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PHYSICAL REVIEW C 72, 045801 (2005)
Neutron stars and strange stars in the chiral SU(3) quark mean field model
P. Wang,1,2 S. Lawley,1,2 D. B. Leinweber,1 A. W. Thomas,2 and A. G. Williams11Special Research Center for the Subatomic Structure of Matter (CSSM) and Department of Physics, University of Adelaide 5005, Australia
2Jefferson Laboratory, 12000 Jefferson Ave., Newport News, Virginia 23606 USA(Received 6 June 2005; published 27 October 2005)
We investigate the equations of state for pure neutron matter and for nonstrange and strange hadronic matterin equilibrium, including ,, and hyperons. The masses and radii of these kinds of stars are obtained.For a pure neutron star, the maximum mass is about 1.8Msun, while for a strange (nonstrange) hadronic starin equilibrium, the maximum mass is around 1.45Msun (1.7Msun). The typical radii of pure neutron stars andstrange hadronic stars are about 11.513.0 km and 11.512.5 km, respectively.
DOI: 10.1103/PhysRevC.72.045801 PACS number(s): 26.60.+c, 12.39.x, 21.65.+f, 21.80.+a
I. INTRODUCTION
Hadronic matter under extreme conditions has attracted alot of interest in recent years. On the one hand, many theoreticaland experimental efforts have been devoted to the discussionof heavy ion collisions at high temperatures. On the otherhand, the physics of neutron stars has become a hot topic thatconnecting astrophysics with highdensity nuclear physics. In1934, Baade and Zwicky [1] suggested that neutron stars couldbe formed in supernovae. The first theoretical calculation ofa neutron star was performed by Oppenheimer and Volkoff[2] and independently by Tolman [3]. Observing a range ofmasses and radii of neutron stars will reveal the equationsof state (EOS) of dense hadronic matter. Determination ofthe EOS of neutron stars has been an important goal formore than two decades. Six double neutronstar binaries areknown so far, and all of them have masses in the surprisinglynarrow range of 1.36 0.08Msun [4,5]. A number of earlytheoretical investigations on neutron stars were based on thenonrelativistic Skyrme framework [6]. Since the Waleckamodel [7] was proposed and applied to study the propertiesof nuclear matter, the relativistic mean field approach has beenwidely used in the determination of the masses and radii ofneutron stars. These models lead to different predictions forneutronstar masses and radii [8,9]. For a recent review, seeRef. [10]. Though models with maximum neutronstar massesconsiderably smaller than 1.4Msun are simply ruled out, theconstraint on EOS of nuclear matter (for example, the densitydependence of pressure of a hadronic system) has certainly notbeen established from the existing observations.
In the process of neutronstar formation, equilibrium canbe achieved. As a consequence, hyperons will exist in neutronstars, especially in stars with high baryon densities. Thesehyperons will affect the EOS of hadronic matter. As a result,the massradius relationship of strange hadronic stars will bequite different from that of pure neutron stars. The simplestway to discuss the effects of hyperons is to study strangehadronic stars including only hyperons. This is because is the lightest hyperon and the N interaction is knownbetter than other hyperonnucleon interactions. However, onemust also consider hyperons with negative charge in neutronstars because the negatively charged hyperons can substitutefor electrons. There have been many discussions of strange
hadronic stars including hyperons, and , or even thewhole baryon octet [1120].
At high baryon density, the overlap effects of baryonsare very important, and the quark degrees of freedom withinbaryons should be considered. Some phenomenological models are based on the quark degrees of freedom, such as thequarkmeson coupling model [21], the cloudy bag model [22],the quark mean field model [23], and the NambuJonaLasinio(NJL) model [24]. Several years ago, a chiral SU(3) quarkmean field model was proposed [25,26]. In this model,quarks are confined within baryons by an effective potential.The quarkmeson interaction and meson selfinteraction arebased on SU(3) chiral symmetry. Through the mechanismof spontaneous symmetry breaking, the resulting constituentquarks and mesons (except for the pseudoscalars) obtainmasses. The introduction of an explicit symmetry breakingterm in the meson selfinteraction generates the masses ofthe pseudoscalar mesons which satisfy the partially conservedaxialvector current (PCAC) relations. The explicit symmetrybreaking term in the quarkmeson interaction gives reasonablehyperon potentials in hadronic matter. This chiral SU(3) quarkmean field model has been applied to investigate nuclear matter[27], strange hadronic matter [25], finite nuclei, hypernuclei[26], and quark matter [28]. Recently, we improved the chiralSU(3) quark mean field model by using the linear definition ofeffective baryon mass [29]. This new treatment is applied tostudy the liquidgas phase transition of an asymmetric nuclearsystem and strange hadronic matter [30,31]. By and large theresults are in reasonable agreement with existing experimentaldata.
