Possibility of hadron-quark coexistence in massive neutron stars

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1 Possibility of hadron-quark coexistence in massive neutron stars Tsuyoshi Miyatsu Department of Physics, Soongsil University, Korea July 17, 2015 Nuclear-Astrophysics: Theory and Experiments on nd HaPhy POSTECH, Pohang T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 1/26

2 Table of contents 1 Introduction 2 Uniform hadronic matter with hyperons using relativistic Hartree-Fock approximation in SU(3) flavor symmetry 3 Quark effect on properties of neutron stars (hybrid stars) 4 Summary T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 2/26

3 Introduction T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 3/26

4 Nuclear physics and astrophysics Many-body calc. Equation of state for nuclear matter TOV Neutron-star properties by Lattimer. T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 4/26

5 Nuclear physics and astrophysics Many-body calc. Equation of state for nuclear matter M theory > M measured TOV Neutron-star properties by Lattimer. T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 4/26

6 Properties of neutron stars Tolman-Oppenheimer-Volkoff (TOV) equation for hydrostatic equilibrium of a spherical object in general relativity (GR): P (MeV/fm 3 ) dp (R) = G[P (R) + ϵ(r)][m(r) + 4πR3 P (R)], dr R[R 2GM(R)] R ϵ: energy density M(R) = 4πr 2 ϵ(r)dr, P : pressure 0 G: gravitational constant P = ε Mass (M O ) M/M O 2.0 neutron star white dwarf ε (MeV/fm ) Radius 10 2 (km) Radius (km) T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 5/26

7 Properties of neutron stars Tolman-Oppenheimer-Volkoff (TOV) equation for hydrostatic equilibrium of a spherical object in general relativity (GR): dp (R) = G[P (R) + ϵ(r)][m(r) + 4πR3 P (R)], dr R[R 2GM(R)] R ϵ: energy density M(R) = 4πr 2 ϵ(r)dr, P : pressure 0 G: gravitational constant P (MeV/fm 3 ) P = ε M/M O ε (MeV/fm 3 ) Radius (km) T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 5/26

8 Properties of neutron stars Tolman-Oppenheimer-Volkoff (TOV) equation for hydrostatic equilibrium of a spherical object in general relativity (GR): dp (R) = G[P (R) + ϵ(r)][m(r) + 4πR3 P (R)], dr R[R 2GM(R)] R ϵ: energy density M(R) = 4πr 2 ϵ(r)dr, P : pressure 0 G: gravitational constant P (MeV/fm 3 ) P = ε M/M O ε (MeV/fm 3 ) Radius (km) T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 5/26

9 Properties of neutron stars Tolman-Oppenheimer-Volkoff (TOV) equation for hydrostatic equilibrium of a spherical object in general relativity (GR): dp (R) = G[P (R) + ϵ(r)][m(r) + 4πR3 P (R)], dr R[R 2GM(R)] R ϵ: energy density M(R) = 4πr 2 ϵ(r)dr, P : pressure 0 G: gravitational constant P (MeV/fm 3 ) P = ε M/M O ε (MeV/fm 3 ) Radius (km) T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 5/26

10 Properties of neutron stars Tolman-Oppenheimer-Volkoff (TOV) equation for hydrostatic equilibrium of a spherical object in general relativity (GR): dp (R) = G[P (R) + ϵ(r)][m(r) + 4πR3 P (R)], dr R[R 2GM(R)] R ϵ: energy density M(R) = 4πr 2 ϵ(r)dr, P : pressure 0 G: gravitational constant P (MeV/fm 3 ) P = ε M/M O ε (MeV/fm 3 ) Radius (km) T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 5/26

11 Properties of neutron stars Tolman-Oppenheimer-Volkoff (TOV) equation for hydrostatic equilibrium of a spherical object in general relativity (GR): dp (R) = G[P (R) + ϵ(r)][m(r) + 4πR3 P (R)], dr R[R 2GM(R)] R ϵ: energy density M(R) = 4πr 2 ϵ(r)dr, P : pressure 0 G: gravitational constant P (MeV/fm 3 ) P = ε M/M O ε (MeV/fm 3 ) Radius (km) T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 5/26

