outline 1.introduction and theoretical model 2.isospin...

Outline： 1.Introduction and theoretical model 2.Isospin dependence of phase diagram 3.Quark matter symmtry energy 4.Applications to hybrid stars 5.Conclusion

He Liu(刘鹤) Supervisor:Jun Xu(SINAP)

Collaborators: Lie-Wen Chen(SJTU), Kai-Jia Sun(SJTU) Shanghai Institute of Applied Physics, Chinese Academy of Sciences

LiuHe SINAP Nusym2016@Beijing 1

1.Introduction and theoretical model

I.Introduction

L.W.Chen et al., PRL (2005) J.Xu et al., PRC (2010)

a nuclues the neutron-rich heavy-ion beams a neutron star

QCD phase diagram in 3D the symmetry energy a hybrid star

2


II.Theoretical model The Lagrangian of NJL model

Global Symmetric group(if K=GIV=GIS=0)

where

QCD has a global symmetric group and anomal.

If GIV=GIS=0,NJL model can fit to the symmetric group of QCD.

IVISKMTVNJL

aaa

S qiqqqGqmiq LLLLL

8

0

25

2 ])()[(2

)(

(1)U(1)U)3(SU)3(SU)3(U)3(U AVAV RLG

3

1

25

2 ])()[(a

aaISIS qiqqqG L

3

1

25

2 ])()[(a

aaIVIV qqqqG L

8

0

25

2 ])()[(2 a

aaV

V qqqqG L

]})1(det[])1({det[ 55 qqqqKKMT L fit to QCD anomaly)1(AU

(1)U)3(SU)3(SU VAV G )1(AU

LiuHe SINAP Nusym2016@Beijing

the scalar-isovector coupling term

the vector-isovector coupling term

3


Mean-Field Approximation

if ,the chiral symmetry will be broken.

222)( ffffffffffff qqqqqqqqqq

0f

50100150200250300350400

0.0

0.2

0.4

0.6

0.8

1.0

50100

150200

250

(MeV)

T(MeV)

Quark Condensate

50 100 150 200 250 300 350 400

50

100

150

200

250

(MeV)

T/M

eV

00.050000.10000.15000.20000.25000.30000.35000.40000.45000.50000.55000.60000.65000.70000.75000.80000.85000.90000.95001.000

<uubar>

chiral symmetry breaking

chiral symmetry restorationchiral symmetry breaking

chiral symmetry restoration

The NJL model can successfully interpret the dynamics of spontaneous breaking of chiral symmetry.

fff qq


0/ uu

0/ uu (MeV)

(MeV)

(MeV)T

(MeV)T

4


)(lnln),( NHeZ TrVT

VT

T

Thermodynamic Potential

where

0

s

NJL

d

NJL

u

NJL

)(222 3 duiISkjiSii GKGmM

)(22~3 duiIViVii GG

)1/(1 )~( iiEi ef

)1()2(

20 3

3

iii

ici ff

EMpdN

)1/(1 )~( iiEi ef


5

2.Isospin dependence of phase diagram


In relativistic heavy-ion collision, the ratio of electric/baryon charge should

be fixed and the should also be satisfied.

，AZdu

du /2

.0s

0s

For the Au+Au collision experiment，

0 50 100 150 200 250 300 350 4000

50

100

150

200

250

T (M

eV)

(MeV)

GIS=0

NJL Model

CP(354,45)

s=0

u-quarkd-quark

critical piont

0 50 100 150 200 250 300 350 4000

50

100

150

200

250

T (M

eV)

(MeV)

GIS=0.14GS

NJL Model

CP(354,45)

u-quarkd-quark

s=0

critical piont

The chiral phase transitions of u quark and d quark just become to slightly separate and have only one critical point.

critical point critical point

6



50 100 150 200 250 300 350 400

50

100

150

200

250

(MeV)(MeV)

T(M

eV)

-24.00-23.20-22.40-21.60-20.80-20.00-19.20-18.40-17.60-16.80-16.00-15.20-14.40-13.60-12.80-12.00-11.20-10.40-9.600-8.800-8.000-7.200-6.400-5.600-4.800-4.000-3.200-2.400-1.600-0.80000

We introduce the chemical potential matrix , and

SBs

IBd

IBu

3/,3/,3/

2du

I

),,(ˆ sdudiag

The small effect of isopin is due to the small isospin chemical potential near phase transition district at low temperatures.

-25.0

-20.0

-15.0

-10.0

-5.0

0

7



0 50 100 150 200 250 300 350 4000

50

100

150

200

250

CP(347,93)

deconfinement

T (M

eV)

(MeV)

GIS=0

pNJL Model

s=0

u-quarkd-quark

critical piont

pNJL results

The potential of Polyakov loop

)]}(4)(361ln[54{),,(

332

/

TaeTbTu

),,( TuNJLpNJL

LL

01

deconfined phase

confined phase

The Polyakov loop doesn't strongly affect the isospin dependence of the phase diagram but moves the critical point to higher temperatures.

The phase boundary of deconfinement transition is mostly independent of the isovector couplingconstant.

K.Fukushima, PRD (2008)

0 50 100 150 200 250 300 350 4000

50

100

150

200

250

CP(347,93)

deconfinement

T (M

eV)

(MeV)

GIS=0.14*GS

pNJL Model

s=0

u-quarkd-quark

critical piont

critical point

critical point

8



In the relativistic heavy ion collisions, the isospin chemical potential is small near the phase transition region at low temperature, but in the calculation of hybrid stars,we found that there is a large isopin chemical potential in the mixing region.

