Chiral SU(3) quark mean-field model for hadronic systems

P. Wanga, D. B. Leinwebera, A. W. Thomasab and A. G. Williamsa

aSpecial Research Center for the Subatomic Structure of Matter (CSSM) andDepartment of Physics, University of Adelaide 5005, Australia

bJefferson Laboratory, 12000 Jefferson Ave., Newport News, VA 23606 USA

We perform a study of infinite hadronic matter and finite nuclei at finite temperature and density with animproved method of calculating the effective baryon mass. A detailed study of the predictions of the model ismade in comparison with the available data and the level of agreement is generally very good.

1. INTRODUCTION

The complex action of QCD at finite chem-ical potential makes it difficult to study finite-density QCD properties directly from first prin-ciples lattice calculations. There are many phe-nomenological models based on hadronic degreesof freedom, such as the Walecka model [1], theZimanyi-Mozkovski model [2] and the nonlinearσ − ω model [3], as well as models which ex-plicitly include quark degrees of freedom, for ex-ample, the quark meson coupling model [4], thecloudy bag model [5], the quark mean-field model[6] and the NJL model [7]. With the developmentof these phenomenological models, the interac-tions between hadrons become more and morecomplex. The symmetries of QCD can be usedto determine largely how the hadrons should in-teract with each other. With this in mind, thechiral SU(2) × SU(2) effective quark model wasproposed. It has been used widely to investigatenuclear matter and finite nuclei at both zero andfinite temperature.

Recently, we proposed a chiral SU(3) quarkmean-field model based on quark degrees of free-dom [8,9]. In this model, quarks are confined inthe baryons by an effective potential. The quark-meson interaction and meson self-interaction arebased on SU(3) chiral symmetry. Through themechanism of spontaneous symmetry breakingthe resulting constituent quarks and mesons (ex-cept for the pseudoscalars) obtain masses. Theintroduction of an explicit symmetry breaking

term in the meson self-interaction generates themasses of the pseudoscalar mesons which sat-isfy the relevant PCAC relations. The explicitsymmetry breaking term in the quark-meson in-teraction gives reasonable hyperon potentials inhadronic matter. This chiral SU(3) quark mean-field model has been applied to investigate nu-clear matter [10], strange hadronic matter [8], fi-nite nuclei, hypernuclei [9], and quark matter [11].By and large the results are in reasonable agree-ment with existing experimental data.

A surprising result, which is quite differentfrom most other models is that at some criticaldensity the effective baryon mass drops to zero[12]. There is a first order phase transition aroundthis critical density where the physical quantitieschange discontinuously. We will see later that thisbehavior is primarily a consequence of the nonlin-ear ansatz for the effective baryon mass which isrelated to the subtraction of the centre of mass(c.m.) motion. In Ref. [13], the authors pro-vided an exact solution of the effect of c.m. mo-tion and found that it was only very weakly de-pendent on the external field strength for the den-sities of interest. As a result, the c.m. correctioncan be included in the zero point energy. In thispaper, we will use this alternative definition ofthe effective baryon mass in medium to reexam-ine the properties of nuclear systems. A success-ful model should describe well the properties ofnuclear matter, not only at zero temperature butalso at finite temperature. We will also apply thechiral SU(3) quark mean-field model to study the

Nuclear Physics B (Proc. Suppl.) 141 (2005) 273–278

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liquid-gas phase transition and Comloub instabil-ity of asymmetric nuclear systems and compareour results with the recent experimental analysis.

The paper is organized as follows. The model isintroduced in Sec. II. In Sec. III, we study nuclearsystems at finite temperature and density. Thenumerical results are presented in Sec. IV.

2. THE MODEL

Our considerations are based on the chiralSU(3) quark mean-field model (for details seeRefs. [8,9]), which contains quarks and mesonsas basic degrees of freedom. In the chiral limit,the quark field q can be split into left and right-handed parts qL and qR: q = qL + qR. UnderSU(3)L× SU(3)R they transform as

qL → q′L = L qL, qR → q′R = R qR . (1)

The spin-0 mesons are written in the compactform

M(M+) = Σ ± iΠ =1√2

8∑

a=0

(sa ± ipa)λa, (2)

where sa and pa are the nonets of scalar and pseu-doscalar mesons, respectively, λa(a = 1, ..., 8) are

the Gell-Mann matrices, and λ0 =√

23 I. The

alternatives, plus and minus signs correspond toM and M+. Under chiral SU(3) transformations,M and M+ transform as M → M ′ = LMR+ andM+ → M+′