12 Properties of neutron stars Tolman-Oppenheimer-Volkoff (TOV) equation for hydrostatic equilibrium of a spherical object in general relativity (GR): dp (R) = G[P (R) + ϵ(r)][m(r) + 4πR3 P (R)], dr R[R 2GM(R)] R ϵ: energy density M(R) = 4πr 2 ϵ(r)dr, P : pressure 0 G: gravitational constant P (MeV/fm 3 ) P = ε M/M O black hole ε (MeV/fm 3 ) Radius (km) T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 5/26

13 Internal structure of a neutron star Radius (km) outer crust outer core (~2ρ 0) (2ρ ~) 0 inner crust (<0.5ρ ) 0 (n,p,e,μ) - - inner core hyperons pion condensates kaon condensates quarks and gluons Understanding of the equation of state (EOS) for dense matter is important to clarify compact astrophysical phenomena. Crust nonuniform nuclear matter Outer crust: n, e, nuclei (A) Inner crust: neutron drip out of nuclei, nuclear pasta structure core uniform nuclear matter Outer core: n, p, e and µ, Inner core: exotic degrees of freedom (hyperons (Y), quarks, ) The EOS which must be adaptable to a wide-density range is required. T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 6/26

14 Theoretical calculation (EoS) P (MeV/fm 3 ) this study(tf) KMS12(BPS+BBP) P = ε outer crust inner crust core ε (MeV/fm 3 ) T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 7/26

15 Theoretical calculation (MR-relation) 1 Baryon Mass Density (g/fm 3 ) n Particle Fraction Y i e p A 0.1 outer crust inner crust µ Ξ n B (fm -3 core ) T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 8/26

16 Particle Fraction Y i Theoretical calculation (MR-relation) Baryon Mass Density (g/fm 3 ) ( fm 3 ) n e A outer crust inner crust Nucleon distributions ( fm 3 ) ( n B (fm -3 core 2 fm 3 ) ) p ( fm 3 ) ( fm 3 ) µ Ξ T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 8/26

17 Particle Fraction Y i Theoretical calculation (MR-relation) Baryon Mass Density (g/fm 3 ) ( fm 3 ) n e A neutron-star radius outer crust inner crust Nucleon distributions ( fm 3 ) ( n B (fm -3 core 2 fm 3 ) ) p ( fm 3 ) ( fm 3 ) µ Ξ T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 8/26

18 Theoretical calculation (MR-relation) 1 Baryon Mass Density (g/fm 3 ) n Particle Fraction Y i e p A 0.1 outer crust inner crust µ Ξ n B (fm -3 core ) T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 8/26

19 Theoretical calculation (MR-relation) 1 Baryon Mass Density (g/fm 3 ) n Particle Fraction Y i e p A 0.1 outer crust inner crust µ Ξ n B (fm -3 core ) T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 8/26

20 Theoretical calculation (MR-relation) 1 Baryon Mass Density (g/fm 3 ) n Particle Fraction Y i 0.1 e maximum mass A of a neutron star outer crust inner crust p µ Ξ n B (fm -3 core ) T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 8/26

21 Hyperon puzzle The inclusion of other degrees of freedom softens an equation of state, such as hyperon admixture, meson condensation, and quark matter. Thus, the maximum mass of neutron stars is reduced. In fact, it is difficult to explain the mass of J by the EoSs which have been proposed so far. We have to make the equation of state with hyperons which is consistent with the massive neutron stars. M/M O Hartree CQMC (np) QMC (np) QHD+NL (np) CQMC (npy) QMC (npy) QHD+NL (npy) Radius (km) T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 9/26

22 Hyperon puzzle Particle Fraction Y i 1 The inclusion n of other degrees of QMC freedom softens an equation of state, such p as hyperon admixture, 0.1 meson condensation, and quark e matter. Thus, the maximum mass of neutron stars is reduced µ In fact, itλisσdifficult Ξ to Ξ 0 explain Σ 0 the Σ + mass of J by the EoSs which have beenn proposed so far. B (fm -3 ) We have to make the equation of state with hyperons which is consistent with the massive neutron stars. M/M O Hartree CQMC (np) QMC (np) QHD+NL (np) CQMC (npy) QMC (npy) QHD+NL (npy) Radius (km) T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 9/26

