1. Crossover Workshop （ 2012.7.12-13, 名大） T.Takatsuka (Iwate Univ.) □ Motivations □ New way of approach □ Some results and remarks Equation of state for hadron-quark transition region and characteristics of hybrid stars* * In colaboration with K.Masuda (Univ.of Tokyo) and T.Hatsuda (Univ.of Tokyo /RIKEN)

2. □ Motivations ○ New phases in NSs → softening of NS- EOS ○ Hyperons surely participate in NS cores （ N 、 e^- 、 μ^- ） → （ N 、 Y 、 e^- 、 μ^- ） ; standard picture → EOS is dramatically softened → Even M_{obs}=1.44M_{solar} cannot be sustained 。 serious contradictio ｎ （  missing something ! ） ⇒ Universal ３ -body force can solve the problem [1] More than 2M_{solar} is okay （ e.g. 「 2πΔ ＋ SJM 」） [2] ーーーーーーーーーーーーーーーー [1] As a review, T. Takatsuka, Prog. Theor. Suppl. No.156 (2004) 84. [2] T. Takatsuka, S. Nishizaki and R. Tamagaki, Proc. Int. Symp. “FM50” (AIP Conference proceedings, 2008) 209. [3] P.B. Demorest, et al., Nature, 467 (2010) 1081.

3. Dramatic softening of EOS Necessity of “Extra Repulsion” TNI3 TNI3u: Universal inclusion of TNI3 repulsion

4. Mass v.s. Central Density NS-mass from 2-body force + ”universal” 3-body force (2πΔ- type + SJM). M_{max} >2M_{solar} is possible. How about NSs With Q-Matt.core?  Our motivation

5. □ Obs. Of a NS with 2M_{solar} ！ Higher central density than expected so far → Possibility of NSs with quark matter core （ Hybrid stars) shows up as an important problem ⇒ someone says OK and someone NO What is the situation ? How to understand ! ＜ Our aim here ＞ ① We propose a new strategy and analyze the possibility and the maximum-mass to be allowed ② Focus our attention to the mechanism of large maximum-mass appearance

6. A way of approach H-phase H-Q trans. Qphase ? uncertain □ H: point particle + interaction → G-Matrix, Variational □ Q: q-matter + asymptotic freedom □ HQ Phase transition Cross point (Maxwell, Gibbs) → not necessarily reliable □ Matching Conditions ○ p increasing with ρ ○ thermodynamics (e, p) ○ coincide at x_H and x_Q

7. □ Hadron Matter phase ○ Matter composed of N (n, p), Y(Λ, Σ^-) and Leptons (e^-, μ^-) ○ effective interaction approach based on G-matrix calculations, (effective int. V for NN, NY, YY) Introduction of 3-body force U (TNI, phenomenological Illinoi-type, expressed as effective 2-body force) ○ V+U satisfy the saturation property and symmetry energy at nuclear density ○ (hard, soft) is classified by the incompressibility κ TNI3u → κ=300MeV, TNI2u → κ=250MeV S. Nishizaki, Y. Yamamoto and T. Takatsuka, Prog. Theor. Phys. 105 (2001) 607; 108 (2002) 703

8. □ Quark Matter phase ○ (2 + 1) -flavor NJL model with vector interaction (1) ○ Hatsuda-Kunihiro parameter set (Phys. Rep - 247 (1994) 221) Λ = 631.4 MeV, G_SΛ^2 = 1835, G_DΛ^2 = 9.29 m_u = m_d = 5.5 MeV, m_s = 135.7 MeV ○ g_V is not well determined, but it issuggested that g_V can be comparable or even larger than g_S → we take g_V / G_S ～ (0 -1.5) λ^α ↓

9. □ Marching ○ From a view of “H-Q Crossover” (2) (3) *) Asakawa-Hatsuda P.R. D55(1997) 4488 ○ energy density ε(ρ) is derived from