In this paper, we will study the neutron star and strangestar in the chiral SU(3) quark mean field model. The paper isorganized in the following way. In Sec. II, we briefly introducethe model. In Sec. III, we apply this model to investigate theneutron star and strange hadronic star. The numerical resultsare discussed in Sec. IV. We summarize the main results inSec. V.
II. THE MODEL
Our considerations are based on the chiral SU(3) quarkmean field model (for details see Refs. [25,26]), which containsquarks and mesons as the basic degrees of freedom. In the
05562813/2005/72(4)/045801(8)/$23.00 0458011 2005 The American Physical Society
http://dx.doi.org/10.1103/PhysRevC.72.045801
WANG, LAWLEY, LEINWEBER, THOMAS, AND WILLIAMS PHYSICAL REVIEW C 72, 045801 (2005)
chiral limit, the quark field can be split into left andrighthanded parts L and R: = L +R . UnderSU(3)L SU(3)R they transform as
L L = LL, R R = RR. (1)The spin0 mesons are written in the compact form
M
M= i = 1
2
8a=0
(sa ipa)a, (2)
where sa and pa are the nonets of scalar and pseudoscalarmesons, respectively, a(a = 1, . . . , 8) are the GellMannmatrices, and 0 =
23 I . The alternatives indicated by the
plus and minus signs correspond to M and M, respectively.Under chiral SU(3) transformations, M and M transform asM M = LMR and M M = RML. The spin1mesons are arranged in a similar way as
l
r= 1
2(V A) = 1
2
2
8a=0
(va aa
)a, (3)
with the transformation properties l l = LlL andr r = RrR. The matrices ,,V, and A can bewritten in a form where the physical states are explicit. For thescalar and vector nonets, we have the expressions
= 12
8a=0
saa
=
12
( + a00
)a+0 K
+
a012
( a00
)K0
K K0
, (4)
V = 12
8a=0
vaa
=
12
( + 0
)+ K
+
12
( 0
)K0
K K0
. (5)
Pseudoscalar and pseudovector nonet mesons can be writtenin a similar fashion.
The total effective Lagrangian is written
Leff = L0 +LqM +L +LVV +LSB +Lms +Lh +Lc,(6)
where L0 = i is the free part for massless quarks.The quarkmeson interaction LqM can be written in a chiralSU(3) invariant way as
LqM = gs(LMR + RML) gv(L lL + R rR)
= gs2
(8
a=0saa + i 5
8a=0
paa
)
gv2
2
(
8a=0
vaa 58
a=0aaa
). (7)
From the quarkmeson interaction, the coupling constantsbetween scalar mesons, vector mesons, and quarks have thefollowing relations:
gs2= gua0 = gda0 = gu = gd = =
12gs ,
(8)gsa0 = gs = gu = gd = 0,gv
2
2= gu = gd = gu = gd = =
12gs,
(9)gs = gs = gu = gd = 0.
In the mean field approximation, the chiralinvariant scalarmeson L and vector meson LVV selfinteraction terms arewritten as [25,26]
L = 12k0
2( 2 + 2) + k1( 2 + 2)2
+ k2( 4
2+ 4
)+ k3 2
k44 144ln
4
40+
34ln
2
20 0, (10)
LVV = 12
2
20
(m2
2 +m22 +m22)
+ g4(4 + 622 + 4 + 24), (11)where = 6/33; 0 and 0 are the vacuum expectation valuesof the corresponding mean fields , which are expressed as
0 = F, 0 = 12
(F 2FK ). (12)
The vacuum value 0 is about 280 MeV in our numericalcalculation. The Lagrangian LSB generates the nonvanishingmasses of pseudoscalar mesons
LSB = 2
20
[m2F +
(2m2KFK
m22F
)
], (13)
leading to a nonvanishing divergence of the axial currentswhich in turn satisfy the PCAC relations for and K mesons.Pseudoscalar and scalar mesons and also the dilaton field obtain mass terms by spontaneous breaking of chiral symmetryin the Lagrangian of Eq. (10). The masses of u, d, and s quarksare generated by the vacuum expectation values of the twoscalar mesons and . To obtain the correct constituent massof the strange quark, an additional mass term has to be added:
Lms = msqSq, (14)where S = 13 (I 8
3) = diag(0, 0, 1) is the strangeness
quark matrix. Based on these mechanisms, the quark constituent masses are finally given by
mu = md = gs20 and ms = gs0 +ms. (15)
The parameters gs = 4.76 and ms = 29 MeV are chosento yield the constituent quark masses mq = 313 and ms =490 MeV. The hyperon potential felt by baryon j in i matter is
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NEUTRON STARS AND STRANGE STARS IN THE . . . PHYSICAL REVIEW C 72, 045801 (2005)
defined as
U(i)j = Mj Mj + gj + gj. (16)
To obtain reasonable hyperon potentials in hadronic matter,we include an additional coupling between strange quarks andthe scalar mesons and [25]. This term is expressed as
Lh = [h1( 0) + h2( 0)]ss. (17)Therefore, the strange quark scalarcoupling constants aremodified and do not exactly satisfy Eq. (8). In the quark meanfield model, quarks are confined in baryons by the LagrangianLc = c [with c given in Eq. (18), below]. We note thatthis confining term is not chiral invariant. Possible extensionsof the model that would restore chiral symmetry in this termhave been discussed in Ref. [32].