23 Neutron Stars and Quark Stars Charge neutrality: q HP = B q B ρ B + l q l ρ l, q QP = i q i ρ i, with q B(l)[i] and ρ B(l)[i] are the electric charge and the number density for baryons (leptons) [quarks], respectively. β equilibrium in weak interaction: chemical potential µ n = µ Λ = µ Σ 0 = µ Ξ 0, µ n + µ e = µ Σ = µ Ξ, µ n µ e = µ p = µ Σ +, µ e = µ µ, µ u = µ d = µ s = 1 3 (µ n 2µ e ), 1 2 (µ n + µ e ). T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 10/26

24 Hadronic and quark matter Under β equilibrium condition ε/n B - M N (MeV) Binding energy Pressure RHF 150 RH-SU(3) RH-SU(6) 100 QM00 QM02 50 QM10 QM n B (fm -3 ) P (MeV/fm 3 ) n B (fm -3 ) RHF RH-SU(3) RH-SU(6) QM00 QM02 QM10 QM12 Quark matter can generate at high densities. Quark matter can be bound by itself (quark stars). T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 11/26

25 Uniform hadronic matter with hyperons using relativistic Hartree-Fock approximation in SU(3) flavor symmetry T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 12/26

26 Hadronic matter Description of uniform nuclear matter: Nonrelativistic phenomenological potential, meson exchange, Skyrme interaction, Relativistic mean-field, chiral perturbation, effective field theory, Relativistic mean-field (RMF) calculation: high-density region Hartree approximation only direct interaction Hartree-Fock approximation direct and exchange interactions Baryon self-energty Σ tadpole exchange T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 13/26

27 Lagrangian density Lagrangian density for uniform hadronic matter: L H = L B + L M + L int. We consider the octet baryons (B): proton (p), neutron (n), Λ, Σ +0, and Ξ 0. In addition, not only the mesons which is composed of light quarks (σ, ω, π, and ρ) but also the strange quarks (σ and ϕ) are taken into account. L int = [ ψ B g σb (σ) σ + g σ B (σ ) σ g ωb γ µ ω µ + f ωb 2M σ µν ν ω µ B g ϕb γ µ ϕ µ + f ϕb 2M σ µν ν ϕ µ g ρb γ µ ρ µ I B + f ρb 2M σ µν ν ρ µ I B f ] πb m γ 5γ µ µ π I B ψ B, with M being the scale mass (= M p ). T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 14/26

28 Lagrangian density Lagrangian density for uniform hadronic matter: L H = L B + L M + L int. We consider the octet baryons (B): proton (p), neutron (n), Λ, Σ +0, and Ξ 0. In addition, not only the mesons which is composed of light quarks (σ, ω, π, and ρ) but also the strange quarks (σ and ϕ) are taken into account. field-dependent L int = [ C.C. using the CQMC model ψ B g σb (σ) σ + g σ B (σ ) σ g ωb γ µ ω µ + f ωb 2M σ µν ν ω µ B g ϕb γ µ ϕ µ + f ϕb 2M σ µν ν ϕ µ g ρb γ µ ρ µ I B + f ρb 2M σ µν ν ρ µ I B f ] πb m γ 5γ µ µ π I B ψ B, with M being the scale mass (= M p ). T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 14/26

29 Lagrangian density Lagrangian density for uniform hadronic matter: L H = L B + L M + L int. We consider the octet baryons (B): proton (p), neutron (n), Λ, Σ +0, and Ξ 0. In addition, not only the mesons which is composed of light quarks (σ, ω, π, and ρ) but also the strange quarks (σ and ϕ) are taken into account. L int = [ field-dependent C.C. using the CQMC model ψ B g σb (σ) σ + g σ B (σ ) σ g ωb γ µ ω µ + f ωb 2M σ µν ν ω µ B g ϕb γ µ ϕ µ + f ϕb 2M σ µν ν ϕ µ g ρb γ µ ρ µ I B + f ρb 2M σ µν ν ρ µ I B f ] πb m γ 5γ µ µ Hartree-Fock approximation π I B ψ B, (tensor couplings) with M being the scale mass (= M p ). T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 14/26