10. Energy / Particle v.s. density neuton star matter

11. Pressure v.s. density

12. Pressure v.s. energy density

13. Mass v.s. Central - density

14. Mass v.s. Radius

16. □ Summary and Remarks 1.EOS in H-Q transition region is constructed on the basis of the idea of hadron-quark crossover 2.Hybrid stars with M>2M_{solar} are possible as long as (1) The crossover proceeds through relatively low density region, e.g., ρ= (3 – 4) ρ_0 with Γ = ρ_0, (2) The quark matter is strongly interacting, g_V / G_S ～ (1 – 1.5) ( by increasing g_V further → M_{max} ～ 2.4M_{solar} ) 3. M>2M_{solar} is possible even for TNI2u H-EOS ( κ = 250MeV), as well as for TNI3u ( κ = 300MeV) → reconciles the existence of massive NSs ( M > 2M_{solar} ) with the experimental nuclear incompressibility κ = ( 240 +- 20 ) MeV 4. Our interpolated EOS contains strangeness only above 4ρ_0 → hybrid stars with M > 1.9M_{solar} have no strangeness → free from “too-rapid cooling” (contradicting observations) due to Y-Durca ν-emission process under the NO Y-superfluidity 5. Our massive hybrid stars originate from the transition from the soft EOS to stiff EOS in ρ ～ (2 - 4) ρ_0 due to the H-Q crossover → importance of independent exploration of such crossover by laboratory experiments.

17. Appendix

18. □ Hyperon Mixing in NS cores → Hyperon (Y) surely participate in Neutron Star (NS) Cores → Standard picture for NS constituens: Old (n, p, e^-, μ^-) → Now (n, p, Y, e^-, μ^-)

19. L-Vidana et al, P.R. C62 (2000) 035801 M. Baldo et al, P.R. C61 (2000) 055801

20. ○ Hyperons are always present → profound consequence for NS-mass H. Dapo, B-J. Schaefer and J. Wambach, Phys. Rev. C81 (2010) 035803

21. 2πΔ （ Fujita-Miyazawa)-Type 3-body Force Extended to N+Δ space B* B*=(Δ, Σ^-, Σ^0)

23. (a)2B come in short distance (b)Deformation (resistance) (c)Fusion into 6-quark state (by R. Tamagaki) Repulsion from SJM (string-junction quark model) -----flavor independent (universal) Prog. Theor. Phys. 119 (2008) 965.

24. Matching X=ρ/ρ _ o

25. □ Quark Matter phase ○ Flavor symmetric (u, d, s)-quark gass, by a simple MIT-Bag model ○ η= effective parameter (deviation from asymptotic freedom) η=1 ← free q-gass η<1 ← + OGE correction η>1 ← + some repulsive effects (assumed) ○ EOS for H-phase would be reliable up to (3-6)ρ_0 → x_H=(3-6)ρ_0 ○ q-matter would come into existence beyond x_Q ≃ (8 ～ 10) c.f. ρ ～ 8.4 (11.2)ρ_0 for r_0=0.55 (0.50)fm

26. Some results (preliminary) CASE η B**(1/4) x_H x_Q M_{max} R ρ_c ① 1. 200 4 8 1.62 10.8 7.2 ② 1. 200 4 10 1.59 10.8 6.9 ③ 1. 200 2 8 1.62 11.2 5.6 ④ 1.5 100 2 8 2.61 12.4 4.2 ⑤ 1.5 0 2 8 2.58 12.4 4.3 B**(1/4) in MeV, x_H=ρ_H/ρ_0, x_Q=ρ_Q/ρ=0 M_{max} in M_{solar}, R in km ρ_c in ρ_0 (= nuclear density; 0.17/fm^3)

27. 10 1 0.1 0.01 0 5 10 15 2.5 2.0 1.5 1.0 0.5 0 10 15 20 dM/dR<0 の 通常型に加え dM/dR>0 領域 をもつ非通常型 の出現 dM/dR>0 領域 密度分布の特徴