0 2 4 6 8 10 12 14 16 18 20-120

-100

-80

-60

-40

-20

0

I (MeV

)quarkmixed

B

Gis=0Giv=0

hadron

T=00ISG0IVG

9



With the increasing scalar-isovector coupling constant,the phase boundaries as well as the critical points of u and d quarks become to separate and their difference reaches the maximum around GIS=0.14GS.

The isospin splitting of the chiral phase transition boundary also can be observed for positive GIV.

NJL results(if )

pNJL results(if )

There is a similar result fromthe pNJL model

MeV30-I

MeV30-I

0

50

100

150

200

250

0

50

100

150

200

0 50 100 150 200 250 300 350 4000

50

100

150

200

0 50 100 150 200 250 300 350 400

GIS=0.14*GS

CP(348,27)CP(387,22)

CP(364,24)

T (

MeV

)

GIS=0

NJL Model

s=0

u-quarkd-quark

=-30MeV

=-30MeV

CP(364,51)

deconfinement

GIS=0

pNJL Model

s=0

Critical point

CP(388,9)CP(347,15)

NJL Model

s=0

=-30MeV

=-30MeV

deconfinement

GIS=0.14*GS

pNJL Model

s=0

CP(359,8)

(MeV)

GIV=0.5*GS

NJL Model

s=0

=-30MeV

=-30MeV

deconfinement

GIV=0.5*GS

pNJL Model

s=0 CP(357,21)CP(379,4)

10


)(222)(222

udISsudSdd

duISsduSuu

GKGmMGKGmM


0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250 300 350 4000

50

100

150

200

250

300

50 100 150 200 250 300 350 400

NJL Model

u-quarkd-quark

GIS=0.14*GS

s=0

=-30MeV

T=51MeV

GIV=0.5*GS

GIS=0.14*GS

(MeV)

M (M

eV)

GIV=0.5*GS

11

3.Quark matter symmtry energy

Isospin asymmetric quark matter:

Similar to the case of nuclear matter, the binding energy of quark matter consisting of u, d, and s quarks can be expanded in isospin asymmetry as

where

can be extracted approximately:

The energy density of the system

)(),(),(),,( 420 sBsymsBsB EEE

du

udB

3/30

2

2 ),,(!2

1),(

sB

sBsymEE

2),0,(),,(),(

sBsB

sBsymEEE

022222 )()(4)( duIVduISsdusduV GGKG

ii

iisduSG )~()( 222

ffffsB Z

ln),,(

)1()2(

2),,(,, 0

3 iiisdui

CsB ffEdpN

BsBE /),,(


where is introduced to ensure in the vacuum.0 0

12


Esym decreases with increasing constant of both the scalar-isovector and vector-isovector couplings constants.


13


A similar result can be obtained from the pNJL model. A larger isovector coupling constant leads to a smaller symmetry energy.


14


)1/(1 )~( iiEi ef

)1/(1 )~( iiEi ef

32

2

33121

iii

iiiF

32

2

33121

iii

iiiF

TEi

iie /)~(

TEi

iie /)~(

The larger quark matter symmetry energy in the pNJL model than in the NJL model is mainly due to their different kinetic energy contributions.

NJL pNJL


15

4.Applications to hybrid stars

a schematic presentation of the different structures of neutron stars

quark coreu,d,s

crustn,p,e,hadron gas

quark-hybrid star

hadron-quark mixed zone

QH

QH

PPTT

0Gibbs condition(two phase equilibrium condition)

Charge neutrality condition the equilibrium conditionBaryon number conservation


16


For positive isovector coupling constants, quarks of different flavors appear at different densities and their fractions are quite different. But for the negative vector-isovector coupling constants, the quarks appear at a larger density.


SISIS GRG * SIVIV GRG *

17


A negative GIV leads to the later appearance of quarks and thus a stiffer EOS at intermediate densities, although at higher densities a positive GIV somehow leads to a larger pressure. As a result, a negative vector-isovector coupling gives the largest maximum mass of hybrid stars.


SISIS GRG * SIVIV GRG *

18

5.Conclusion

Conclusion

1.In the relativistic heavy ion collisions, the isospin chemical potential is small near the phase transition region.However,if the isospin chemical potential can reach as large as about 30 MeV, we find that the phase boundaries as well as the critical points of u quark and d quark become to separate with the increasing isovector coupling constant.

2.The isospin splittings of quark condensate, constituent quark mass, and chiral phase transition as well as the critical point are more sensitive to the scalar-isovector coupling, while the quark matter symmetry energy is more sensitive to the vector-isovector coupling.

3.A positive scalar-isovector coupling constant can lead to an unstable isospin asymmetric quark matter and hybrid star matter. The particle fraction as well as the equation of state in hybrid stars depends on the isovector couplings as well.

Thanks for your attention!


19

Appendix.A

a.the hadron phase

e

enp

ep

pnB

0)(31

32

esdu

euds

c.the quark phase

b.the hadron-quark phase transition

ensd

enu

esdup

sdupnB

QH

QH

YY

YY

PPTT

32

31

32

31

0)2(33

)(3

))(1(

0

the charge neutrality condition

the charge neutrality condition

the equilibrium condition

the equilibrium condition

Gibbs condition(two phase equilibrium condition)

the baryon number conservation )(31

sduB

the baryon number conservation


20

Appendix.B


022222 )()(4)( duIVduISsdusduV GGKG

ii

iisduSG )~()( 222

)1()2(

2),,(,, 0

3 iiisdui

CsB ffEdpN

Kinetic energy part

)1/(1 )~( iiEi ef

)1/(1 )~( iiEi ef

32

2

33121

iii

iiiF

32

2

33121

iii

iiiF

TE

iiie /)~(

TEi

iie /)~( NJL pNJL

outline 1.introduction and theoretical model 2.isospin...

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