The Dirac equation for a quark fieldij under the additionalinfluence of the meson mean fields is given by
[i + c(r) + mi ]ij = ei ij , (18)where = 0 , = 0, the subscripts i and j denote thequark i (i = u, d, s) in a baryon of type j (j = N,,,)and c(r) is a confinement potential, i.e., a static potentialproviding the confinement of quarks by meson mean fieldconfigurations. In the numerical calculations, we choosec(r) = 14kcr2, where kc = 1 (GeV fm2), which yieldsbaryon radii (in the absence of the pion cloud [33]) around0.6 fm. The quark mass mi and energy e
i are defined as
mi = gi gi +mi0 (19)and
ei = ei gi gi gi, (20)where ei is the energy of the quark under the influence ofthe meson mean fields. Here mi0 = 0 for i = u, d (nonstrangequark) andmi0 = ms for i = s (strange quark). The effectivebaryon mass can be written as
Mj =i
nij ei E0j , (21)
where nij is the number of quarks with flavor i in a baryonwith flavor j, with j = N{p, n}, {, 0}, {0, }, and; and E0j is only very weakly dependent on the externalfield strength. We therefore use Eq. (21), with E0j a constant,independent of the density, which is adjusted to give a best fitto the free baryon masses. Here we use the linear definitionof effective baryon mass instead of the earlier square rootansatz. As we explained in Ref. [29], the linear definition ofeffective baryon mass has been derived using a symmetricrelativistic approach [34], while to the best of our knowledge,no equivalent derivation exists for the square root case.
III. HADRONIC SYSTEM
Based on the previously defined quark mean field model, thethermodynamical potential for the study of hadronic systems
is written as
=
j=N,,,
2kBT(2 )3
0
d3k{ln(1 + e(Ej (k)j )/kBT )
+ ln(1 + e(Ej (k)+j )/kBT )} LM, (22)where Ej (k) =
M2j + k2 and Mj is the effective baryon
mass. The quantity j is related to the usual chemical potentialj by j = j gj gj gj. The mesonic Lagrangian
LM = L + LVV + LSB (23)describes the interaction between mesons, which includes thescalar meson selfinteraction L , the vector meson selfinteraction LVV , and the explicit chiral symmetrybreakingterm LSB defined previously in Eqs. (10), (11), and (13).The Lagrangian LM involves scalar (, , and ) and vector(, , and ) mesons. The interactions between quarks andscalar mesons result in the effective baryon masses Mj .The interactions between quarks and vector mesons generatethe baryonvector meson interaction terms. The energy pervolume and the pressure of the system can be derived as = 1
TT
+ jj and p = , where j is the densityof baryon j. At zero temperature, can be expressed as
=
j=N,,,
1
242
{j[2j M2j
]1/2[22j 5M2j
]
+ 3M4j ln[j +
(2j M2j
)1/2Mj
]} LM. (24)
The equations for mesons i can be obtained by the formulai
= 0. Therefore, the equations for , , and are
k02 4k1( 2 + 2) 2k2 3 2k3 2
34
+ 2
20m2F
(
0
)2m
2 m
(
0
)2m
2 m
+
j=N,,,
Mj
jj = 0, (25)
k02 4k1( 2 + 2) 4k2 3 k3 2
34
+ 2
20
(2m2kFk
12m2F
)
(
0
)2m
2 m
+
j=,,
Mj
jj = 0, (26)
k0 (2 + 2) k3 2+(
4k4 + 1 + 4 ln 0
43
ln 2
20 0
)3
+ 220
[m2F +
(2m2kFk
12m2F
)
]
20
(m2
2 +m22 +m22) = 0, (27)
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WANG, LAWLEY, LEINWEBER, THOMAS, AND WILLIAMS PHYSICAL REVIEW C 72, 045801 (2005)
TABLE I. Hyperon potentials in MeV.