30 Lagrangian density Lagrangian density for uniform hadronic matter: L H = L B + L M + L int. We consider the octet baryons (B): proton (p), neutron (n), Λ, Σ +0, and Ξ 0. In addition, not only the mesons which is composed of light quarks (σ, ω, π, and ρ) but also the strange quarks (σ and ϕ) are taken into account. L int = [ field-dependent C.C. using the CQMC model ψ B g σb (σ) σ + g σ B (σ ) σ g ωb γ µ ω µ + f ωb 2M σ µν ν ω µ B g ϕb γ µ ϕ µ + f ϕb 2M σ µν ν ϕ µ g ρb γ µ ρ µ I B + f ρb 2M σ µν ν ρ µ I B f ] πb m γ 5γ µ µ Hartree-Fock approximation π I B ψ B, (tensor couplings) pion contribution (pseudovector) with M being the scale mass (= M p ). T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 14/26

31 Matter properties SU(6): quark model ideal mixing θ v = and z = 1/ 6 g ϕn = 0. SU(3): flavor θ v = and z = (ESC08). g ϕn = 3z tan θv 1 + 3z tan θ v g ωn, Results for saturation properties w 0 n 0 E sym M N /M N K 0 L Symmetry (MeV) (fm 3 ) (MeV) (MeV) (MeV) SU(6) SU(3) ε B /n B - M N (MeV) red: input values Binding energy 0 SU(3) SU(6) neutron matter δ = 1.0 δ = nuclear matter n B (fm -3 ) T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 15/26

32 Equations of state (hadronic matter) P (MeV/fm 3 ) P = ε w/o Y ESC08Y MYN13 SU(3)Y ε (MeV/fm 3 ) T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 16/26

33 Equations of state (hadronic matter) 600 P (MeV/fm 3 ) P = ε w/o Y ESC08Y MYN13 SU(3)Y ε (MeV/fm 3 ) T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 16/26

34 Equations of state (hadronic matter) P (MeV/fm 3 ) stiff or hard P = ε w/o Y ESC08Y MYN13 SU(3)Y ε (MeV/fm 3 ) T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 16/26

35 Equations of state (hadronic matter) P (MeV/fm 3 ) stiff or hard P = ε w/o Y ESC08Y MYN13 SU(3)Y ε (MeV/fm 3 ) T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 16/26

36 Equations of state (hadronic matter) P (MeV/fm 3 ) stiff or hard P = ε soft ε (MeV/fm 3 ) w/o Y ESC08Y MYN13 SU(3)Y T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 16/26

37 Particle Fraction Y i Neutron-star matter (a) ESC08Y e µ (b) SU(3)Y e µ Λ p p Ξ n n n B (fm -3 ) Ξ Charge neutrality β-equilibrium central density of a neutron star T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 17/26

38 Mass (M O ) Mass-radius relations (without hyperons) 2.04M Radius (km) Steiner et al. (2σ) Steiner et al. (1σ) PSRJ PSRJ w/o Y ESC08Y MYN13 SU(3)Y T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 18/26

39 Mass (M O ) Mass-radius relations (without hyperons) 2.04M 1.95M (ESC08Y) (without σ and ϕ mesons) Radius (km) Steiner et al. (2σ) Steiner et al. (1σ) PSRJ PSRJ w/o Y ESC08Y MYN13 SU(3)Y T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 18/26

40 Mass (M O ) Mass-radius relations (without hyperons) 2.04M 2.03M (ESC08Y) 1.95M (ESC08Y) (without σ and ϕ mesons) Radius (km) Steiner et al. (2σ) Steiner et al. (1σ) PSRJ PSRJ w/o Y ESC08Y MYN13 SU(3)Y T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 18/26

