Interplay between Spin Polarization and Color Superconductivity in High Density Quark Matter

Here, it is suggested that a four-point interaction of the tensor type may lead to spin polarization in quark matter at high density. It is found that the two-flavor superconducting phase and the spin polarized phase correspond to distinct local minima of a certain generalized thermodynamical potential. It follows that the transition from one to the other phase occurs passing through true minima with both a spin polarization and a color superconducting gap. It is shown that the quark spin polarized phase is realized at rather high density, while the two-flavor color superconducting phase is realized at a lower density region.


§1. Introduction
Under extreme conditions, such as high temperature and/or high baryon density, it is interesting to study the behavior of quark and gluonic matter and/or hadronic matter governed by quantum chromodynamics (QCD) in the context of the physics of the relativistic heavy ion collisions and of the interior of compact stars. At low temperature and high density region 1) such as the interior of a neutron star, it is widely investigated and pointed out that various phases may be realized, for example, various meson condensed phases in hadronic matter, 2) the two-flavor color superconducting (2SC) and the color-flavor locked (CFL) phases, 3)-5) the quarkyonic phase 6) and the quark ferromagnetic phase 7) in quark matter, and so forth. Especially, we may conjecture that the quark ferromagnetic phase exists at high density quark matter, since the existence of magnetar was reported. 8) In a preceding work, 7) the pseudovector-type interaction between quarks was considered and it was pointed out that the quark spin polarization was realized. However, it was shown that, the quark spin alignment disappears, if the quark mass is zero, for example, in the chiral symmetric phase. 9) Thus, it is interesting to investigate whether there is a possibility of spontaneous spin polarization under another interaction between quarks.
Recently, the present authors have indicated that the quark spin polarization may occur at high density quark matter even in the chiral symmetric phase due to the tensor-type four-point interaction between quarks, which leads to the quark spin polarization. 10), 11) In that paper, it has been shown that a second-order phase typeset using PTPT E X.cls Ver.0.9 transition occurs from normal quark phase to quark spin polarized phase. However, in quark matter at high baryon density, it is believed that the two-flavor color superconducting phase appears. Thus, it is interesting to investigate the stability of the quark spin polarized phase against the 2SC phase.
In this paper, we investigate the color-symmetric superconducting phase 12) and the quark spin polarized phase and clarify which phase is stable by using the Nambu-Jona-Lasinio (NJL) model 13) including quark-pair interaction 14) and the tensor-type four-point interaction. 10), 11) The quark-pair interaction is derived by the Fierz transformation from the original NJL model Lagrangian. As for the color superconducting phase, many possibilities, such as the gapless superconducting phase which is still controversial for realistic situations, have been considered. However, in this paper, the simple 2SC phase derived by the NJL model is treated. We thus consider the quark-pairing gap ∆ and the quark spin polarization F such that ∆ = 0 and/or F = 0 leads to the 2SC phase and/or quark spin polarized phase, respectively. In this paper, the system is treated by the mean field approximation and the BCS state is introduced. 15) This paper is organized as follows: In the next section, the basic Hamiltonian is introduced and expressions of the Hamiltonian under some basis sets are given. In Appendix A, one of the expressions of the Hamiltonian is given in which good helicity states are used. In §3, the BCS state is defined, where detail derivation is given in Appendix B, and the thermodynamic potential is derived. In §4, numerical results and discussions are given together with Appendix C and the last section is devoted to a summary and concluding remarks. §2. NJL model with tensor interaction

Basic Hamiltonian
Let us start with the following Lagrangian density: Here, ψ C = Cψ T with C = iγ 2 γ 0 being the charge conjugation operator. Also, τ 2 is the second component of the Pauli matrices representing the isospin su(2)-generator and λ A are the antisymmetric Gell-Mann matrices representing the color su(3) cgenerator. As is well known, the Lagrangian density L 0 + L S corresponds to the original NJL model. We add L c which can be derived by the Fierz transformation from L S and represents the quark-pair interaction. Also, we introduce L T which represents the tensor-type interaction between quarks.
