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
1.Introduction and theoretical model
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
1.Introduction and theoretical model
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
LiuHe SINAP Nusym2016@Beijing
0/ uu
0/ uu (MeV)
(MeV)
(MeV)T
(MeV)T
4
1.Introduction and theoretical model
)(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
LiuHe SINAP Nusym2016@Beijing
5
2.Isospin dependence of phase diagram
LiuHe SINAP Nusym2016@Beijing
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
2.Isospin dependence of phase diagram
LiuHe SINAP Nusym2016@Beijing
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
2.Isospin dependence of phase diagram
LiuHe SINAP Nusym2016@Beijing
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
2.Isospin dependence of phase diagram
LiuHe SINAP Nusym2016@Beijing
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
2.Isospin dependence of phase diagram
LiuHe SINAP Nusym2016@Beijing
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
LiuHe SINAP Nusym2016@Beijing
)(222)(222
udISsudSdd
duISsduSuu
GKGmMGKGmM
2.Isospin dependence of phase diagram
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 /),,(
LiuHe SINAP Nusym2016@Beijing
where is introduced to ensure in the vacuum.0 0
12
3.Quark matter symmtry energy
Esym decreases with increasing constant of both the scalar-isovector and vector-isovector couplings constants.
LiuHe SINAP Nusym2016@Beijing
13
3.Quark matter symmtry energy
A similar result can be obtained from the pNJL model. A larger isovector coupling constant leads to a smaller symmetry energy.
LiuHe SINAP Nusym2016@Beijing
14
3.Quark matter symmtry energy
)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
LiuHe SINAP Nusym2016@Beijing
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
LiuHe SINAP Nusym2016@Beijing
16
4.Applications to hybrid stars
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.
LiuHe SINAP Nusym2016@Beijing
SISIS GRG * SIVIV GRG *
17
4.Applications to hybrid stars
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.
LiuHe SINAP Nusym2016@Beijing
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!
LiuHe SINAP Nusym2016@Beijing
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
LiuHe SINAP Nusym2016@Beijing
20
Appendix.B
LiuHe SINAP Nusym2016@Beijing
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