Hadron-Quark Crossoverand Massive Hybrid Stars
Kota Masudawith Tetsuo Hatsuda (RIKEN) and Tatsuyuki Takatsuka (RIKEN)[1]``Hadron-Quark Crossover and Massive Hybrid Stars with Strangeness”,
Masuda, Hatsuda and Takatsuka, ApJ 764, 12 (2013)[2]``Hadron-Quark Crossover and Massive Hybrid Stars”,
Masuda, Hatsuda and Takatsuka, PTEP 073D01 (2013)
熱場の量子論とその応用, 26th Aug., 2013

Introduction: Massive Neutron Star

1/16

Typical value of the observed massfor double NS binaries ⇠ 1.4M

1.44M

In 2010, NS (PSR J1614-2230, NS-WD binary)with M = (1.97 ± 0.04)M was found

1.97M

Timing residual (μs)

Shapiro delay

Orbital phase (turns)
Demorest et al. (2010)

Key Questions:
Any EOS which can explain 2 M NS？
Ozel et al. (2012)

The fate of the quark matter inside a heavy NS？

Hadronic EOSs

2/16
(1)

(2)

AV18+TBF TNI2 SCL3⇤⌃

!#)('$"  MethodHyperons

BHF

BHF
RMF 

M/M

(3)

  ΛΣ

2NF

AV18

Reid

3NF

Yes

Yes

NoHyperons

Yes

Yes

Yes

%&• Hyperons soften EOS

(1) Baldo et al. (2000), Schulze et al. (2010)(2) Nishizaki et al. (2001,2002)(3) Tsubakihara et al. (2010)

Maximum mass is less than 1.44M

Hadronic EOSs

2/16!#)('$"   *TNI2u``NNN”Yes

Yes``NNY”``NYY”``YYY”

NoYes

 

M/M

TNI2

  ΛΣ























%&

• Universal 3-body force stiffens EOS• However maximum mass cannot exceed 2M

Maximum mass is larger than 1.44M

Pressure (P)Hadron-Quark Crossover

3/16

BEC-BCS Crossover
hadroncrossover

quark

Baryon density (ρ)

This research seeks the possibility of crossover
Ref.)
Baym (1979)
Celik, Karsch and Satz (1980)
Fukushima (2004)
Hatsuda, Tachibana,Yamamoto and Baym (2006)

Method of Interpolation
Phenomenological interpolation: P (⇢)
1 ± tanh( ⇢ ⇢¯ )f± =2

P = pH ⇥ f + pQ ⇥ f +@("/⇢)⇢

Pressure (P)P = ⇢2

hadron⇢0crossover⇢¯Condition for ⇢¯ : f+ < 0.1 at ⇢0

quark

Baryon density (ρ)⇢¯ > ⇢0 + 2

4/16

EOS at ⇢⇢¯

5/16(2+1)-flavor NJL Lagrangian (u,d,s, e , µ )LN JL = q(i@8gv
Gs Xa 2a 2m)q +[(q q) + (qi 5 q) ](q̄ µ q)2+GD [detq(1 +2 a=025 )q + h.c.]Parameter setcutoff (MeV)

631.4

Gs ⇤2 GD ⇤5 mu,d (M eV ) ms (M eV )

3.67

9.29

5.5

135.7

0  gv  1.5Gs

Hatsuda and Kunihiro (1994)

Conditions:
1. beta-equilibrium
2. charge neutrality

Recent estimate of gV

=d2 Tc (µ)
Tcdµ2 µ2 = 0gV ⇠ GSgV

0 : repulsivegV /GS = 0 :no-repulsion
gV /GS = 1.0 :medium repulsion
gV /GS = 1.5 :strong repulsion

gV /GS

Bratovic et al., Phys. Lett. B719 (2013)

Interpolated EOS

6/16

H-EOS: TNI2u, Q-EOS: NJL
gv = GS(¯⇢, ) = (3⇢0 , ⇢0 )500 

400



300200 100 00

123456

• In the crossover region, interpolated EOS is larger than H-EOS.
• Rapid stiffening of the EOS in the crossover region