41 Mass (M O ) Mass-radius relations (without hyperons) 2.04M 2.03M (ESC08Y) 1.95M (ESC08Y) ϕ (without σ and ϕ mesons) Radius (km) Steiner et al. (2σ) Steiner et al. (1σ) PSRJ PSRJ w/o Y ESC08Y MYN13 SU(3)Y T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 18/26

42 Mass (M O ) Mass-radius relations (without hyperons) 2.04M 2.03M (ESC08Y) 1.95M (ESC08Y) ϕ (without σ and ϕ mesons) Radius (km) Steiner et al. (2σ) Steiner et al. (1σ) 1.87M (lower limit in naive SU(3) symmetry) PSRJ PSRJ w/o Y ESC08Y MYN13 SU(3)Y T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 18/26

43 Quark effect on neutron stars: hybrid stars (baryons, leptons, and quarks) T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 19/26

44 Hybrid stars 1 Can quarks be generated in the core of neutron stars? 2 To study phase transition from hadron phase to quark phase. 3 To make constraints on the critical density n (c) B, the critical chemical potential µ (c) B, from the astronomical observations (so-called two-solar-mass neutron stars). Quark matter: MIT bag model + OGE ε Q = [ ] Q q=u,d,s Ωq + µ q n q + B(n [ B (, β), ) nq 2 B(n Q, β) = B + (B 0 B ) exp β g q ], n 0 with B 0 = 400 MeV fm 3 and B = 50 MeV fm 3. We vary the parameter β to change the transition density of quark phase. T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 20/26

45 Phase transition Maxwell condition (Maxwell equal area rule) Gibbs criteria : P HP (µ n, µ e ) = P QP (µ n, µ e ) = P MP (µ n, µ e ) ϵ MP = (1 χ) ϵ HP + χϵ QP, q MP = (1 χ) q HP + χq QP = 0, with χ being the volume fraction of quark phase. n B and µ c B increase T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 21/26

46 Particle fractions (ESC08Y and SU3Y) Particle Fraction Y i Particle Fraction Y i 10 0 ESC08Y (2) n p e u µ µ Ξ s β = s d (3) n (5) n d p p e u e u µ s d (4) n p n B (fm -3 ) (3) n d (5) p u p n d 10-1 s u e e SU3Y (4) µ e µ µ Λ Ξ β s = Λ n d d (6) n p p u e u Λ Ξ s β = β = n B (fm -3 ) e µ µ s s Λ u d β = β = β = β = T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 22/26

47 Equations of state (ESC08Y) P (MeV/fm 3 ) P = ε Critical Maximum In general, 1 (GeV) µ (c) ε (MeV/fm 3 ) hadron (ESC08Y) 1.6 (GeV) B T. Sasaki et al., arxiv: [hep-ph]. End (1) β=0.000 (2) β=0.025 (3) β=0.050 (4) β=0.100 (5) β=0.150 (6) β=0.200 µ (c) B (MeV) T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 23/26

48 Mass-radius relations (ESC08Y) Mass (M O ) Radius (km) Steiner et al. (2σ) Steiner et al. (1σ) PSRJ PSRJ hadron (ESC08Y) (1) β=0.000 (2) β=0.025 (3) β=0.050 (4) β=0.100 (5) β=0.150 (6) β=0.200 µ (c) B (MeV) T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 24/26

49 Mass-radius relations (ESC08Y) Mass (M O ) Radius (km) Steiner et al. (2σ) Steiner et al. (1σ) PSRJ PSRJ hadron (ESC08Y) (1) β=0.000 (2) β=0.025 (3) β=0.050 (4) β=0.100 (5) β=0.150 (6) β=0.200 µ (c) B (MeV) T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 24/26

50 Mass-radius relations (ESC08Y) Mass (M O ) Critical end point (CEP) Radius (km) Steiner et al. (2σ) Steiner et al. (1σ) PSRJ PSRJ hadron (ESC08Y) (1) β=0.000 (2) β=0.025 (3) β=0.050 (4) β=0.100 (5) β=0.150 (6) β=0.200 µ (c) B (MeV) T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 24/26

