Neutron stars within the SU(2) parity doublet model

a r X i v :0805.3301v 1 [n u c l -t h ] 21 M a y 2008

Neutron stars within the SU(2)parity doublet model

V.Dexheimer 1,G.Pagliara 2,L.Tol′o s 1,J.Scha?ner-Bielich 2,S.Schramm 1,3

1

Frankfurt Institute for Advanced Studies,J.W.Goethe-Universit¨a t,D-60438Frankfurt am Main,Germany

2

Institut f¨u r Theoretische Physik,J.W.Goethe Universit¨a t,D-60438,Frankfurt am Main,Germany and 3

Center for Scienti?c Computing,J.W.Goethe-Universit¨a t,D-60438,Frankfurt am Main,Germany

The equation of state of beta-stable and charge neutral nucleonic matter is computed within the SU(2)parity doublet model in mean ?eld and in the relativistic Hartree approximation.The mass of the chiral partner of the nucleon is assumed to be 1200MeV.The transition to the chiral restored phase turns out to be a smooth crossover in all the cases considered,taking place at a baryon density of just 2ρ0.The mass-radius relations of compact stars are calculated to constrain the model parameters from the maximum mass limit of neutron stars.It is demonstrated that chiral symmetry starts to be restored,which in this model implies the appearance of the chiral partners of the nucleons,in the center of neutron stars.

I.INTRODUCTION

E?ective models for the computation of the equation of state of nucleonic matter at ?nite density must take into account two very important physical aspects:the symmetry properties of QCD,in particular chiral symmetry and its spontaneous breaking in vacuum,and the properties of strongly interacting matter at saturation.Actually this is a di?cult task.While from one side the standard sigma model contains a mechanism for the restoration of the chiral symmetry,it does not describe nuclear matter saturation [1].On the other side,Walecka-type models [2]can successfully describe the properties of nuclear matter but they do not contain the symmetries of QCD.To overcome this problem,many di?erent extensions of the simple sigma model have been proposed including vector mesons [3],the dilaton ?eld [4],non linear realizations of chiral symmetry both in SU(2)and SU(3)[5,6,7,8]and chiral models with a hidden local symmetry for vector mesons [9,10,11,12,13].The model that we want to investigate here is the SU(2)parity doublet model considered before in Refs.[14,15,16,17,18,19,20,21,22,23,24].It has been shown,in Ref.[22],that this model can describe correctly both chiral symmetry properties and the properties of nuclear matter at saturation.

The essential new ingredient of this model is an explicit mass term in the Lagrangian,which is chirally invariant due to the special transformation properties of the nucleon ?eld and its chiral partner.The value of this mass parameter,m 0,contributes to the mass of the nucleons and therefore,if its value is very large,the breaking of chiral symmetry is responsible only for the mass splitting between the two nucleons.A still open question concerns the identi?cation of the chiral partner of the nucleon:the most likely candidate is the well known N ′

(1535)resonance.This possibility has been investigated in the previous works on symmetric matter [14,15,16,17,18,19,20,21]and recently it has been extended to beta-stable matter for the study of the properties of neutron stars [24].As already pointed out in Ref.[22]and as we will discuss in this paper,the assignment for the partner of the nucleon is still uncertain and it is also possible that it is a broad resonance,not yet identi?ed by the experiments,with a mass smaller than the mass of the N ′

(1535).

The main scope of this paper is to investigate further this hypothesis and we will consider a possible lower mass for the ”true”chiral partner of the nucleon in the following,choosing a value of 1200MeV.Here,in particular,we are interested in studying the properties of the beta-stable equation of state within the parity model and its applications to neutron stars.We will compute the equation of state both at mean ?eld level and using the relativistic Hartree approximation.The latter approach is more complete since it accounts for Dirac sea e?ects.Interestingly,within the relativistic Hartree approximation,the value of the bare mass m 0is slightly smaller than the value obtained within the mean ?eld approximation.Finally,we will present results showing that the choice of a small value for the mass of the chiral partner of the nucleon allows this particle to be formed at the center of neutron stars.This could have interesting e?ects on the transport properties of the matter (like viscosities or neutrino opacities)with possible phenomenological applications.

