given by 3.2. Spin polarized phase without color superconducting gap ∆ In this subsection, we derive the thermodynamic potential with ∆ = 0. In this case, it is only necessary to diagonalize the Hamiltonian matrix (2.6), namely, where q = p, e τ = F τ p 2 1 + p 2 2 /p and g τ = F τ p 3 /p. The eigenvalues of κ are easily obtained as where F τ are defined in (2.5) or (2.2). Since the Hamiltonian is diagonalized, the thermodynamic potential with ∆ = 0, which is written as Φ F , can be easily obtained as where the factor 3 represents the degree of freedom of color. Here, the sum with respect to the momentum should be replaced to the integration of momentum: (1/V )· p → d 3 p. Hereafter, we assume |F τ | < µ because we are interested in the phase transition from CFL phase to SP phase. In the case 0 ≤ F τ < µ, the ranges of integration are obtained as In the case −µ < F τ ≤ 0, the ranges of integration are obtained as for η = −1, Regardless of the sign of F τ , positive or negative, F τ is regarded as the absolute value of F τ since we consider both η = 1 and η = −1.
The simultaneous gap equations with respect to F 3 and F 8 are obtained through ∂Φ F /∂F 3 = 0 and ∂Φ F /∂F 8 = 0 as Let us calculate the thermodynamic potential Φ 0 with F k = 0 in (3.14) and Φ F with ∆ = 0 in (3.22) or (3.25) numerically. If Φ F < Φ 0 , the SP phase is realized. However, in rather smaller µ, the CFL phase should be realized. The numerical results are summarized in Table I. We adopt the parameters as Λ = 0.631 GeV, G c = 6.6 GeV −2 and G = 20 GeV −2 . As is seen in Table I, in the region of µ ≤ 0.45 GeV, Φ 0 < Φ F is satisfied and the CFL phase is realized. At µ = 0.4558 GeV (= µ c ), Φ 0 ≈ Φ F is satisfied. Thus, the phase transition may occur. In the region of µ ≥ 0.46 GeV, Φ 0 > Φ F is satisfied and the realized phase is the SP phase.
As for the solutions of the simultaneous gap equations in (3.26), it seems that a relation F 3 ≈ √ 3F 8 may be satisfied.
As for another case, for example, F 3 = 0 and F 8 = 0, then F u = F d and F s = −2F u are satisfied. So, another local minimum may be obtained at (F 3 = 0, F 8 ).
Of course, the pressure p A can be expressed by the thermodynamic potential Φ A through the thermodynamical relation: (4.1) In Fig.1, the pressures of the CFL phase and the SP phase are depicted as a function of the chemical potential µ with a comparison of that of the free quark matter. In µ < µ c (µ > µ c ), the pressure of CFL phase is larger (smaller) than that of SP phase. Thus, the realized phase is CFL (SP) phase. Also, the quark number density ρ q can be derived from the thermodynamic potential by thermodynamical relation: The baryon number density ρ B can be expressed as ρ B = ρ q /3. In Table II, we summarize the baryon density and baryon density divided by the normal nuclear density ρ 0 = 0.17 fm −3 . Table I [GeV ] Finally, let us estimate the spin polarization. Here, we consider the helicity instead of spin. The quark number density for the flavor τ and helicity η can be derived as where the factor 3 means color degree of freedom. First, in the case of F τ being positive and 0 < F τ < µ, the particle number density with helicity ±, n (±) τ > , can be calculated as On the other hand, in the case −µ < F τ < 0, the particle number density n (η) τ < is calculated as (4.5) For the case F τ > µ , there is no contribution to the spin polarization from ǫ (+) pτ due to integration range. Thus, we obtain only n It should be here noted that the fermi surface has a form of torus in the case F τ > µ, namely, ( p 2 1 + p 2 2 − F τ ) 2 + p 2 3 = µ 2 , whose volume is obtained as 2πF τ × πµ 2 where F τ and µ correspond to the major and minor radius of torus. For the case F τ < −µ, as is similar to Eq.(4.6), we obtain the following: Next, let us consider the case |F τ | < µ. The spin polarization per unit volume is obtained by the difference between the quark number with the plus helicity and that with minus helicity, as is shown in Fig.2. Further, we assume that F u > 0, F d < 0 and F s < 0. The spin polarization of each flavor, S τ , can be expressed as Similarly, Of course, the total spin polarization S is written as (4.10) Figure 3 shows the numerical results. For d-and s-quarks, the spin polarization has almost the same magnitude. On the other hand, for u-quark, the spin polarization is opposite to d-and s-quarks. The total spin polarization is nearly equal to be zero. The reason is as follows: From the numerical calculation, the condensate F 3 and F 8 satisfy the relation F 3 ≈ √ 3F 8 which makes the thermodynamic potential be minimum. Under this relation, F u = −2F d and F d = F s are satisfied. Thus, from (4.8) ∼ (4.10), S = 0 is derived. §5. The order of phase transition and the second order perturbation with respect to V SP In order to investigate the order of phase transition from CFL phase to SP phase, we treat the interaction term V SP in the perturbation theory on the vacuum of the CFL phase. We neglect the contribution of negative-energy particle represented by (c pτ α ,c † pτ α ). Thus, in V SP in (2.7), it is enough to pick up only the following term: The correction of the first-order perturbation with respect to H 1 , namely Φ|H 1 |Φ , vanishes because the helicity η is different from each other like Φ|d pη;a d † p−η;a |Φ = 0, which is led from Φ|c † pητ α c p−ητ α |Φ . Thus, we need