Results (1): Effects of Q-EOS

7/16⇢, ) = (3⇢0 , ⇢0 ) gv = GS
M-R relation (¯!#)('$"&*")('$ *"!   +   *    ΛΣ

M/MM/M ΛΣ

%&

#%

• Maximum mass exceeds 2 solar mass, no matter what kind of H-EOS is taken

Results (2): Effects of parameters

8/16

How maximum mass depends on ⇢¯,⇢¯/⇢0 = 2/⇢0 = 1
gv = Gs

gv = 1.5Gs

gv = Gs

gv = 1.5Gs

3⇢0

2.05

2.17--

4⇢0

1.89

1.97

--5⇢0

1.73

1.79

1.74

1.80

6⇢0

1.60

1.64

1.62

1.66

Crossover occurs at relatively low densities and quarks are strongly interacting

2M

Results (3): Sound Velocity

9/16⇢, ) = (3⇢0 , ⇢0 ) gv = GS
M-R relation (¯&*)('$*,"*! +     !" +M/M #%



##

• The emergence of strangeness softens EOS• Due to the interpolation, the sound velocity increases rapidly in the crossover region

Results (4): Strangeness Core⇢r relation (¯⇢, ) = (3⇢0 , ⇢0 ) gv = GS

1.44M!

2M!

gV = GS

2M!

M!

1.44M!

Typical NSs with universal 3-body force do
not include strangeness inside themselves
possibility of solving cooling problem

10/16Color Superconductivity (CSC)• Chiral Condensate

11/16• Diquark CondensateµEFermi SeapDirac SeaAlford• NJL modelHLCSC = LNJL +2XXTT(q̄i 5 ⌧A A0 C q̄ )(q Ci 5 ⌧A A0 q)

A=2,5,7 A0 =2,5,7

3
H = Gs4(Fierz)

Results (5): Gap parameter

12/16

gv = 0   

      

Results (5): Gap parameter

12/16

gv = 0



Dispersion
relation (bd-gs)
              gapless phaseResults (6): Effects of CSC

13/16

Diquark condensation with J P = 0+

X

HLCSC = LNJL +2
M-R relation

X(q̄i 5 ⌧A A0 C q̄ T )(q T Ci 5 ⌧A A0 q)

A=2,5,7 A0 =2,5,7H=(¯⇢, ) = (3⇢0 , ⇢0 ) gv = GS

3Gs4      

M/MM/M    



without CSCwith CSC

• CSC softens EOS, but the effects of CSC is very smallProto-NS with 2 solar mass puzzle
At the birth stage, NSs are composed of so-called supernova matter.
• constant lepton fraction• constant entropy per baryon

Yl = 0.3 0.4
s = 1 1.5

M/M

NB
Finite temperature extension of our model

P = f (⇢) ⇥ pH (⇢, T )|Yl ,s + f+ (⇢) ⇥ pQ (⇢, T )|Yl ,s

14/16

Results (7): Finite T
g v = Gs

15/16(¯⇢, ) = (3⇢0 , ⇢0 )"    " (N 0 , M 0 ) 



" M/M !!#    





$$  

N 0 = 2.49 ⇥ 1057

M 0 = 1.92M < 1.97M
neutron star merger ？

Summary

16/16

Summary(1) Crossover occurs at relatively low densities(2) Quarks are strongly interacting at and above the crossover region
Interpolated EOS can become stiffer due to the presence of quark matter
Observation of very massive neutron star cannot exclude the existenceof the quark matter core
Conventional belief＊ Other Characteristics:
Interpolated EOS with the repulsive 3-body force among nucleons and hyperons
have a impact on the cooling problem of neutron star with hyperon core
＊ Perspective
1. Finite temperature extension of the present model (proto-neutron star)
2. Constraints on the EOS from other observables such as neutron star radius and coolingThank you !