51 Mass-radius relations (ESC08Y) Mass (M O ) Critical end point (CEP) Radius (km) Steiner et al. (2σ) Steiner et al. (1σ) PSRJ PSRJ hadron (ESC08Y) (1) β=0.000 (2) β=0.025 (3) β=0.050 (4) β=0.100 (5) β=0.150 (6) β=0.200 µ (c) B (MeV) T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 24/26

52 Summary T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 25/26

53 Summary A new EoS for neutron stars: Field-dependent coupling constants using the CQMC model Mean-field calculation within relativistic Hartree-Fock approximation Coupling constants are determined in SU(3) flavor symmetry. The maximum mass of a neutron star: 2.03M Hybrid stars: Quark matter can be seen in mixed (coexistence) phase. The maximum mass of a neutron star: 2.0M with µ (c) B = 1450 MeV T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 26/26

54 Thank You for Your Attention. T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 26/26

55 Appendix T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 26/26

56 Explicit Quark Degrees of Freedom The chiral quark-meson coupling (CQMC) model Baryons are described by the cloudy bag model: M B = E bag (m q, R B = ) q α q (m q, R B ) z B R B πr 3 B B bag + E g (m q, R B ) + E π(m q, R B ), gluon pion At hadron level, (after self-consistent calculation): M B ( σ, σ ) = M B g σb ( σ) σ g σ B ( σ ) σ. q q q σ, ω, ρ, π meson σ, ϕ q q q T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 26/26

57 Explicit Quark Degrees of Freedom The chiral quark-meson coupling (CQMC) model Baryons are described by the cloudy bag model: M B = E bag (m q, R B = ) q α q (m q, R B ) z B R B m q ( σ, σ ) = m q g q σb σ gq σ B σ πr 3 B B bag + E g (m q, R B ) + E π(m q, R B ), gluon pion At hadron level, (after self-consistent calculation): M B ( σ, σ ) = M B g σb ( σ) σ g σ B ( σ ) σ. q q q σ, ω, ρ, π meson σ, ϕ q q q T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 26/26

58 CQMC model Effective baryon mass: M B ( σ, σ ) = M B g σb σ g σ B σ, QHD-type, = M B g σb ( σ) σ g σ B ( σ ) σ QMC and CQMC. Density-dependent coupling constants for scalar mesons: [ g σb ( σ) = g σb b B 1 a ] B 2 (g σn σ), [ ] g σb ( σ ) = g σ Bb B 1 a B 2 (g σ Λ σ ). where g σn and g σ Λ are respectively the σ-n and σ -Λ coupling constants at zero density. The effect of the variation of baryon structure at the quark level can be described with the parameters a B and a B. In addition, the extra parameters, b B and b, are necessary to express the effect of hyperfine interaction between two B quarks. If we set a B = 0 and b B = 1, g σb (σ) becomes identical to the σ-b coupling constant in QHD. This is also true of the coupling g σ B(σ ). T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 26/26

59 Coupling constants for hyperons Scalar (σ and σ ) mesons: g σy : U (N) Λ g σ Y: U (Ξ) Ξ = 28 MeV, U (N) Σ U(Ξ) Λ 2U(Λ) Ξ T.M., M.-K. Cheoun, and K. Saito, Phys. Rev. C 88 (2013) 1, = +30 MeV, and U(N) Ξ = 18 MeV 2U(Λ) Λ, U(Λ) Λ 5 MeV (Nagara event) Vector (ω, ρ, and ϕ) mesons: SU(3) flavor symmetry 1 g ωλ = g ωσ = 1 + g ωn, g ωξ = 1 3z tan θv 3z tan θ v 1 + g ωn, 3z tan θ v tan θ v 3z + tan θv g ϕλ = g ϕσ = 1 + g ωn, g ϕξ = 3z tan θ v 1 + g ωn, 3z tan θ v g ρn = 1 2 g ρσ = g ρξ, g ρλ = 0, with θ v = and z = based on the Nijmegen extended-soft-core (ESC08) model. T. A. Rijken, M. M. Nagels, and Y. Yamamoto, Prog. Theor. Phys. Suppl. 185, 14 (2010). T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 26/26

