Quark spin polarization and spontaneous magnetization in high density quark matter

Transcription

1 Quark sin olarization and sontaneous magnetization in high density quark matter Physics Division, Faculty of Science, Kochi University, Kochi , Jaan João da Providência and Constança Providência CFisUC, Deartamento de Física, Universidade de Coimbra, Coimbra, Portugal Hiroaki Matsuoka Graduate School of Integrated Arts and Science, Kochi University, Kochi , Jaan Masatoshi Yamamura Deartment of Pure and Alied Physics, Faculty of Engineering Science, Kansai University, Suita , Jaan Henrik Bohr Deartment of Physics, B.307, Danish Technical University, DK-2800 Lyngby, Denmark By using the Nambu-Jona-Lasinio model with a tensor-tye four-oint interaction between quarks, it is shown that there exists a ossibility of a sin olarized hase in quark matter at finite temerature and density. When there exists the sin olarization, the sontaneous magnetization may occur if the effect of the anomalous magnetic moment of quark is taken into account. An imlication to the comact star objects with strong magnetic field is discussed when the sin olarization occurs. The 26th International Nuclear Physics Conference Setember, 2016 Adelaide, Australia Seaker. c Coyright owned by the author(s under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0. htt://os.sissa.it/

2 Quark sin olarization and sontaneous magnetization 1. Introduction In recent year, one of interests for hysics governed by the quantum chromodynamics (QCD may be to clarify the hase structure with resect to temerature, quark and/or isosin chemical otential, external magnetic field and so on [1]. In the region with low temerature and high baryon density, various hases are exected in quark matter, such as inhomogeneous chiral condensed hase [2], quarkyonic hase [3], color suerconducting hase [4] and so forth. Recently, it has been shown that there is a ossibility of the existence of sin olarized hase by using the Nambu-Jona- Lasinio (NJL model [5], which is regarded as an effective model of QCD [6, 7, 8], with a tensortye four-oint interaction between quarks at zero temerature and finite baryon density [9, 10, 11]. Also, under the external magnetic field, a ossible QCD ground state has been investigated at zero and finite temerature in the case of one flavor [12]. The investigation of a ossible hase structure under the magnetic field is one of interesting subjects about QCD hase structure [13]. The existence of a strong magnetic field is found in certain kinds of comact stars such as neutron star and magnetar [14]. Further, in the ultrarelativistic heavy-ion collision exeriments, it is indicated that a strong magnetic field may be created in the early stage of nucleus-nucleus collisions [15]. In this aer, we show that the sin olarized hase may be exist even at finite temerature in the region of high baryon density in quark matter due to the tensor-tye four-oint interaction between quarks [16]. Further, it is shown that a sontaneous magnetization aears by introducing an external magnetic field, only if sin olarization occurs and the effect of an anomalous magnetic moment of quark is taken into account [17]. 2. Sin olarized hase originated from the tensor-tye four-oint interaction Let us start from the following two-flavor NJL model Lagrangian density with a tensor-tye four-oint interaction between quarks: L = ψiγ ρ ρ ψ + G S { ( ψψ 2 + ( ψiγ 5 τψ 2} G T 4 { } ( ψγ ρ γ σ τψ ( ψγ ρ γ σ τψ + ( ψiγ 5 γ ρ γ σ ψ( ψiγ 5 γ ρ γ σ ψ, (2.1 where ψ reresents the quark field with the two-flavor. Here, the first line reresents the original NJL model Lagrangian density and the second line reresents the tensor-tye interaction introduced in this model. Under a mean field aroximation, the above Lagrangian density is recast into L MFA = ψ ( iγ ρ ρ M ψ F ( ψσ 3 ψ M2 4G S F2 2G T. (2.2 Here, we define the dynamical quark mass M with the chiral condensate ψψ and the sin olarized condensate F, resectively, as M = 2G S ψψ, F = G T ψσ 3 ψ, Σ 3 = iγ 1 γ 2 = ( σ 3 0, (2.3 0 σ 3 1