Back Up SlideIntroduction: QCD Phase Diagram

1/16

lattice QCD

Fukushima, Hatsuda (2010)

Observation

TheoryMass of Neutron stars (NSs)

Equation of state(EOS):

P = f (")

constraints

Tolman-Oppenheimer-Volkov equation (TOV eq.):✓ 2◆ 1dPr3=M + 4⇡P r (" + P )2M rdrGZ
M = 4⇡r2 "(x)dr

Introduction: Hadronic EOSs

3/16

! 
  APR 

M/M













MethodVariational

2NF

AV183NF

YesHyperonsNo

Akmal et al. (1998)

Crossover at finite temperature

8/29• Phenomenological Interpolation: s(T)s: entropy density, T: temperature

Asakawa, Hatsuda (1995)

s(T ) = sh (T )wh (T ) + sq (T )wq (T )wq (T ) =n 1 + tanhm 1tanh

T Tc

T Tc

+ n 1 + tanh

T Tc

"/T 4
("3P )/T 4

P/T 4

Phenomenological Interpolation

lattice QCD

Karsch (1995)

EOS at ⇢⇢¯

10/29(2+1)-flavor NJL Lagrangian (u,d,s, e , µ )LN JL = q(i@8gvGs Xa 2a 2m)q +[(q q) + (qi 5 q) ](q̄ µ q)2+GD [detq(1 +2 a=0
22Gs hq̄i qi i

M i = mi⌦=TlnZVe↵= ⌦q (M, µ ) + ⌦l + Gse↵⌦q (µ ) =Tµe↵ 0Gap equations:li

1hq̄i qi i + 4GD hq̄i qi ihq̄j qj ihq̄k qk i
gv
2XXZi

Si 1 = p

X2GD hq̄j qj ihq̄k qk i

X †gv
hqi qi iµi ! µe↵i ⌘ µi
23d p
Trln
3(2⇡)

✓5 )q + h.c.]◆1
Si 1 (i!l , !
p) ,TXi!2hqi† qi i

Mi , p0 = i!l = (2l + 1)⇡T@⌦=0@hq̄i qi i

Parameter sets
cutoff (MeV)

631.4

Gs ⇤2 GD ⇤5 mu,d (M eV ) ms (M eV )

3.67

9.29

5.5

135.7

Hatsuda and Kunihiro (1994)

0  gv  1.5Gs(Fierz: GV = 0.5GS )Bratovic et al. (2012)

Conditions:
1. beta-equilibrium
2. charge neutrality

EOS at ⇢Constituent mass⇢¯

7/16

Number fraction

Chiral restoration• s quark starts to appear above 4 ⇢0

• u,d quark : low densities• SU(3) flavor symmetric matter at high densities• s quark: 4 ⇢0• muon does not appear due tos quark
charge neutrality• figures do not depend on the magnitude of vector interaction

EOS at ⇢⇢¯

13/29

Pressure P2000

 

 1500

1000

   500 00

1

2

3

4567

8

910



• EOS becomes stiffer as gv increases due to the universal repulsion

Results (2): Effects of Crossover Density ( ⇢¯ )M-R relation= ⇢0 gv = GS"&%$# &(& '  ⇢¯ = 3⇢0

 '

M/M ⇢¯ = 6⇢0!

• Crossover occurs at relatively low densities

2M

16/29

Results (3): Effects of Vector Int. ( gV )⇢, ) = (3⇢0 , ⇢0 )M-R relation (¯&*)('$*,"*! +    +M/M 









#%

• The maximum mass exceeds 2M only if the vector type repulsion isas strong as the scalar interaction• Radius: about 11km

17/29

Another Interpolation
Phenomenological interpolation: "(⇢)
" = "H ⇥ f + " Q ⇥ f +P = ⇢2@("/⇢)⇢

H-EOS: TNI2u, Q-EOS: NJL
gv = 0.5Gs(¯⇢, ) = (3⇢0 , ⇢0 )1 ± tanh( ⇢ ⇢¯ )
f± =2

21/29

Results (7): Effects of Method
M-R relation

22/29(¯⇢, ) = (3⇢0 , ⇢0 )       



M/M

M/M 

  "(⇢)

P (⇢)




(a)(b)

• The " -interpolation makes EOS stiff more drastically than the P-interpolation.
• Even for (gv , ⇢¯) = (0, 3⇢0 ) , the maximum mass can exceed 1.97M

CSC Lagrangian
HLCSC = LNJL +2X

XCTT

1 =C q̄ )(q Ci 5 ⌧AZXT2

`

✓

1p
=
2

Hs55 ,

2 =3d p
Trln
3(2⇡)

✓qqC

Hs77 ,S1S

1=+S0A0q)