The paper is organized as follows:in Section II we will present the Lagrangian of the parity model for asymmetric matter.In Section III and IV we will compute the equation of state and the neutron star structure at mean ?eld level and in relativistic Hartree approximation,respectively.Finally in Section V we draw our conclusions.

II.THE PARITY MODEL FOR ASYMMETRIC MATTER

In the parity doublet model one uses the so-called“mirror assignment”for the positive and negative parity nucleon states(N+and N?),in which they belong to the same multiplet.Under the SU L(2)×SU(2)R transformations L and R,the two nucleon?eldsψ1andψ2transform as:

ψ1R?→Rψ1R,ψ1L?→Lψ1L,(1)

ψ2R?→Lψ2R,ψ2L?→Rψ2L.(2) This allows for a chirally invariant mass term in the Lagrangian that reads:

m0(ˉψ2γ5ψ1?ˉψ1γ5ψ2)=m0(ˉψ2Lψ1R?ˉψ2Rψ1L?ˉψ1Lψ2R+ˉψ1Rψ2L),(3) where m0represents a bare mass parameter.

To study the equation of state of beta-stable matter,in the Lagrangian of Ref.[22]we add the vector-isovector meson?→ρwhich couples to the isospin current:

L=ˉψ1i?/ψ1+ˉψ2i?/ψ2+m0 ˉψ2γ5ψ1?ˉψ1γ5ψ2 +aˉψ1(σ+iγ5τ·π)ψ1

+bˉψ2(σ?iγ5τ·π)ψ2?gωˉψ1γμωμψ1?gωˉψ2γμωμψ2

?gρˉψ1γμτ·ρμψ1?gρˉψ2γμτ·ρμψ2+L M,(4) where a,b,gωand gρare the coupling constants of the mesons?elds(σ,π,ωandρ)to the baryonsψ1andψ2and the mesonic Lagrangian L M contains the kinetic terms of the di?erent meson species,and potentials for the scalar and vector?elds:

L M=1

2

?μ πμ?μ πμ?

1

4

ρμνρμν+

1

2

m2ρρμρμ

+g44[(ωμωμ)2]+

1

4

(σ2+ π2)2+?σ,(5)

whereωμν=?μων??νωμandρμν=?μρν??νρμrepresent the?eld strength tensors of the vector?elds.The parametersλ,ˉμand?are as in Ref.[22]:

λ=

m2σ?m2π

2

,

?=m2πfπ,(6)

with mπ=138MeV,fπ=93MeV.The vacuum expectation value of the sigma?eld isσ0=fπand its vacuum mass mσis taken as a parameter.The vacuum mass of theω?eld is mω=783MeV,while the g4term for theω?eld represents a free parameter with?nite values causing a softening of the equation of state.Concerning the?→ρ?eld, we choose,among the possible SU(2)chiral invariant terms for the vector meson self-interaction,the one that has no self-interaction for theωmeson and noω?ρmixing term[25].This choice leads to a sti?er EoSs besides being in agreement with the observed small mixing of the two mesons.The vacuum mass of the?→ρmeson is mρ=761MeV.

III.MEAN FIELD APPROXIMATION

In a?rst approximation,to study dense cold matter,we neglect the?uctuations around the constant vacuum expectation values of the mesonic?eld operators.Only the time-like component of the isoscalar vector mesonˉω≡ω0 and the time-like third component of the isovector vector mesonρ30(where the upper index refers to isospin component and the lower index refers to the Lorentz component)of the?→ρ?eld remains(see for instance Ref.[26]).Additionally, parity conservation demandsˉπ=0.The mass eigenstates for the parity doubled nucleons,the N+and N?are

TABLE I:Parametrization and results for the two possible con?gurations within the MFT approximation considering M N

?