3 Quark sin olarization and sontaneous magnetization where σ 3 is the third comonent of the Pauli sin matrix. We can easily derive the Hamiltonian density under the mean field aroximation and by diagonalizing the Hamiltonian density, we can also derive the energy eigenvalues or single-article energies of quark easily as E (η = ( M2 + ηf 2, (η = ±1 (2.4 where = ( 1, 2, 3 is momentum. The thermodynamic otential Ω at finite temerature T and quark chemical otential µ can be exressed as where H MFA = H vacuum = n (η = S = E (η,η,τ,α ( Ω = H MFA µ(n P N AP + H vacuum T S, (2.5 n (η E (η,η,τ,α (( 1 + ex [,η,τ,α n (η E (η + n (η + M2 + F2, 4G S 2G T, N P = 1 µ lnn (η n (η,η,τ,α, n (η = /T + (1 n (η, N AP =,η,τ,α 1 (( 1 + ex E (η + µ ln(1 n (η + n (η ln n (η n (η,, /T + (1 n (η ln(1 n (η ]. (2.6 Here, τ and α are indices for isosin and color of quark, resectively. Then, the thermodynamic otential can be reexressed as [ ( ( ( ( ] Ω = E (η + T ln 1 + ex E(η µ + T ln 1 + ex E(η + µ T T,η,τ,α + M2 4G S + F2 2G T. (2.7 When the sum with resect to three-momentum is relaced to the integration, we have to introduce the three-momentum cutoff Λ. Also, the isosin and color degrees of freedom give factor 2 and 3, resectively. As a result, the sum is relaced as following way: where T = ,τ,α 3 π 2 Λ 0 d T Λ 2 2 T 0 d 3 T, Λ/GeV G S /GeV 2 G T /GeV Table 1: Parameters used here. 2

4 Quark sin olarization and sontaneous magnetization Figure 1: The contour ma of the thermodynamic otential is deicted. The horizontal and vertical axes reresent the dynamical quark mass M and the sin olarization F, resectively, with various quark chemical otential µ and temerature T. The darker color reresents the lower value of the thermodynamic otential. We use the arameters in Table 1. The couling constant G S and the three-momentum cutoff Λ are determined by the quark condensate and the ion decay constant in vacuum. However, the couling strength of the tensor-tye interaction cannot be determined by the hysical quantity in vacuum. In this aer, we regard G T as a free arameter. 1 It is shown in Figure 1 that chiral symmetry is broken for small chemical otential and low temerature, namely the minimum oint of the thermodynamic otential has M 0 and F = 0. However, if the chemical otential or temerature becomes high, chiral symmetry is restored, namely, M = 0 and F = 0. In the large chemical otential and low temerature region, the sin olarized condensate F aears without M. However, for higher temerature, it disaears. In Figure 2, the hase diagram in this model is resented on the T -µ lane, where the horizontal and vertical axes reresent the quark chemical otential µ and temerature T, resectively. It is indicated that, for the boundary of the chiral 1 If the term G( ψψ 2 is Fierz-transformed alone, we obtain (G/4( ψψ 2 (G/8( ψγ ρ γ σ τψ 2 + [8]. Thus, if we rewrite the above exression into G S ( ψψ 2 (G T /4( ψγ ρ γ σ τψ 2 +, then G T = 2G S is derived. 3

5 Quark sin olarization and sontaneous magnetization [GeV] M = 0, F = 0 M 0 Figure 2: The hase diagram is shown. The horizontal and vertical axes reresent quark chemical otential and temerature, resectively. condensed and the normal hases, there is a critical endoint for the hase transition near µ = 0.31 GeV and T = GeV. On the other hand, the hase transition from the normal quark hase to the sin olarized hase is always of second order. 3. Sontaneous magnetization In this section, a ossibility of sontaneous magnetization of quark matter is investigated by using the NJL model with the tensor-tye four-oint interaction between quarks. For this urose, from beginning, the effect of the anomalous magnetic moment of quark, µ A, is introduced at the level of the mean field aroximation. Here, we use a fact that the anomalous magnetic moment is exresses as [18] µ A 1 = ( µ u 0, 0 µ d µ u = 1.85 µ N, µ d = 0.97 µ N, [GeV] F 0 µ N = e 2m = GeV/T. (3.1 We introduce the effects of the anomalous magnetic moment in the Lagrangian density within the mean field aroximation. As a result, we obtain the following Lagrangian density : L A = L MFA i 2 ψµ Aγ ρ γ σ F ρσ ψ, (3.2 where F ρσ = ρ A σ σ A ρ and F 12 B z = B. Here, L MFA is nothing but Eq.(2.2 with M = 0 because we consider the large chemical otential region. We only take ρ and σ as ρ = 1, σ = 2 and ρ = 2, σ = 1 because the magnetic field has only z-comonent. Then, in the mean field aroximation, Eq.(3.2 is recast into L A = i ψγ ρ D ρ ψ ψ(f 3 τ 3 + µ A B1Σ 3 ψ F2 2G T = i ψγ ρ D ρ ψ ψ FΣ 3 ψ F2 2G T. (3.3 Here, since F 3 τ 3 = F1 where 1 is the 2 2 identity matrix for the isosin sace, which are denoted in Eq.(2.3, then we introduced the flavor-deendent variables F f as F f = F + µ f B, namely F u = F + µ u B, F d = F + µ d B. (3.4 4

