A Way of Approach to Ultra Dense Matter and

Neutron Stars with Quark Matter Core

T. Takatsuka (Iwate Univ.)In collaboration with T. Hatsuda (Univ. Tokyo/ RIKEN) and K. Masuda (Univ. Tokyo)

Int. Workshop on “Formation of Compact Stars” (March 7-9, 2012, Waseda Univ.)

□ Motivation□ Dramatic effects of hyperons□ Strategy to hybrid star EOS□ Possibility of Hybrid Stars

□ Motivation

○ Obs. of 2M_{solar}-NS[1] → stringent cond. on NS-EOS (stiffer)○ Various new-phases proposed → tend to soften EOS (→ significant)○ Especially, hyperon (Y)-mixing → Dramatic Softening, contradict NS-mass observations (even for 1.44 M_{solar})○ This serious problem can be solved if we introduce the universal 3-body force repulsion (not only for NN part, but also YN and YY parts) [2]○ Even 2M_{solar}-NS is possible[3]------------------------------------- [1] P.B. Demorest, et al., Nature, 467 (2010) 1081. [2] T. Takatsuka, Prog. Theor. Phys. Suppl. No 156 (2004) 84. [3] T. Takatsuka, S. Nishizki and R. Tamagaki, Proc. Int. Symp. “FM50” (AIP Conference proceedings, 2008) 209.

□ Hyperon Mixing in NS cores < K.E. only > < with interactions >

→ 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^-)

□ Our approach [1] 1) G-matrix-based effective interactions V applicable to {N+Y}-matter

2) 3-body force U(TNI; phenomenological one of Illinoi’s type, expressed as an effective 2-body force)

3) Parameters in TNI are determined so that the EOS from V+U satisfies the saturation property, symmetry energy and nuclear incompressibility κ: κ=250(300)MeV for TNI2(TNI3)---------------------------------------- [1] T. Takatsuka, Prog. Theor. Phys. Suppl. No. 156 (2004) 84

Dramatic softening of EOS Necessity of “Extra Repulsion”

TNI3 TNI3u: Universal inclusion of TNI3 repulsion

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

○ Hyperons are always present → profound consequence for NS-mass

H. Dapo, B-J. Schaefer and J. Wambach, Phys. Rev. C81 (2010) 035803

2πΔ （ Fujita-Miyazawa)-Type 3-body Force Extended to N+Δ space

B*B*=(Δ, Σ^-, Σ^0)

(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.

Mass v.s. Central DensityNS-mass from 2-body force + ”universal” 3-body force (2πΔ-type + SJM).

M_{max} >2M_{solar} is possible.

How about NSsWith Q-Matt.core?

Our motivation

○ Massive NS → means that the central density (ρ_c) would be very high → hadrons begin to overlap,quarks tend to be deconfined and eventually realize q-matter core → 2-solar mass NS (hybrid) is made possible or not ? → Some works say “YES” and Some works say “NO”, Why?○ Our aim here is to discuss the problem by a new strategy.

A way of approach

H-phase H-Q trans. Q-phase <Matching>

?

uncertain 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

□ 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

□ 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

Matching 　　　　　　 X=ρ/ρ_o

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)

□ 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.

Appendix

Pressure v.s. Baryon Number Density

H H-Q Q

Energy (Mass) Density v.s. Baryon Number Density

H H-Q Q

Results (Linear Interpolation for HQ-Phase) 2011.6.26CASE 　 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

□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.