In this paper, we concentrate on quark matter at high baryon density where the chiral symmetry is restored in the density region considered here. Thus, the chiral condensate, ψ ψ , is assumed to be equal to zero in this high density region. Under the mean field approximation, the above Lagrangian density is recast into where h.c. represents the Hermitian conjugate term of the preceding one. Here, we used 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 value F corresponds to the order parameter of the spin alignment which leads to quark ferromagnetization. The expectation value ∆ corresponds to the quark-pair condensate which means the existence of the color superconducting phase if ∆ = 0. Here, in order to insure color symmetry, we assume that all quark-pair condensates have the same expectation values, ∆ 2 = ∆ 5 = ∆ 7 . The mean field Hamiltonian density with quark chemical potential µ is easily obtained as In the Dirac representation for the Dirac gamma matrices, the Hamiltonian matrix of the spin polarization part, where α i = γ 0 γ i and β = γ 0 . For good helicity states, this Hamiltonian matrix is diagonalized with F = 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: 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 = ±, isospin index τ = ± and color α. Further, φ τ = 1 for τ = 1 (up quark) and φ τ = −1 for τ = −1 (down quark). Also, ǫ τ τ ′ and ǫ αα ′ α ′′ represent the complete antisymmetric tensor for the isospin and color indices. We define p = p 2 1 + p 2 2 + p 2 3 , that is, the magnitude of momentum.
For later convenience, we rewrite the above Hamiltonian as As for V cs , we can easily sum up with respect to τ and τ ′ which leads to (2 . 9)

Expressions of the Hamiltonian under other basis sets
If the quark-pair condensate ∆ is equal to zero, the operator given by equation . Since the quark number N is already diagonal, let us diagonalize the Hamiltonian matrix κ. First, we set up τ 3 = 1 because the contribution of τ 3 = −1 is the same as the contribution of τ 3 = 1 as was seen in (2 . 9), which results in a factor 2. Then, the Hamiltonian matrix κ can be simply expressed as where q = p, e = F p 2 1 + p 2 2 /p and g = F p 3 /p. The eigenvalues of κ are easily obtained as where the operators V (p) and W are given in the Appendix A, the mean field Hamiltonian in which both quark spin polarization and quark-pair condensate are simultaneously considered can be expressed as Here, the summation with respect to isospin indices τ and τ ′ is explicitly rewritten. The above mean field Hamiltonian is the starting point for discussing the quark spin polarized phase and color superconducting phase. §3. BCS state and thermodynamic potential We introduce the BCS state following Ref.15): where |0 is the vacuum state with respect to a pητ α and K pη = K −pη and K pη = K −pη are satisfied. Here, the contribution of quark-pairing with negative energy is not considered in quark matter. Then, the state |Ψ is a vacuum state with respect to new operators d pητ α : where (α, β, γ) is a cyclic permutation for color indices and τ ′ = −τ . For the above operators, the state |Ψ satisfies the following relation: The state |Ψ is identical with the BCS state. In order to calculate the expectation value of the mean field Hamiltonian (2 . 13), it is necessary to obtain the expectation values of a † pητ α a pητ α , a † pητ α a † −pητ ′ α ′ and so on with respect to the BCS state |Ψ . The detailed calculations are given in Appendix B. Thus, by summing up with respect to color and isospin indices, we obtain the expectation value of the mean field Hamiltonian for the BCS state as In order to determine the BCS state, namely, to determine the variational variables K pη and K pη , we introduce new variational variables θ pη andθ pη instead of K pη and K pη : Thus, the Hamiltonian and color superconducting gap ∆ can be expressed in terms of θ pη andθ pη as Here, in the first line in (3 . 6) and the third line in (3 . 7), the isospin indices are summed up and an extra factor 2 appears with respect to the corresponding expression in 12). Also, in the second line in (3 . 7), color indices, β and γ, are summed up and an extra factor 2 also appears.
Next, we impose the minimization condition for Ψ |H M F − µN |Ψ with respect to ∆, θ pη andθ pη . First, we impose the minimization condition with respect to ∆: This minimization condition leads to the gap equation in (3 . 7) exactly. Secondly, we minimize Ψ |H M F − µN |Ψ with respect to θ pη andθ pη : which lead to the following equations: (3 . 10) Thus, we obtain tan 2θ Since the variational parameters θ pη andθ pη are determined completely, namely, K pη and K pη are determined, the BCS state is obtained.