6 Quark sin olarization and sontaneous magnetization Thus, we can derive the thermodynamic otential in the same way develoed in the section 2, excet for the existence of the external magnetic field B. The energy eigenvalues are obtained as E f z,ν,η = ( F f + η 2 Q f Bν, { ν = 0, 1, 2, for η = 1, ν = 1, 2, for η = 1, (3.5 with Q u = 2e/3 and Q d = e/3. Further, because the Landau quantization is carried out with resect to T, the integration with resect to T is relaced to the sum with resect to integers ν. As a result, the thermodynamic otential Φ at zero temerature with M = 0 can be exressed as F d 3 Φ = 3 F 2π Q f B f =u,d;η=± 2π νmax f η [ ] E f z,ν,η µ + F2 + (vacuum term, (3.6 2G ν=ν f η T min where F, ν f η min and ν f η max are determined by the condition E f z,ν,η µ. Keeing in mind that we take B 0 finally in order to derive the sontaneous magnetization through the thermodynamic relation, the sum with resect to the Landau level ν can be relaced into the integration with resect to the continuum variable aν with a small quantity a. A detailed derivation is found in Ref.[17]. After the calculation, the results are summarized as follows: Φ = Φ > = 1 2 F f µ 3 f =u,d 2π Φ = Φ < = f =u,d π 2 F 4 f µ ln + F2 [ µ 2 F f 2 2G + O(B2, for F > µ, ( µ 3 F f µ F f µ 3 F f arctan 4 3 µ 2 F 2 f F f µ 2 F 2 f ]+ F2 2G + O(B2, for 0 < F µ, (3.7 Thus, the sontaneous magnetization er unit volume, M, can be derived thorough the thermodynamic relation, namely, M = Φ/ B B=0 = f =u,d Φ/ F f B=0 F f / B. From (3.7, we can derive the sontaneous magnetization originated form the anomalous magnetic moment µ u, µ d and the sin olarization F: M = Φ > B M = f =u,d = µ3 B=0 4π (µ u + µ d, (3.8 Φ < F f F f B = F 2G (µ u + µ d, (3.9 B=0 where the ga equation Φ/ F = f =u,d Φ/ F f + (F 2 /(2G/ F = 0 is used when Eq.(3.9 is derived. It should be noted that if F = 0, the sontaneous magnetization disaears because F = 0 in Eq.(3.9. In Figure 3, the sin olarization F and the sontaneous magnetization M are deicted as a function of the quark chemical otential µ. For µ < µ c GeV, the magnetization does not occur because the sin olarization does not aear, F = 0. It is shown that, at µ = µ c, the sontaneous magnetization aears connected with the sin olarization F. 5