U(N)N U
(N) U
(N) U
(N) U
() U
() U
() U
() U
() U
()
64.0 28.0 28.0 8.0 24.5 24.5 20.6 30.6 30.6 50.3
where jj is expressed as
jj =Mj2
kFj0
dkk2
M2j + k2
= M3j
22
kFjMj
1+ k2FjM2j
lnkFjMj
+1+ k2Fj
M2j
,
(28)
with kFj =2j M2j .
For the equilibrium, the chemical potentials for thebaryons satisfy the following equations:
= 0 = 0 = n = p + e = p + , (29)+ = p, (30)
= = n + e. (31)There are only two independent chemical potentials which aredetermined by the total baryon density and neutral charge:
B = p + n + + + + 0 + + 0 + , (32)
p + + e = 0. (33)To get the massradius relation, one has to resolve the
TolmanOppenheimerVolkoff (TOV) equation:
dp
dr= [p(r) + (r)][M(r) + 4r
3p(r)]
r(r 2M(r)) , (34)
where
M(r) = 4 r
0(r)r2dr. (35)
With the equations of state, functions such asM(r), (r), p(r),can be obtained.
IV. NUMERICAL RESULTS
The parameters of this model were determined by the mesonmasses in vacuum and the saturation properties of nuclearmatter. The improved linear definition of effective baryon massis chosen in our numerical calculations. The binding energy ofsymmetric nuclear matter is 16 MeV, and the saturation density0 is 0.16 fm3. The incompression modulus at 0 is 303 MeV.The hyperon potentials are listed in Table I. For the hyperon,the empirical value of U (N) is 28 MeV at the saturationdensity of nuclear matter 0 [35]. ForU
(N) , recent experiments
suggest that U (N) may be 14 or less [36]. In matter,the typical values of U ()j (j = ,) are around 20 MeVat density = 0/2 [37]. In matter, U ()j (j = ,) arearound 40 MeV at density = 0 [37]. From Table I, one
can see that though only two parameters h1 and h2 are adjustedin this model, most of the hyperon potentials are reasonable.
We first discuss the equations of state of neutron matter andstrange hadronic matter which are needed for the calculation ofneutron stars. For pure neutron stars, only neutrons are present.For strange hadronic stars, with increasing baryon density,other kinds of baryons will appear. In Fig. 1, we show thefractions of octet baryons versus density with equilibrium.With increasing baryon density, the neutron fraction decreasesslowly from 1. If the density is lower than about 0.19 fm3,the fraction of electrons is the same as that of protons, whichmakes the system charge neutral. The muon appears whenthe density is in the range 0.190.98 fm3. The maximumfractions of muons and electrons appear at B 0.4 fm3.Their fractions decrease with increasing fractions of hyperons.When the density is larger than about 0.4 fm3, the hyperons appear and the fraction of neutrons decreases faster.After the density is larger than about 0.57 fm3, hyperonsstart to appear. The fraction of hyperons decreases withincreasing density after hyperons appear where the densityis about 0.84 fm3.
The density dependence of the effective baryon masses andscalar mean fields are shown in Fig. 2. The field decreasesquickly with the increasing baryon density when the densityis small, B < 0.4 fm3. This is because at small baryondensity, the nucleon is dominant and there are no hyperons.With increasing density, more and more hyperons appear. Asa result, the field decreases quickly. At a broad range ofdensities, the value of changes little.
0.0 0.5 1.0 1.5 2.0
0.01
0.1
1

e
+
0
0
p
n
i/
B
B
( fm3 )
FIG. 1. Fractions of proton, neutron, ,, and of strangehadronic stars vs baryon density with equilibrium.
0458014
NEUTRON STARS AND STRANGE STARS IN THE . . . PHYSICAL REVIEW C 72, 045801 (2005)
0.0 0.4 0.8 1.2 1.60.0
0.2
0.4
0.6
0.8
1.0
N
/
0,
Mj*
/M
j
B
( fm3 )
FIG. 2. Effective baryon masses and meson mean fields vs baryondensity with equilibrium.