28. Q-Bag-type Polytrope EOS TNI3u 1000 500 0 5 10 15 Q-Bag-type Polytrope EOS

29. Pressure v.s. Baryon Number Density HH-QQ

30. Energy (Mass) Density v.s. Baryon Number Density HH-QQ

31. Results (Linear Interpolation for HQ-Phase) 2011.6.26 CASE H-EOS Q-EOS HQ-PHASE NS MODEL Cq B**1/4 ρ_{H} ρ_{Q} M_{max} R ρ_{C} 1-1 TNI3u 0.35 100 6 10 1.8 １ 5 10.1 11.7 -2 TNI3u 0. 100 6 10 1.810 10.1 8.1 -3 TNI3u 0.35 200 6 10 1.767 10.4 12.9 -4 TNI3u 0.35 100 4 10 1.835 10.1 11.7 -5 TNI3u 0.35 100 6 15 1.530 8.6 15.8 -6 TNI3u 0.35 100 4 15 1.763 10.1 11.1 -7 TNI3u 0.35 100 2 10 2.055 11.2 9.4 -8 TNI3u 0. 200 2 10 1.537 10.6 6.5 -9 TNI3u 0. 200 2 15 1.688 10.8 5.9 2-1 TNI2u 0.35 100 6 10 1.621 8.97 14.8 -2 TNI2u 0. 100 6 10 1.549 8.79 11.4 -3 TNI2u 0.35 200 6 10 1.372 9.77 11.3 -4 TNI2u 0.35 100 4 10 1.726 9.35 13.8 -5 TNI2u 0.35 100 6 15 1.409 8.14 13.4 -6 TNI2u 0.35 100 4 15 1.611 9.15 13.6 -7 TNI2u 0.35 100 2 10 2.022 10.8 10.6 -8 TNI2u 0. 200 2 10 1.480 10.2 7.1 -9 TNI2u 0. 200 2 15 1.634 10.4 6.4 3-1 TNI3 0.35 100 6 10 1.530 8.59 15.8 -2 TNI3 0. 100 6 10 1.409 8.14 13.4 -3 TNI3 0.35 200 6 10 1.171 8.43 16.8 -4 TNI3 0.35 100 4 10 1.656 8.34 10.4 -5 TNI3 0.35 100 6 15 1.171 8.43 16.8 -6 TNI3 0.35 100 4 15 1.551 8.97 14.8 -7 TNI3 0.35 100 2 10 2.142 11.6 8.8 -8 TNI3 0. 200 2 10 1.612 11.0 5.9 -9 TNI3 0. 200 2 15 1.776 11.1 5.9 ＊ B**1/4 in MeV, ρ_{H} and ρ_{Q} in ρ_0, M_{max} in M_{solar}, R in κ_{m}, ρ_{C} in ρ_0

32. □Some remarks (at the present stage) ○ Maximum Mass M_{max} of NSs with quark matter core depends strongly on the stiffness of EOS and the pertion of hadron phase included and is larger for higher Q-H transition density. In this sense, M_{max} is mainly controlled by the EOS of hadron phase. ○ In the matching procedure, special care should be taken for the thermo-dynamic relation between e and p. Otherwise, in some case, we encounter a class of “ unusual” NS models with dM/dR >0 (dρ_c/dR >0). The stability condition to satisfy average Γ>4/3 for these NSs is under investigation. ○ Our study is in progress, by using more detailed quark matter EOS (by NJL-model;[6]) and more refined matching procedure, i.e., “superposition” of hadron and quark phases at H-Q region, instead of “linear interpolation” used here. [6] T. Hatsuda and T. Kunihiro, Phys. Reports 247 (1994) 221.

33. □ まとめ ( to summarize ) Hyperon effects should be taken into account. The maximum mass of hybrid star strongly depends on the Q-matter EOS, as well as the stiffnes of H-matter EOS. The mass exceeding 2-solar mass would be possible only when the EOS of Q-matter is stiffer than that of free quark gass. Further study is in progress, by using more detailed quark matter EOS (by NJL-model ;[ ４ ] ) and more refined matching procedure, i.e., “superposition” of hadron and quark phases in the H-Q region. ---------------------------- [ ４ ] T. Hatsuda and T. Kunihiro, Phys. Reports 247 (1994) 221.