7 Quark sin olarization and sontaneous magnetization F [GeV] [GeV] M [C/ms] [GeV] (a Sin Polarization : F (b Sontaneous Magnetization : M Figure 3: (a The sin olarization F and (b the sontaneous magnetization er unit volume M are deicted as functions with resect to the quark chemical otential µ. Finally, let us roughly estimate the strength of the magnetic field due to the sontaneous magnetization. If the sontaneous magnetization er unit volume M can be regarded as the magnetic diole moment, we can simly calculate the strength of the magnetic filed yielded by the sontaneous magnetization. As is well known in the classical electromagnetism, the magnetic flux density B(r created by the magnetic diole moment m can be exressed as B = µ 0 4π [ m 3r(m r + r3 r 5 ], (3.10 where µ 0 reresents the vacuum ermeability. In our case, m = (0,0,M V, where V reresents a 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 ole of hybrid star is roughly estimated. Figure 4 (a and (b shows the surface magnetic flux density as a function of the quark chemical otential µ in the case (a r q = 1 km and (b r q = 2.15 km where quark matter occuies (a 0.1 % and (b 1 % of the total volume of star. It is known that the strength of the magnetic field at the surface of magnetar is near G. Thus, in our calculation, if quark matter exists and the sin olarization occurs, a strong magnetic field about or Gauss may be created. B z( = 0 [G] B ( = 0 [G] z r q= 1 km r q= 2.15 km 0 [GeV] 0 [GeV] (a Magnetc flux density (b Figure 4: The z-comonent of the magnetic flux density is shown in a logarithmic scale as a function with resect to the quark chemical otential µ. (a The case that the quark matter occuies 0.1 % in the hybrid comact star and (b the case that the quark matter occuies 1%.. 6

8 Quark sin olarization and sontaneous magnetization 4. Summary In this aer, it has been shown that a ossibility of the sin olarized hase in quark matter at finite temerature and density is indicated by using the NJL model with the tensor-tye four-oint interaction between quarks as an effective model of QCD. When there exists the sin olarization, the sontaneous magnetization may aear if the effect of the anomalous magnetic moment of quark is taken into account. It is interesting to investigate what hase is realized under the external strong magnetic field created by the ultrarelativistic heavy-ion collision exeriments and exected in the inner core of the comact stars such as neutron star and magnetar. The results of this subject will be reorted elsewhere [19]. Acknowledgements One of the authors (Y.T. is artially suorted by the Grants-in-Aid of the Scientific Research (No from the Ministry of Education, Culture, Sorts, Science and Technology in Jaan. References [1] See, for examle, K. Fukushima and T. Hatsuda, Re. Prog. Phys. 74, (2011. [2] M. Buballa and S. Carignano, Prog. Part. Nucl. Phys. 81, 39 (2015, and references cited therein. [3] L. McLerran and R. D. Pisarski, Nucl. Phys. A 796, 83 (2007. [4] M. Alford, A. Schmitt, K. Rajagoal and T. Schafer, Rev. Mod. Phys. 80, 1455 (2008. [5] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961; ibid 124, 246 (1961. [6] S. P. Klevansky, Rev. Mod. Phys. 64, No. 3, 649 (1992. [7] T. Hatsuda and T. Kunihiro, Phys. Re. 247, 221 (1994. [8] M. Buballa, Phys. Re. 407, 205 (2005. [9] H. Bohr, P. K. Panda, C. Providência and J. da Providência, Int. J. Mod. Phys. E 22, (2013. [10] Y. Tsue, J. da Providência, C. Providência and M. Yamamura, Prog. Theor. Phys, 128, 507 (2012. [11] Y. Tsue, J. da Providência, C. Providência, M. Yamamura and H. Bohr, Prog. Theor. Ex. Phys. 2013, 103D01 (2013; ibid 2015, 013D02 (2015. [12] E. J. Ferrer, V. de la Incera, I. Portillo and M. Quiroz, Phys. Rev. D 89, (2014. [13] J. O. Andersen and W. R. Naylor, Rev. Mod. Phys. 88, (2016, and references cited therein. [14] A. K. Harding and D. Lai, Ret. Prog. Phys. 69, 2631 (2006, and references cited therein. [15] D. E. Kharzeev, L. D. McLerran and H. J. Warringa, Nucl. Phys. A 803, 227 (2008. [16] H. Matsuoka, Y. Tsue, J. da Providência, C. Providência, M. Yamamura and H. Bohr, Prog. Theor. Ex. Phys. 2016, 053D02 (2016. [17] Y. Tsue, J. da Providência, C. Providência, M. Yamamura and H. Bohr, Prog. Theor. Ex. Phys. 2015, 103D01 (2015. [18] R. G. Felie, A. P. Martinez, H. P. Rojas and M. Orsaria, Phys. Rev. C 77, (2008. [19] Y. Tsue, J. da Providência, C. Providência, M. Yamamura and H. Bohr, Int. J. Mod. Phys. E 25, (