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A Way of Approach to Ultra Dense Matter and Neutron Stars with Quark Matter Core

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

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)

□ 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

universal3-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) 035801

M. 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*=(Δ, Σ^-, Σ^0)

B*

Repulsion from SJM (string-junction quark model) -----flavor independent (universal)

- 2B come in short distance
- Deformation (resistance)
- Fusion into 6-quark state

(by R. Tamagaki)

Prog. Theor. Phys. 119 (2008) 965.

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

WithQ-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: 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

H-phase H-Q trans. Q-phase

<Matching>

?

uncertain uncertain

□ 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

MatchingX=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 theQ-matter EOS, as well as the stiffnes of H-matter EOS.

The massexceeding2-solarmasswould 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 quarkmatter EOS (by NJL-model ;[４] ) and more refinedmatching 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)2011.6.26

CASE H-EOS Q-EOS HQ-PHASENS MODEL

Cq B**1/4 ρ_{H} ρ_{Q} M_{max} R ρ_{C}

1-1 TNI3u 0.35 100 6 10 1.8１5 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

3-1TNI3 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.