1◆()↵ab=◆3 =(i!` , p)
T+4GD u d s✓ 1S0+

T

A=2,5,7 A0 =2,5,7q = C q̄⌦(T, µu,d,s ) =(q̄i 5 ⌧AA024/29X◆Hs22+ GS

color

M ± µ̃ 0µ̃ = µ1µ321p µ82 3X2ii1gV2"↵ c "abc

S0±1 = p3H = Gs
4by Fierz

Xc 5,ini!2+=1 X+|
2H
color

0()† 0c|2CSC Lagrangian (2)
X Z ⇤

1p= 2
4⇡ i=1,36µrd + Md
Bp
B@
0p
µrd

30 rµs M s
Bp
B@
0220Gs14GD u d s + gV2X0Md

33gµu + M u0pprµs + M s⇣⇣⌘⌘dpp2 |"i | + 2T ln 1 + e |"i /T |

0µbu

µgu

Mu

22Mup

30pCCA0 rµd M dBp
B@0101C0CApbµu + M u

3µgs + Ms
BpB@01prµd + M d0µgu

3033Mupp
µgs

0Ms

100pg
µu + M u

10p

1µbd + Mupµbd

Md

1CCA

1CCAX

2ii

@p
@p
@p
===0
@ i
@ i@µiGap equations:
00Buballa (2004)

i0µrs + Ms
BpB@0n2i!2µrs01pgµs + M s
100Ms

2c|

color

p

20 gµs M s
Bp
B@01 X|
2H20p2µbu + Mu
p0µbd

Mu

11Mu
pµbu

21CCA
1C0CA
pbµd + M d

3
H = Gs4Results (5): Case 1

12/16

gv = 0

 

Δ325



20


 Δ2



30



Gap Parameters [MeV]     15   10500Δ125CFL

5075100
MS /μ [MeV]
2gCFL

125

150

Ms2 /(µ ) Alford (2004)

Results (6): Case 2 H = Gs

13/16

gv = 0          2SC phase       udududCrossover vs. 1st order Transition (Example) 
      



In the case of gV = 1.0, 1.5GS , H-EOS and Q-EOS do not cross at all densities

``H”
``QM”

Crossover 1st order Transition

Pressure (P)

``QM”

``H”
Baryon chemical potential (µ)

Pressure (P)Baryon chemical potential (µ)

15/16
``QM”

Pressure (P)Pressure (P)Crossover vs. 1st order Transition``H”
hadroncrossoverquark

``QM”
``H”
Baryon density (ρ)

Baryon density ( )``QM” stiffens EOS``QM” softens EOS

M > 2MM < 2M

Neutron Star Observation
Observables:
mass function

binary periodPbprojection of the pulsar’s semimajor axis on the line of sight x ⌘ asini/c
eccentricity
etime of periastronT0longitude of periastron!0(m2 sini)3
f=M2+General relativity effects:
the advance of periastron of the orbit
Doppler + gravitational redshift
the orbital decay Ṗbrange parameter r
shape parameter sPlane+of+the+sky!˙⌫
centre+of+mass

periastron

!0i
ascending+nodeShapiro delay:

1 + ecos⌫= 2rlog
1 ssin(! + ⌫)Observer

Mass fraction f + 2 general relativity effects
Mass estimation

Universal 3-body force
TNI model

Urbana UIX modelH-EOS: Universal 3-body force
3-body force is needed for saturation property

Akmal et al. (1998)→ From the point of view of NS observation, 3-body force is needed for the stiffness of EOS→ 3-body force between YN and YY can delay the appearance of the exotic components
Universal 3-body force

TNI3

TNI3u

• TNI model:
G-matrix
NN : Reid soft-core potential
YN,YY: Nijmegen type-D
hard-core potential
TNI2(3):
κ=250(300)MeV
Nishizaki et al. (2002)

Cooling Problem
Rapid cooling is occurred by hyperons (Y-Durca)

κ=280MeV⇤ ! p + l + ⌫¯l , p + l ! ⇤ + ⌫l

⌃ ! ⇤ + l + ⌫¯l , ⇤ + l ! ⌃ + ⌫l

Y-mixedTsuruta et al. (2009)