Thus, we can derive the thermodynamic potential Φ(∆, F, µ) at zero temperature from (3 . 6) with (3 . 11) as where ε . Here, we explicitly introduce and write a three momentum cutoff parameter Λ. The above derivation is nothing but that of the usual method in the BCS theory. Therefore, it is possible to derive the same result using the equations of motion by eliminating so-called dangerous terms in the algebraic method. 15) It may be interesting to observe that Eq.(3 . 12) means, physically, that only two "effective colors" participate in the pairing process, the third "color" remaining inert. The contribution of the active "colors" to the energy p < µ and for ε (η) p > µ, while the contribution of the inert "color" is simply ε In this section, first, we investigate the possible phases with F = 0 and ∆ = 0 or with F = 0 and ∆ = 0 at high density, namely, two-flavor color superconducting phase or spin polarized phase, separately. First, as an extreme situation in the quark spin polarized phase with ∆ = 0, the thermodynamic potential reduces to where it should be noted that (ε p < µ, respectively, when (4 . 1) is derived from (3 . 12). Of course, the sum over momentum is replaced by the momentum-integration as Then, the thermodynamic potential derived here is identical with the one which has been previously obtained by the present authors, that is, Eq.(3.1) in Ref. 10). * ) Thus, the thermodynamic potential with ∆ = 0 has already been given in 10) in the analytical form as If the minimum of the thermodynamical potential exists in the range of F < µ, the gap equation for F is derived from ∂Φ(∆ = 0, F, µ)/∂F = 0 which leads to In the other case, namely F = 0 case, the two-flavor color superconducting phase may be realized with ∆ = 0. When F = 0, the quasiparticle energy ε (η) p = p is obtained for η = ±. The thermodynamic potential (3 . 12) is then evaluated as After replacement of the sum over momentum into momentum-integration in (4 . 2), the analytical form of the thermodynamic potential is obtained with a three-momentum cutoff parameter Λ: The gap equation becomes The quark number density ρ is calculated from the thermodynamical relation as In our model Hamiltonian, the model parameters are G, G c and Λ. In the original NJL model, the coupling strength G S appears where G S = 5.5 GeV −2 is adopted 16) although this parameter does not appear explicitly in the model considered here. Then, following Ref.14), the strength of the quark-pair interaction, G c , is taken * ) as (G c /2)/G S = 0.6. Thus, we adopt G c = 6.6 GeV −2 . As for the three-momentum cutoff Λ, a standard value 16) is adopted here, namely, Λ = 0.631 GeV. If the tail stretching beyond the Fermi momentum of occupation number should be fully taken into account in the BCS theory, a larger value of three-momentum cutoff should be adopted such as Λ = 0.8 GeV. 15) However, it will be later seen that the tail of the occupation number is already sufficiently taken into account in the case Λ = 0.631 GeV. As for G in the strength of the tensor-type interaction between quarks, we put G = 20 GeV −2 , which was used in our previous paper. 10) As was discussed in Ref. 10), if the effect of the vacuum polarization is taken into account, the coupling constant G should be replaced to the renormalized coupling G r in which 1/G r = 1/G − Λ 2 /π 2 . Then, G r = 20 GeV −2 corresponds to the bare coupling G = 11.1 GeV −2 for Λ = 0.631 GeV or G = 8.7 GeV −2 for Λ = 0.8 GeV. Thus, the strengths of the quark-pairing and tensor-type interactions are comparable. The parameter set used here is summarized in Table 1.
First, let us estimate the thermodynamic potential numerically in two phases, namely, color superconducting phase with F = 0 and ∆ = 0 and quark spin polarized phase with F = 0 and ∆ = 0, separately. The state with (F = 0, ∆ = ∆ 0 ) gives a local minimum of the thermodynamic potential, where ∆ 0 is the solution of gap equation (3 . 13) with F = 0 or Eq.(4 . 7). However, the state with (F = F 0 , ∆ = 0) gives a local minimum or a saddle point of the thermodynamic potential corresponding to the value of µ, where F 0 is the solution of gap equation ∂Φ(∆ = 0, F, µ)/∂F = 0 or Eq.(4 . 4). This is shown in Appendix C. We compare the pressure in the color superconducting phase (F = 0, ∆ = ∆ 0 ) with the one in the quark spin polarized phase (F = F 0 , ∆ = 0). The pressure p is given by (4 . 9) In Fig.1 (a), the pressures for normal (thin curve), two-flavor color superconducting (dash-dotted curve) and quark spin polarized (solid curve) phases are shown in the case of Λ = 0.631 GeV as functions with respect to quark chemical potential µ.