In Fig. 3, the pressure versus baryon density is shown.When the density is low, the three curves are close to eachother. With increasing baryon density, the contribution fromprotons and hyperons is not negligible. For nuclear matter,some of the protons and neutrons are in states of high energyand momentum when the density is high, because of the Pauliexclusion principle. For strange hadronic matter, nucleons canbe replaced by hyperons that are in states with lower kinetic
0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
p(
fm4
)
B
( fm3 )
FIG. 3. Pressure of hadronic matter p vs baryon density B .Dotted, dashed, and solid curves are for pure neutron stars, andfor nonstrange and strange hadronic stars with equilibrium,respectively.
0 2 4 6 8 10 12
0.0
0.5
1.0
1.5
2.0
2.5
3.0
p(
fm4
) ( fm4 )
FIG. 4. Pressure of hadronic matter p vs energy density . Curvesare the same as in Fig. 3.
energy. Thus the momenta of the hyperons are lower, and fora given baryon density, the strange hadronic matter has lowerpressure and hence a softer equation of state. As plottedin Fig. 3, at a given baryon density, the pressure of strangehadronic matter is the smallest one among these three curves.The pressure p versus energy density is shown in Fig. 4.Again, one can see that the equation of state of strange hadronicmatter is softer than those in the other two cases.
In Fig. 5, we plot the derivative of pressure with respectto energy per unit volume. At low density, for example, less
0.0 0.5 1.0 1.5 2.0
0.0
0.1
0.2
0.3
0.4
dp/d
B
( fm3 )
FIG. 5. Derivative of pressure with respect to energy density vsbaryon density. Curves are the same as in Fig. 3.
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WANG, LAWLEY, LEINWEBER, THOMAS, AND WILLIAMS PHYSICAL REVIEW C 72, 045801 (2005)
than 20, dp/d is smaller than 0.2. For nonstrange matter,when the baryon density is larger than about 40, dp/dapproaches 1/3, which means that the baryons are relativistic.For strange hadronic matter, when hyperons appear, dp/dbecomes smaller compared with the other two curves. Thenonsmooth change of the curve can be understood from Fig. 4,where the slope of the curve for strange hadronic matterchanges discontinuously. This behavior is due to the effectof hyperons (see Fig. 1). Our results are comparable withRef. [38], where strange hadronic matter was studied in arelativistic mean field model. For strange matter, though thetotal baryon density can be as large as 100, the density ofeach kind of baryon is not high enough to make the baryonsbehave as highly relativistic particles.
We now study neutron stars with the obtained EOS. Becausethe nucleon crust makes an important contribution to a starsradius, especially for stars with low central density [18,39],we will replace the obtained EOS at low density by those forthe crust. At low density, the EOS of Negele and Vautherin[40] are close to those of Baym et al. [41]. In the numericalcalculation, the crust data in Ref. [40] at density smaller than0.1 fm3 are used. By solving the TOV equation, the baryondensity vs radius can be obtained as shown in Fig. 6. Thecentral densities c are chosen to be 30 and 50 where 0(0.16 fm3) is the saturation density of symmetric nuclearmatter. With increasing radius, the density of strange hadronicstars decreases a little faster than that of pure neutron stars,which results in a smaller radius. The radii of stars are not verysensitive to their central density c when c is in some region.For example, from the figure, one can see that for c = 30and c = 50, the difference of the star radii is less than 1 km.Figure 6 also shows that when the density is lower than about0.1 fm3, a nucleon crust exists on the surface of a star with aradius of about several hundred meters. Calculations show that
0 3 6 9 12 150.0
0.2
0.4
0.6
0.8
1.0
c
= 30
c
= 50
B(
fm3
)
R ( km )
FIG. 6. Baryon density of hadronic stars vs radii. Curves are thesame as in Fig. 3.
0.0 0.4 0.8 1.2 1.60.0
0.5
1.0
1.5
2.0
M/
Msu
n
B
( fm3 )
FIG. 7. Masses of hadronic stars vs their central baryon densities.Curves are the same as in Fig. 3.
the nucleon crust has little effect on the stars mass. However,the radius of a neutron star will be increased by the crustcontribution, especially in the case of stars with low centraldensity.
We plot the star mass ratio M/Msun versus central baryondensity in Fig. 7. The maximum mass of pure neutronstars is about 1.8Msun with a central density 1.05 fm3.Beyond that density, the star becomes unstable. The maximummass changes to 1.7Msun and 1.45Msun when proton andhyperons are included. When the central density is smaller than0.4 fm3, there is no hyperon. Therefore, the solid anddashed lines are the same in the small density region. In therange 30 < c < 60, the masses of pure neutron stars,and of nonstrange and strange hadronic stars with equilibrium are 1.48Msun < M < 1.8Msun, 1.32Msun < M