QUARK MATTER SYMMETRY ENERGY AND QUARK STARS

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Peng-cheng Chu(初鹏程 )(INPAC and Department of Physics, Shanghai Jiao Tong

University. kyois@sjtu.edu.cn)

Collaborators：Lie-wen Chen (SJTU)

OutlineSymmetry energy introduction in quark matter.Isospin density-dependent-quark-model.EOS in the Isospin DDQM of Beta-equilibrium strange quark matterProperties of compact star based on the Isospin DDQM.Summary and outlook.

Main References:G.X.Peng,H.C.Chiang,J.J.Yang,L.Li. Phys.Rev.C 61 015201

F . Weber ,Progress in Particle and Nuclear Physics 54 (2005) 193-288G.X.Peng,A.Li,U.Lombardo Phys. Rev.C 77,065807 (2008)

Thomas D.Cohen, R.J.Furnstahl, and David K.Griegel Phys.Rev.C 45 X.J.Wen, X.H.Zhong, G.X.Peng, P.N.Shen, P.Z.Ning Phys.Rev.C 72 015204

Motivation to learn

Strange quark matter may be the ground state~

The loop diagram of my work

Maybe 2 solar mass of a compact star

The EoS of quarksThe phenomenological

models of quarks

Constraints of QCD chiral symmetry

Color confinement

The symmetry energy of quarks

Symmetry energy of quark matter.

In quark matter:

The symmetry energy

In hadron matter:

[-3,3] [-1,1][-1,1]

In symmetric quark matter , make

So we deduce the non-interaction symmetry energy of quark matter

Define the symmetry energy as

We can get the non-interaction symmetry energy of quark matter

So we choose the second and discuss the symmetry energy in 3 different models

Density-dependent-quark-model.

Since bag model incorporate the bag constant ,many ways of effective term can be introduced to meet the principle.

Write the Hamiltonian density as:

Use the effective mass to make the form like a non-interacting system:

mq is the effective mass G.X.Peng,H.C.Chiang,J.J.Yang,L.Li. Phys.Rev.C 61 015201

The two hamiltonian density must have the same eigenenergy

Isospin density-dependent-quark-model.

If we considered as an invariant interacting term for q=u,d or s

Notice that:

Hellmann-Feynman theorem

Give a renormalization-group invariant about quark condensate.

Thomas D.Cohen, R.J.Furnstahl, and David K.Griegel Phys.Rev.C 45

is used in sum-rules as –(225±25MeV)^3 for each flavor of quarks

Isospin density-dependent-quark-model.

Isospin density-dependent-quark-model "+" for d quark, else for u quarkFor s quark , DI = 0.

With this treatment and doing volume

integral :

We check the range of DI

G.X.Peng,H.C.Chiang,J.J.Yang,L.Li. Phys.Rev.C 61 015201,1999

In Density-dependent-quark-model.

Follow the postulate above ,the symmetry energy is:

The mass is effective mass

The symmetric energy vs. baryon number density in CDDM(D^1/2=160Mev, ms=80Mev)

The equivalent mass when we take the iso-spin dependence is:

Then the symmetry energy under this isospin DDQM is:

The mass is effective mass

"+" for d quark, else for u quark.

Symmetry energy of quark matter

When DI=0, CDDM and isospin DDQM has the same form

DI = 1.0 Isospin DDQM

The symmetry energy vs. baryon number density in Isospin DDQM

Symmetry energy in NJL model

0 4 2i i i i j j k km m G q q K q q q q

We can get the symmetry energy in NJL model

Where i for u,d and s quarks.

22 2 2

mC k dk

Λ=602.3MeV , G =1.835/602.3^2 , K = 12.36/602.3^5

Symmetry energy in NJL model

2 2 2 2 2 202

3 2 ( ) 4iF

NJL i u d s u d ski u d s

k m k dk G C C C KC C C

rhoQs = rhoB

rhoQs = 0.

Thermodynamic treatment to EOSNow we calculate the EoS of beta equilibrium quark matter

based on the isospin density dependent quark model.