= 1200MeV and m0=790MeV.

mσ(MeV)318.56302.01

gω 6.08 6.77

gρ 4.22 4.18

FIG.1:Equations of state of beta-stable matter computed within the parity doublet model for the P1and P2parameter sets. For comparison also the equation of state of the relativistic mean?eld model GM3is shown(see text).The chiral symmetry restoration produces a softening of the equation of state at large energy densities.

determined by diagonalizing the mass matrix,Eq.(3),forψ1andψ2.Writing the coupling constants a and b as

functions of the mass of the positive parity nucleons M N

+=939MeV,the mass of the negative parity nucleons M N

?

,

the vacuum value of the scalar condensateσ0and the the bare mass term m0,the e?ective masses of the baryons are given by:

M?N

±

= 4?m20 σ22σ

V

=?L M+ iγi

2m2ωω20+g44ω40+

1

4

σ4(9)

+?σ+

1

FIG.2:The ratio σ/σ0is shown as a function of the baryon density for the P1and P2parameter sets for both symmetric matter and neutron star matter.In the case of neutron star matter,due to the beta stability and charge neutrality conditions,the beginning of the restoration of chiral symmetry takes place slightly before with respect to the case of symmetric matter.The transition is in all cases a smooth crossover.

where i ∈{n +,n ?,p +,p ?}denotes the nucleon type (positive and negative parity neutrons and positive and negative parity protons),γi is the fermionic degeneracy,k F i are the Fermi momenta,E ?

i

(k )=

k 2F +M ?i

2the corresponding e?ective chemical potential where I 3is the third component of the isospin (1/2for the positive and negative parity proton and -1/2for the positive and negative

parity neutron).The single particle energy of each parity partner i is given by E i (k )=E ?

i (k )+g ωω0+g ρρ30I 3.

Altogether there are six unknown parameters:g ω,g ρ,m σ,g 4,m 0,M N ?.

The

?rst

three are determined

by

the basic

nuclear

matter

saturation

properties,

i.e.,the

stable

minimum of the grand canonical potential for μB =923MeV has to meet three conditions:

E/A (μB =923MeV)?M N =?16MeV ,

ρ0(μB =923MeV)=0.16fm ?3,

a sym =32.5MeV ,

(10)

which are the measured values for the binding energy per nucleon,the baryon density and a phenomenologically reasonable value for the symmetry energy at saturation.The nuclear matter compressibility at saturation,de?ned as

K =9ρB 2?2

E/A ?ρB

ρB =ρ0

=9ρB ?μB ?σ ˉ

σ,

FIG.3:Number density fractions for the di?erent particles for the two equations of state here considered.The beta stability and charge neutrality conditions split the thresholds for the appearance of the negative parity neutron and of the negative parity proton.

0=?m 2ωˉω

?4g 44ˉω3

+g ω

i

ρi (ˉσ,ˉω,ˉρ)=0,

0=

?m 2ρˉρ

?g ρ(ρn +(ˉ

σ,ˉω,ˉρ)+ρn ?(ˉσ,ˉω,ˉρ)?ρp +(ˉσ,ˉω,ˉρ)?ρp ?(ˉσ,ˉω,ˉρ)).(12)

The energy density is obtained from the grand canonical potential:

?=?L M +

i

γi

(2π)3=

γi k 3

F

i

(2π)3

M ?

i 4π2

k F i E ?

F i

?M ?i 2ln

k F i +E ?

F

i

FIG.4:Left panel:mass radius relations for the di?erent equations of state.The horizontal line,representing the1.44M⊙of the Hulse-Taylor pulsar,allows to rule out the P2parametrization.Right panel:masses as functions of the central baryon density.The stars on the curves,which stand for the onsets of chiral symmetry restoration,indicate that for stars having a mass larger than1.2M⊙(for the P1case),the chiral partners of the nucleons are produced in the center of the stars.The maximum mass and the respective radius for the P1and P2parametrizations are shown in brackets next to the labels.

the appearance of the chiral partners and the chiral symmetry restoration,the isospin asymmetry has an e?ect also on the chiral restoration density.Moreover it a?ects also the order of the phase transition:for asymmetric matter the phase transition is smoother than for symmetric matter.The density for the beginning of the chiral symmetry restoration turns out to be very low,～2ρ0,for the P1and P2equations of state while it would have been higher if we have used a more massive chiral partner.It can also be seen in Fig.2that the e?ect of the vector-isoscalar meson self-coupling in the chiral restoration is practically negligible.In Fig.3,the number density fractions for the various particles are shown as functions of the baryon density.Notice the di?erent thresholds for the appearance of negative parity protons and negative parity neutrons.