60 Coupling constants for hyperons Scalar (σ and σ ) mesons: g σy : U (N) Λ g σ Y: U (Ξ) Ξ = 28 MeV, U (N) Σ U(Ξ) Λ 2U(Λ) Ξ T.M., M.-K. Cheoun, and K. Saito, Phys. Rev. C 88 (2013) 1, = +30 MeV, and U(N) Ξ = 18 MeV 2U(Λ) Λ, U(Λ) Λ 5 MeV (Nagara event) Vector (ω, ρ, and ϕ) mesons: SU(3) flavor symmetry 1 g ωλ = g ωσ = 1 + g ωn, g ωξ = 1 3z tan θv 3z tan θ v 1 + g ωn, 3z tan θ v tan θ v 3z + tan θv g ϕλ = g ϕσ = 1 + g ωn, g ϕξ = 3z tan θ v 1 + g ωn, 3z tan θ v g ρn = 1 2 g ρσ = g ρξ, g ρλ = 0, g SU(3) ωn g ωn g ESC08 ωn with θ v = and z = based on the Nijmegen extended-soft-core (ESC08) model. T. A. Rijken, M. M. Nagels, and Y. Yamamoto, Prog. Theor. Phys. Suppl. 185, 14 (2010). T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 26/26

61 Equation of state Pressure P H as a function of energy density ϵ H : From the thermodynamic condition, pressure of hadronic matter is given by ( ) P H = n 2 ϵh B, n B n B where the total baryon density n B and with ϵ H = B 2J B + 1 (2π) 3 kfb 0 d k [ T B (k) + 1 ] 2 V B(k), T B (k) = M BM B (k) + kk B E B (k), V B (k) = M B (k)σs B (k) + k B Σv B (k) E B (k) Σ 0 B (k). T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 26/26

62 Hyperons in neutron stars µ B (MeV) (a) ESC08Y (b) SU(3)Y Λ µ n + µ e = µ Ξ Ξ µ n + µ e = µ Ξ Ξ µ n = µ Λ µ n µ n + µ e n B (fm -3 ) µ Λ µ Ξ Σ does not appear. (repulsive) T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 26/26

63 Thermodynamic potential Ω = q=u,d,s Ω q + Ω bag, with the thermodynamic potential for quarks, Ω q, and bag, Ω bag = B(n Q B ). Ω q = g ( q 2Jq + 1 ) [ F ( ) 2α c (µ 2 ) µ q, m q G ( ) ] µ q, m q, 24π 2 π where g q (J q ) is the color (spin degeneracy) factor of quark species, and F ( ( ) µ q, m q = µ q µ 2 q m2 q µ 2 q 5 ) 2 m2 q + 3 µ q + µ 2 m4 q ln 2 q m2 q, m q G ( µ q, m q ) H ( µ q, m q ) = 3H ( ) 2 2 µ q, m q 2 (µ 2 q q) m2, µ q + µ = µ q µ 2 q m2 q m2 q ln 2 q m2 q, m q with m u(d) = 5 MeV and m s = 150 MeV. T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 26/26

64 Quark matter EoS Energy density for quark matter: ε Q = = q=u,d,s q=u,d,s ε q + Ω bag [ Ωq + µ q n q ] + B(n Q B ), Pressure for quark matter: p Q = = ( ) ( ) 2 n Q ε Q B n Q B n Q B ) n q (n Q ε q B ε q + p bag, q=u,d,s n Q B T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 26/26

65 Quark matter EoS Density-dependent bag constant: B(n Q, β) = B + (B 0 B ) exp [ β ( nq n 0 ) 2 ] with B 0 = 400 MeV fm 3 and B = 50 MeV fm 3. We vary the parameter β to change the transition density of quark phase. Pressure of bag term: p bag = n Q db(n Q B ) B B(n Q B ). dn Q B, T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 26/26

66 Mass-density relations (ESC08Y) Mass (M O ) PSRJ PSRJ hadron (ESC08Y) (1) β=0.000 (2) β= (3) β=0.050 (4) β= (5) β=0.150 (6) β= n B (fm -3 ) T. Miyatsu Possibility of hadron-quark coexistence in massive neutron stars 26/26