Published on

07-Apr-2017View

213Download

1

Embed Size (px)

Transcript

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 high-density 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 neutron-star 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 forneutron-star masses and radii [8,9]. For a recent review, seeRef. [10]. Though models with maximum neutron-star 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 neutron-star 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 mass-radius 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 hyperon-nucleon 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 mod-els are based on the quark degrees of freedom, such as thequark-meson coupling model [21], the cloudy bag model [22],the quark mean field model [23], and the Nambu-Jona-Lasinio(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 quark-meson interaction and meson self-interaction 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 self-interaction generates the masses ofthe pseudoscalar mesons which satisfy the partially conservedaxial-vector current (PCAC) relations. The explicit symmetrybreaking term in the quark-meson 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 liquid-gas 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

0556-2813/2005/72(4)/045801(8)/$23.00 045801-1 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- andright-handed parts L and R: = L +R . UnderSU(3)L SU(3)R they transform as

L L = LL, R R = RR. (1)The spin-0 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 Gell-Mannmatrices, 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 spin-1mesons 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 quark-meson 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 quark-meson 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 chiral-invariant scalarmeson L and vector meson LVV self-interaction 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 con-stituent 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

045801-2

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 scalar-coupling 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 self-interaction L , the vector meson self-interaction LVV , and the explicit chiral symmetry-breakingterm 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 baryon-vector 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)

045801-3

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 mass-radius relation, one has to resolve the

Tolman-Oppenheimer-Volkoff (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 inc...