Now we use the above described equations of state to compute the mass-radius relations and the structure of neutron stars by solving the Tolman-Oppenheimer-Volkov equations.We use the parity doublet model equations of state down to baryon densities of0.05fm?3while for lower densities we use the recent equation of state presented in Ref.[35].The results are shown in the left panel of Fig.4for the di?erent cases.In the same plot,we also indicate an horizontal line corresponding to the mass M max=1.44M⊙of the Hulse-Taylor binary pulsar,which is still the largest precisely known neutron star mass.Interestingly,considering the M max limit we can rule out the P2equation of state,indicating that in our model a self-interaction term for theωmeson renders the equation of state too soft. Taking into account the self-interaction of theρmeson or the mixing between theωand theρmeson would make the equation of state even softer.

Let us now study whether chiral symmetry is restored in neutron stars.The results are shown in the right panel of Fig.4where the masses as functions of the central baryon densities are plotted.The stars on the curves indicate the densities corresponding to the onsets of chiral symmetry restoration.We?nd that stars having a mass larger than 1.2M⊙have a core of a partially chiral restored phase for both cases.The same e?ect would not happen considering

=1535MeV.In that case the stars are unstable before the a more massive chiral partner,like for example with M N

?

central density reaches the chiral symmetry restoration threshold(see Ref.[24]).

IV.RELATIVISTIC HARTREE APPROXIMATION

The Relativistic Hartree Approximation(RHA)goes beyond mean?eld by accounting for the e?ect of the baryonic Dirac sea as one sums over the baryonic tadpole diagrams[36].The dressed propagator of a baryon i is obtained by solving the Dyson-Schwinger equation:

G H i=G0i(k)+G0i(k)ΣH i(k)G H i(k),(16) where G0i(k)is the free propagator andΣH i(k)the Hartree self-energy,which contains the scalar(ΣS)and vector (ΣV)parts

ΣH i=ΣS i?γμ(ΣV i)μ.(17) The solution of the Dyson-Schwinger equation is

[G H i(k)]?1=γμˉkμ?M?i,(18)

m 0 (MeV)

K (M e V )

FIG.5:Compressibility at saturation as a function of bare mass m 0for the mean

?eld

and

Hartree approximations for the P1parameter set.If the compressibility has a value of 270MeV,as indicated in Ref.[29],m 0～850MeV.

or,in an equivalent way,

G H i (k )=(γμˉk μ+M ?i )

1

E ?i

( k )δ(ˉk 0i ?E ?i ( k ))θ(k F,i ?| k |i )

=(G H i )F (k )+(G H i )D

(k ),

(19)

with

E ?i ( k )

=

m S 2i

d 4k m S 2i

d 4k m S 2i

i γi

d 4k

ˉk 2?M ?2i +i??ρi,S ,

(20)

(ΣV i )

μ

=i g V 2i

(2π)4

Tr (γ

μ

G H i (k ))

=i

g V 2i

(2π)4

Tr [γμ((G H i )F (k )+(G H i )D

(k ))]

=

g V 2i

(2π)4

ˉk μm S 2i

?ρi,S ,(22)

FIG.6:Equations of state for the doublet model in the four cases discussed in the text for the P1parametrization.

where the additional contribution to the scalar density for each baryon specie ?ρi,S is given by

?ρi,S =?