to calculate the second-order perturbation with respect to H 1 . The correction of energy can be expressed as where |i , E 0 and E i represent the excited state, the vacuum energy and the excited energies, respectively. The term H 1 includes c † pηατ c p−ηατ . For example, in the case ǫ p > µ with τ = u, the necessary term can be expressed in terms of the quasiparticle 3) Therefore, the intermediate states |i are needed: Here, we remember d pη;4 = d pη2u and so on. We summarize the necessary pieces to calculate the second order perturbative correction of energy for ǫ p > µ in Appendix B. Thus, for ǫ p > µ, the second order perturbative correction, E > corr , for the energy can be expressed as As for the case ǫ p < µ, the same results are derived. The intermediate states are adopted as Then, the structure of operators in H 1 for ǫ p < µ is the same as that of the case for ǫ p > µ. Thus, the correction energy, E < corr , for ǫ p < µ has the same form: Finally, we obtain the correction energy by the second-order perturbation as Here, by using x p ,X p ,Ȳ p in (3.8), e τ in (5.1) and F τ in (2.2), and by substituting the above quantities into E corr , we obtain Then, the thermodynamic potential Φ can be obtained up to the second order of V SP , namely, up to the second order of F k as The order of the phase transition form CFL to SP phases is determined through (5.9). Namely, from (5.9) and (5.10), for CFL phase, but with small F 3 and/or F 8 , the thermodynamic potential Φ is obtained: up to the second order of F k . Here, c 3 and c 8 are expressed as , 12 .

(5.12)
If c 3 + 1/(2G) > 0 and c 8 + 1/(2G) > 0, the phase transition from CFL to SP phases is of the first order because F 3 = F 8 = 0 always gives a local minimum of the thermodynamic potential Φ 0 . On the other hand, if c 3 + 1/(2G) > 0 and c 8 + 1/(2G) < 0 and vice versa, or c 3 + 1/(2G) < 0 and c 8 + 1/(2G) < 0, the phase transition from CFL to SP phases is maybe of the second order. As is seen in Table III, in the region of µ ≤ 0.49 GeV, especially, at µ = µ c (= 0.4558 GeV), the coefficients c 3 + 1/(2G) and c 8 + 1/(2G) are positive. Thus, the phase transition may be of the first order. From the above consideration, in Fig.4, the pressure is depicted as a function of the baryon number density divided by normal nuclear density, which has already given in Table II. The realized phase is represented by solid curve. §6. Summary and concluding remarks In this paper, it has been shown that the quark spin polarization for each flavor may occur in three-flavor case at high baryon density, which leads to the quark spin polarized phase, against the color-flavor locked phase due to the four-point tensor-type interaction between quarks in the Nambu-Jona-Lasinio model. In a certain region of quark chemical potential, the CFL phase is favorable energetically. However, as the quark chemical potential increases, the phase transition from CFL phase to spin polarized phase occurs. In our theoretical model, the phase transition occurs around the quark chemical potential being around 0.45 GeV under a certain parameter set. Based on the CFL phase, we have treated the tensor-type interaction term between quarks as a perturbation one. As a results, it has been shown that the phase transition may be of the first order up to the second-order perturbation.
In this paper, it has been shown that, owing to the four-point tensor-type interaction, the spin polarization may occur in the NJL model. The tensor-type interaction may come from the two-gluon exchange term between quarks in QCD. It is interesting to clarify the origin of the tensor-type interaction. In this paper, the total spin polarization is not realized, while the spin polarization with respect to each flavor actually occurs. It may be important to introduce the quark-mass splitting, namely, the strange quark mass should be taken into account, while we have ignored it in this paper. Further, if the chiral symmetry is explicitly broken, namely, the quark masses are not zero even in the chiral symmetric phase, the spin polarization originated from the pseudovector-type four-point interaction between quarks may exist. One of next interesting problems may be to investigate the interplay between the spin polarization from tensor-type interaction and that from the pseudovectortype interaction. As for a realistic calculation of the inner core of neutron stars or quark stars, the charge neutrality condition is important 12) and then, the chemical potential of each quark flavor should be introduced. This is one of future problems we consider, while we expect that the spin polarized phase may appear by the effects developed in this paper. Further, it is interesting to investigate the origin of strong magnetic field, 13) for example, in the core of neutron stars. It is suggested that, in general, the spin polarization leads to the ferromagnetization. In the core of neutron stars, it is expected that there exists the strong magnetic field. Then, if the high density quark matter is realized in the inner core of neutron stars or quark stars, the spin polarization may occur as is shown in this paper. Thus, one direction of investigations in high density quark matter is to understand whether the quark ferromagnetization is realized or not in the quark spin polarized phase. This will be one thing to solve.