, the two-flavor color superconducting (2SC) phase is realized. However, above µ = µ c , the quark spin polarized phase is favored. In Fig. 1 (b), details are depicted around µ ≈ 0.442(= µ c ) GeV. As for the baryon number density, up to ρ B = 4.73ρ 0 , where ρ 0 = 0.17 fm −3 is the normal nuclear matter density, the color superconducting phase is realized. Thus, it is enough to include the effects of the tail of occupation number with the three-momentum cutoff Λ = 0.631 GeV. In Fig.2, the occupation number is depicted as a function of the magnitude of momentum p(= |p|). It seems from Fig.2 that the effects of the tail of occupation number are fully taken into account. Secondly, let us consider the gap ∆ and F simultaneously. Then, up to µ ≈ 0.40 GeV, the color superconducting phase is realized. In the region with µ > ∼ 0.407 GeV which corresponds to the baryon number density ρ divided by the normal nuclear matter density ρ 0 being ρ/ρ 0 = 3.71, the gap equation ∂Φ(∆ = 0, F, µ)/∂F has a nontrivial solution with non-zero F value. Then, the true minimum of the thermodynamic potential moves to the point (F = 0, ∆ ≈ ∆ 0 ) from the point (F = 0, ∆ = ∆ 0 ) as is seen in Fig.3(b). In Fig.3 (c) with µ = µ c = 0.442 GeV, in which Φ(∆ = ∆ 0 , F = 0, µ = µ c ) = Φ(∆ = 0, F = F 0 , µ = µ c ), the true minimum is located at (F ≈ 0.8F 0 , ∆ ≈ 0.8∆ 0 ). This state may be interpreted as a state in a mixed phase, in the sense that this phase is intermediate between a pure spin polarized phase and a pure 2SC one. Further, in the case (d) µ = 0.45 GeV, the true minimum is located at (F ≈ 0.8F 0 , ∆ ≈ 0.7∆ 0 ). In the cases (e) and (f), namely µ > ∼ 0.46 GeV, the point (F = F 0 , ∆ = 0) becomes a true minimum and the spin polarized phase is realized. However, in the case (e), two minima appear. As is seen in the case (e), it is most likely that a jump occurs from (F = F 0 , ∆ = ∆ 0 ) to (F = F 0 , ∆ = 0). Thus, starting from 2SC phase, the second order phase transition starts at the onset of the spin polarization F = F 0 , and finally it seems that the transition occurs from intermediate phase to spin polarized phase. §5. Summary and concluding remarks In this paper, it was shown within the mean field approximation that the quark spin polarized phase may be realized after the two-flavor color superconducting phase and the mixed phase as the baryon density increases. We have first calculated the pressure of the two phases separately, namely, in the 2SC (∆ = 0 and F = 0) and the quark spin polarized (∆ = 0 and F = 0) phases. As a result, at a certain lower density region, the 2SC phase is realized. However, at rather high density region, the quark spin polarized phase is realized. If the large F expansion is carried out in Eq.(3 . 13), then the gap equation for ∆ is expressed approximately as For large F , the above equation has only the trivial solution, ∆ = 0. Thus, it is expected actually that the quark spin polarized phase with ∆ = 0 and F = 0 is realized in the high baryon density region with large F . The situation is the same even if another parameter is reasonably adopted. For example, when we adopt G = 10 GeV −2 which is one-half compared with one used in this paper, the behavior of the phase transition is similar in the case of G = 20 GeV −2 except for the value of chemical potential µ = µ c = 0.605 GeV where Φ(∆ = ∆ 0 , F = 0, µ c ) = Φ(∆ = 0, F = F 0 , µ c ) is satisfied. But since µ c = 0.605 GeV is close to three momentum cutoff Λ, it may be impossible to take conclusions about the phase transition definitely. It might be thought that, as a result, the spin polarized (SP) phase may be more favorable than the two-flavor color superconducting (2SC) phase at high density. It should be noted that the thermodynamic potential can be reexpressed as Φ = − µ 0 dµ N , which is proportional to −µ× (volume surrounded by the Fermi surface). In the quark matter with F = 0 at low baryon density or small quark chemical potential, the thermodynamic potential depends on µ 4 (= µ × µ 3 ), namely Φ ∝ −µ 4 , in which µ 3 is nothing but the volume surrounded by the Fermi surface because the shape of the Fermi surface is sphere. However, at very high baryon density or large quark chemical potential, in which the relation F > µ is satisfied, the shape of Fermi surface becomes torus, namely, ( Here, the volume of torus is 2π 2 µ 2 × F , where µ is smaller radius and F is the other larger radius of torus. In the case F > µ, namely full polarization case, from Eq.(4 . 3), F = (G/(2π)) × µ 3 gives the minimum of the thermodynamic potential. As a result, the thermodynamic potential is proportional to −µ× (volume surrounded by the Fermi surface) ∝ −µ × µ 2 × F ∝ −µ 6 as was shown in our previous paper. 10) Thus, at high baryon density or large quark chemical potential, the spin polarized phase is favored. It might be concluded that the mechanism which makes SP phase more favored than 2SC phase at high densities is the distortion effect of the Fermi surface.