γi

M i

+M 2i (M i ?M ?i )?56

(M i ?M ?i )3 .(23)

As a consequence,the grand canonical potential is modi?ed inducing changes in the pressure,energy density and

the meson ?eld equations.In the parity doublet model,the energy density can be evaluated as

?RHA =?MF T +??,

(24)

with ?MF T being the mean ?eld result of Eq.(13).The contribution to the energy density from the Dirac sea,??,reads

??=? i

γi

M i

+M 3i (M i ?M ?i )

?

7

3

M i (M i ?M ?i )3?

25

?σ

RHA

=

?(?/V )

?σ

?ρi,S =0,(27)

where the mean ?eld contribution is found in Eq.(12).The coupling constants g ω,g ρand m σhave to be re-?tted in order to obtain the nuclear saturation properties.

Let us now apply this formalism to compute the equation of state.We start from the P1set of parameters.In Fig.5we show a comparison between the compressibility obtained in mean ?eld approximation and in relativistic Hartree approximation as a function of the bare mass m 0.Note that when m 0decreases,the scalar potential increases according to Eq.7.To keep the saturation properties,the increase of the scalar potential must be balanced by an increase of the vector potential which in turn induces a higher value of the compressibility (Eq.11).This e?ect is quite pronounced in the mean ?eld approximation as we can notice in the ?gure.On the other hand,in the relativistic Hartree approximation the scalar mesons are enhanced and the vector mesons are suppressed (Ref.[37,38]),causing the equation of state to be softer and,consequently,the compressibility to be smaller (Fig.5).

TABLE II:Parametrization and results for the?ve possible con?gurations within the Hartree approximation considering M N

=1200MeV and g4=0.

?

mσ(MeV)657484362322164

gω9.358.37 6.59 5.740.59

gρ 4.05 4.10 4.25 4.25 4.35

For the applications to neutron star matter,we choose four di?erent bare mass values keeping the mass of the =1200MeV and ignoring a self-coupling for the vector mesons g4=0,for the reasons described chiral partner M N

?

before.The parametrizations are:m0=300,600,790,900MeV(see table II for numerical values of the parameters for these m0s together with m0=750MeV).The four parametrizations are shown in Fig.6.It can again be seen that smaller bare masses generate sti?er equations of states.Although for a very small bare mass of m0=300MeV the compressibility is K=650MeV,which is too high according to phenomenology,this parametrization is still presented in the plots just for illustrative purposes.Finally we use the above described equations of state to compute the mass-radius relations and the structure of neutron stars by solving the Tolman-Oppenheimer-Volkov equation. The results are shown in the left panel of Fig.7for the di?erent cases.In the case of the highest bare mass value, the maximum mass of neutron stars is too low and,therefore,the corresponding equation of state is ruled out.We suggest that the equations of state compatible with the observed neutron star masses correspond to the case of bare masses between600and790MeV.For instance,for m0=750MeV a high maximum mass of1.9M⊙is obtained with a compressibility around400MeV.The results for P2in the relativistic Hartree approximation would be similar to the ones for P1,because the self-interaction of theωmeson does not change qualitatively the results(Ref.[37,38]). However,the equations of state would be softer and,consequently,the maximum masses for the neutron stars would be smaller than in the mean?eld case.For all the cases studied,the chiral symmetry starts to be restored inside neutron stars and chiral partners of the nucleons appear(see the right panel of Fig.7).

We can also use the bare mass parameters of the?ve parametrizations to study vacuum properties,as the pion-nucleon scattering or the decay width of the chiral partner,as was already done for N’(1535).We use the formula of Ref.[14]to compute the decay widthΓof N?→N+πas a function of m0.The result is shown in Fig.8,where it can be seen that the decay width increases quadratically as a function of m0,but,even for the higher bare mass value m0=900MeV,the width is still around100MeV.This value is too small to justify the assumption that the”true”chiral partner of the nucleon is an undetected resonance with a mass of1200MeV.For a more massive chiral partner =1379MeV,which is the limiting case for the formation of chiral partners inside neutron stars,we get as the M N

?

values of the order of300MeV for the width.Probably such a resonance is still not broad enough to have escaped experimental detection.This indicates the importance of improving our model to reconcile the?nite density matter properties with the microphysics of the interaction of the chiral partner with the nucleon and the pion.Working along this line is in progress.