According to the thermodynamic treatment

j j j jBu u u j j

j ju B j u j

d m v m vnu v m n f n fdn n n m n m

3 [ 1 ( 1 )]2

f x x x ln x xx

Define

Then the chemical potential is

2 24 1 12 2

1 (1 ) 2 2 9 ( ) ( )

jI I d u I d du u u j u d

j j u d u u d dB B B

vD D DD n v DD n vu v m n f n f n f

m n n m n n mn n n

Thermodynamic treatment to EOS

Strange quark matter is considered as a mixture of u,d,s quarks and electrons.

chemical equilibrium:

Baryon density :

Charge-neutrality:

Solve this system of equation ,we can get the elements of the EOS

Thermodynamic treatment to EOS

2 24 1 12 2

1 (1 ) DD DD2 29 ( ) ( )

jI u u u dd d d j d

j j u d u u d d

vD D n v n vu v m n f nu f n fm n n m n n mnB nB nB

And for d,s quark:

1 (1 )9

jIs s s j

vD Du v m n f

To solve the equations , we make nB given

Mev fm-3 DI=0 DI=0.3 DI=1.0Energy per baryon(min)

919.980 929.018 929.466

Pressure (zero) -.261608 0.923606E-02 -.330257E-01

Energy per baryon’s minimum value and zero pressure plot appear at the same time

Thermodynamic treatment to EoSNow check DI’s effect in the EoS of quark matter

When DI increases , the minimum of the lines increase too.

Thermodynamic treatment to EOSCompare the relationship between the Fermi momentum and chemical potential

DI=0.1DI=0.06.

DI=0.02.

Thermodynamic treatment to EOSCompare the relationship between the quark fraction and rhoB

The properties of Quark star

Set Ms0 as the parameter of the quark star while making DI=1.1 & D^1/2=147 MeV

Rotating Quark star

HADRON-QUARK PHASE TRANSITION

HADRONIC PHASE: RMF Theory

where the sum on B runs over the baryon octet :

In the RMF model, the meson fields are treated as classical fields, and the field operators are replaced by their expectation values.

Effective mass

The coupling constants set TM1 to calculate.

For neutron star matter consisting of a neutral mixture ofbaryons and leptons, the β equilibrium conditions without

trapped neutrinos are given by

Then we get the chemical potential of baryons and leptons

The charge neutrality condition is given by

At a given baryon density

The total energy density and pressure ofneutron star matter are written by

Also the pressure is given as :

Phase transition may occur in the core of massive neutron star.

The two crucial equations:

The energy density and the baryon density in the mixed phase are given by

Solve the system of equations , we can get the phase transition diagram, which I haven’t done yet.

Particle fraction vs Baryon number density

HYBRID STAR

The mass-radius relation for the hybrid star & Quark star.

OTHER QUARK MODELS

Bag constant for confinement

Quasi-particle bag model

D&T-DQM

Finite temperature may also cause problems.

NJL,PNJL,MIT,CFL,CDM …

Isospin DDQM at Finite temperature:

EoS at Finite Temperature in Isospin DDQM

X.J.Wen, X.H.Zhong, G.X.Peng , P.N.Shen , and P.Z.Ning Phys.Rev.C 72 015204,2005

Summary and outlook.

1. We extend the density-dependent-quark- mass(DDQM) model in which the confinement is modeled by the density-dependent quark masses to include the isospin dependence.

2. We make use of the model we provide to discuss the form of symmetry energy in quark matter. And we discuss the reason why people choose the symmetry parameter.

3. Based on the isospin dependent DDQM model ,we study the symmetry energy of quark matter and the EoS of strange quark matter. And we give the symmetry energy in

NJL model.

4. We give the quark star properties based on the Isospin DDQM and acquire a 2 solar mass quark star.

Get the mass-radius relation for strange stars and study the structure of strange stars with surface effect considered.

We can use RMFT and Isospin DDQM to study the hadron-quark phase transition .

I will study on the stuff about quark matter based on QCD right away.

OUT LOOK

Thank you！

QUARK MATTER SYMMETRY ENERGY AND QUARK STARS

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