For three-momentum cutoff Λ = 0.631 GeV, it is sufficient to take into account the tail of the occupation number since the values of the chemical potential on the phase transition point from 2SC phase to the mixed phase and from the mixed phase to the spin polarized phase are about 0.407 GeV and 0.46 GeV, respectively. In general in the system with finite chemical potential, a chemical-potential dependent cutoff may be used. 17) However, for the sake of comparison, if a rather large fixed value of the three-momentum cutoff, Λ = 0.8 GeV, is adopted where the tail of occupation number is more fully taken into account, the critical chemical potential from 2SC phase to the mixed phase is unchanged. Above µ = µ c = 0.491 GeV, the state with (F = F 0 , ∆ = 0) is favored against the state with (F = 0, ∆ = ∆ 0 ). With this larger fixed three-momentum cutoff, it is concluded that, up to ρ B = 7.42ρ 0 , the color superconducting phase is realized. The use of large three-momentum cutoff leads to the shift of the phase transition density, that is, the phase transition density is higher with larger three-momentum cutoff. However, if a larger Λ is used, it is necessary to re-adjust the coupling constants. Secondly, the quark-pairing gap ∆ and the spin polarization F were considered simultaneously. Under the present treatment, the behavior of phase transition from 2SC phase with ∆ = 0 and F = 0 to the quark spin polarized phase with ∆ = 0 and F = 0 may be clear and the transition occurs passing through the state with (F = 0, ∆ = 0) which can be interpreted as the mixed phase.
In a future work, it is interesting and important to investigate the phase equilibrium at finite temperature and in the presence of an external magnetic field. Also, matter in beta equilibrium will be analyzed, while symmetric matter is considered in the present paper. Also, it is important to investigate the interplay between color-flavor locked (CFL) phase and the spin polarized phase at higher density region in quark matter. Further, it may be interesting to investigate the interplay between two quark spin polarized phases, namely, the quark spin polarized phase originated from the tensor-type four-point interaction and that originated from the pseudovector-type four-point interaction between quarks. In this paper, the mean field approximation for the established BCS theory is adopted under which the thermodynamic potential is considered in the standard manner. Of course, there are other approaches to investigate the phase transition, such as by the use of the CJT potential, 18) the Landau potential 19) and so on. It would be interesting to clarify the difference between the standard BCS mean field approach and the CJT potential approach including the effects of two-loop order as a general problem of the phase transition phenomena. However, that is beyond the scope of the present paper. These topics are left for future investigations.
Then, we can diagonalize the Hamiltonian matrix κ as V † κV . Namely, By introducing new fermion operators (a pητ α , a † pητ α ,ã pητ α ,ã † pητ α ) by  the mean field Hamiltonian in which both quark spin polarization and quark-pair condensate are simultaneously considered can be expressed in Eq.(2 . 13).

(B . 2)
In the same way, for ε (η) p ≤ µ, the following relations are also obtained: Next, let us consider the state with (∆ = 0, F 0 ). The stability matrix M(∆ = 0, F 0 ) is also a diagonal matrix. We easily obtain the inequality As for the other matrix element, ∂ 2 Φ(∆ = 0, F 0 , µ)/∂∆ 2 , numerical estimation, just before and after the point µ = µ c , may also be useful. However, the integral diverges at ε (η) p = µ. This is not surprising, in view of a well-known "anomaly" of the BCS theory, according to which a perturbation expansion in powers of the coupling constant G is not valid, even if this parameter is infinitesimally small. As a result, the thermodynamical potential cannot be expanded in powers of ∆, even if this parameter is infinitesimally small. We find that ∂Φ(∆ = 0, F 0 , µ)/∂∆ = 0, but the second derivative diverges. We wish to see if, in the ∆ direction, the extremum point (∆ = 0, F 0 ) is a maximum or a minimum. Figure 3 shows that the point with (∆ = 0, F 0 ) is a minimum point in the F direction, but is a maximum point in the ∆ direction in the region µ < 0.46 GeV. Thus, in this region, a point (∆ = 0, F 0 ) is a saddle point. However, the case µ > ∼ 0.46 GeV, the extremum point with (∆ = 0, F 0 ) is a minimum point in the ∆ direction. Thus, it is concluded that the point (∆ = 0, F 0 ) gives a true minimum of the thermodynamic potential Φ, which results the quark spin polarized phase.