V.CONCLUSIONS

We have studied the equation of state of nucleonic matter at zero temperature within the SU(2)parity doublet model by using?rst the mean?eld approximation.We assume that the chiral partner of the nucleon is a resonance, not yet detected,which has a mass of1200MeV.The parameters of the model are?xed by?tting the properties of nuclear matter at saturation.To obtain a reasonable value of the nuclear matter compressibility,the mixing parameter between the nucleon and its chiral partner m0turns out to be large,of the order of790MeV,thus indicating that the chiral condensate gives a minor contribution to the mass of the nucleon.We then studied the equation of state of beta-stable and charge neutral matter suitable for the applications to neutron stars.We have shown that a signi?cant softening of the equation of state is realized due to the transition from a chiral broken phase to a partially chiral restored phase at a density of roughly2ρ0,which corresponds also to the threshold for the appearance of the chiral partners of the nucleons.

We then used the equations of state for the computation of the mass-radius relations and structure of neutron stars. Taking into account the neutron star mass measurements,we have ruled out the possibility of having self-interaction terms for vector mesons in the Lagrangian since they render to equation of state too soft.Finally we have shown that

FIG.7:Left panel:mass radius relations for the di?erent choices of the bare mass parameter.Right panel:masses as functions of the central baryon density.The stars on the curves denote the onset of chiral symmetry restoration.

for neutron stars with masses larger than roughly1.2M⊙,chiral symmetry starts to be restored in their core and, therefore,the chiral partners of the nucleons appear.

As a second step we repeated the calculations by using the relativistic Hartree approximation.In this case,slightly smaller values of m0,down to750MeV,can still reproduce reasonable values of the compressibility(in the window of values200?400MeV)due to the suppression on the vector meson sector,which is a characteristic of this kind of approximation.Also within this approximation,we predict that the chiral partners of the nucleon could be formed at the center of neutron stars.

The hypothesis that the chiral partner of the nucleon is a very broad and therefore still undetected resonance with a relatively low mass leads to the population of chiral partners in neutron stars.However,as we have shown,within the present model,we obtain rather small values of the width of this particle.We need to improve the doublet parity model adopting for instance a gauged linear sigma model as done in Ref.[23]in which the physics of the vacuum is described more accurately.Another interesting extension of this work concerns the astrophysical implications of our results:to study proto-neutron stars in order to investigate when,during the time evolution of the stars,the chiral partners appear in the star and if this can have some observable signatures.Also the late cooling of neutron stars could be modi?ed if these new particles appear as they are opening new cooling processes.

Acknowledgments

We thank F.Giacosa for useful discussions.This work is partially supported by:INFN,BMBF project”Hadro-nisierung des QGP und dynamik von hadronen mit charm quarks”(ANBest-P and BNBest-BMBF98/NKBF98).

[1]T.D.Lee and G.C.Wick,Phys.Rev.D9,2291(1974).

[2]B.D.Serot and J.D.Walecka,Int.J.Mod.Phys.E6,515(1997),nucl-th/9701058.

[3]J.Boguta,Phys.Lett.B120,34(1983).

[4]I.Mishustin,J.Bondorf,and M.Rho,Nucl.Phys.A555,215(1993).

[5]R.J.Furnstahl,B.D.Serot,and H.-B.Tang,Nucl.Phys.A598,539(1996),nucl-th/9511028.

[6]R.J.Furnstahl,B.D.Serot,and H.-B.Tang,Nucl.Phys.A615,441(1997),nucl-th/9608035.

[7]P.Papazoglou,S.Schramm,J.Scha?ner-Bielich,H.Stoecker,and W.Greiner,Phys.Rev.C57,2576(1998),nucl-

th/9706024.

[8]P.Papazoglou et al.,Phys.Rev.C59,411(1999),nucl-th/9806087.

[9]P.Ko and S.Rudaz,Phys.Rev.D50,6877(1994).

[10]G.W.Carter,P.J.Ellis,and S.Rudaz,Nucl.Phys.A603,367(1996),nucl-th/9512033.

[11]G.W.Carter,P.J.Ellis,and S.Rudaz,Nucl.Phys.A618,317(1997),nucl-th/9612043.

[12]G.W.Carter and P.J.Ellis,Nucl.Phys.A628,325(1998),nucl-th/9707051.

[13]L.Bonanno,A.Drago,and https://www.sodocs.net/doc/5e8600370.html,vagno,Phys.Rev.Lett.99,242301(2007),0704.3707.

[14]C.DeTar and T.Kunihiro,Phys.Rev.D39,2805(1989).

[15]D.Jido,Y.Nemoto,M.Oka,and A.Hosaka,Nucl.Phys.A671,471(2000),hep-ph/9805306.

[16]D.Jido,T.Hatsuda,and T.Kunihiro,Phys.Rev.Lett.84,3252(2000),hep-ph/9910375.

[17]Y.Nemoto,D.Jido,M.Oka,and A.Hosaka,Phys.Rev.D57,4124(1998),hep-ph/9710445.

300400500

600700800900

m 0 MeV

20

406080100 M e V

FIG.8:The decay width of the process N ?→N +πis shown as a function of m 0for a ?xed value of the mass of the chiral partner M N ?=1200MeV.The width increases quadratically with m 0.

[18]H.-c.Kim,D.Jido,and M.Oka,Nucl.Phys.A640,77(1998),hep-ph/9806275.

[19]D.Jido,H.Nagahiro,and S.Hirenzaki,Phys.Rev.C66,045202(2002),nucl-th/0206043.[20]H.Nagahiro,D.Jido,and S.Hirenzaki,Phys.Rev.C68,035205(2003),nucl-th/0304068.[21]T.Hatsuda and M.Prakash,Phys.Lett.B224,11(1989).

[22]D.Zschiesche,L.Tolos,J.Scha?ner-Bielich,and R.D.Pisarski,Phys.Rev.C75,055202(2007),nucl-th/0608044.[23]S.Wilms,F.Giacosa,and D.H.Rischke (2007),nucl-th/0702076.

[24]V.Dexheimer,S.Schramm,and D.Zschiesche,Phys.Rev.C77,025803(2008),0710.4192.[25]V.Dexheimer and S.Schramm (2008),0802.1999.

[26]N.Glendenning,Compact Stars (Springer-Verlag,1997).

[27]M.V.Stoitsov,M.L.Cescato,P.Ring,and M.M.Sharma,J.Phys.G20,L149(1994),nucl-th/9403003.[28]D.Vretenar,T.Niksic,and P.Ring,Phys.Rev.C68,024310(2003),nucl-th/0302070.[29]D.H.Youngblood et al.,Phys.Rev.C69,034315(2004).[30]M.M.Sharma et al.,Phys.Rev.C38,2562(1988).[31]H.Kouno et al.,Phys.Rev.C51,1754(1995).

[32]Z.Ma,N.Van Giai,H.Toki,and M.L’Huillier,Phys.Rev.C 55,2385(1997).[33]B.K.Agrawal,S.Shlomo,and V.Kim Au,Phys.Rev.C 68,031304(2003).[34]N.K.Glendenning and S.A.Moszkowski,Phys.Rev.Lett.67,2414(1991).

[35]S.B.Ruester,M.Hempel,and J.Scha?ner-Bielich,Phys.Rev.C73,035804(2006),astro-ph/0509325.[36]B.D.Serot and J.D.Walecka,Adv.Nucl.Phys.16,1(1986).[37]A.Mishra,K.Balazs,D.Zschiesche,S.Schramm,H.St¨o cker,and W.Greiner,Phys.Rev.C 69,024903(2004).[38]D.Zschiesche,PhD dissertation,Johann Wolfgang Goethe University Frankfurt am Main,Germany (2004).

一个小游戏引发的深刻思考

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