= RM+L+. The spin-1 mesons arearranged in a similar way as

lµ(rµ) =12

(Vµ ± Aµ) =1

2√

2

8∑

a=0

(va

µ ± aaµ

)λa (3)

with the transformation properties: lµ → l′µ =LlµL+, rµ → r′µ = RrµR+. The total effectiveLagrangian is written:

Leff = Lq0 + LqM + LΣΣ + LV V + LχSB

+L∆ms + Lh + Lc (4)

where Lq0 is the free part for massless quarks.The quark-meson interaction LqM can be writtenin a chiral SU(3) invariant way as

LqM = gs

(ΨLMΨR + ΨRM+ΨL

)

−gv

(ΨLγµlµΨL + ΨRγµrµΨR

). (5)

The meson self-interaction, LΣΣ and LV V , canalso be written in a chiral invariant way [8,9]. TheLagrangian, LχSB , generates the nonvanishingmasses of pseudoscalar mesons, leading to a non-vanishing divergence of the axial currents, whichin turn satisfy the partial conserved axial-vectorcurrent (PCAC) relations for π and K mesons.Pseudoscalar, scalar mesons and also the dilatonfield χ obtain mass terms by spontaneous break-ing of chiral symmetry. The masses of u, d and squarks are generated by the vacuum expectationvalues of the two scalar mesons, σ and ζ. To ob-tain the correct constituent mass of the strangequark, an additional mass term, L∆ms , has to beadded. In order to obtain reasonable hyperon po-tentials in hadronic matter, we include an addi-tional coupling, Lh, between strange quarks andthe scalar mesons, σ and ζ [8]. In the quark mean-field model, quarks are confined in baryons by theLagrangian, Lc = −Ψχc Ψ (with χc given in Eq.(6), below). The Dirac equation for a quark fieldΨij under the additional influence of the mesonmean fields is given by[−iα · ∇ + χc(r) + βm∗

i

]Ψij = e∗i Ψij , (6)

where α = γ0γ , β = γ0 , the subscripts i and jdenote the quark i (i = u, d, s) in a baryon of typej (j = N, Λ, Σ, Ξ) ; χc(r) is a confining potential– i.e., a static potential providing confinement ofquarks by meson mean-field configurations. Thequark mass, m∗

i , and energy, e∗i , are defined as

m∗i = −gi

σσ − giζζ + mi0 (7)

and

e∗i = ei − giωω − gi

φφ , (8)

where ei is the energy of the quark under the in-fluence of the meson mean scalar fields. Using thesolution of the Dirac equation (6) for the quarkenergy, e∗i , it has been common to define the ef-fective mass of the baryon j through the ansatz:

M∗j =

√E∗2

j − < p∗2j cm > , (9)

where E∗j =

∑i nije

∗i + Ej spin is the baryon en-

ergy and < p∗2j cm > is the subtraction of thecontribution to the total energy associated with

P. Wang et al. / Nuclear Physics B (Proc. Suppl.) 141 (2005) 273–278274

spurious center of mass motion. In the expres-sion for the baryon energy nij is the number ofquarks with flavor ”i” in a baryon with flavorj, with j = N p, n , Σ Σ±, Σ0 , Ξ Ξ0, Ξ− , Λand Ej spin is the correction to the baryon energywhich is determined from a fit to the data forbaryon masses.

There is an alternative way to remove the spu-rious c.m. motion and determine the effectivebaryon masses. In Ref. [13], the removal of thespurious c.m. motion for three quarks moving ina confining, relativistic oscillator potential wasstudied in some detail. It was found that when anexternal scalar potential was applied, the effectivemass obtained from the interaction Lagrangiancould be written as

M∗j =

∑

i

nije∗i − E0

j , (10)

where E0j was found to be only very weakly de-

pendent on the external field strength. We there-fore use Eq. (10), with E0

j a constant, indepen-dent of the density, which is adjusted to give abest fit to the free baryon masses.

3. HADRONIC SYSTEM

Based on the previously defined interaction, thethermodynamic potential for hadronic matter iswritten as

Ω = −∑

N=p,n

kBT

(2π)3

∫ ∞

0

d3−→k

ln(1 + e−(E∗

N (k)−νN )/kBT)

(11)

+ln(1 + e−(E∗

N (k)+νN )/kBT)

− LM ,

where E∗N (k) =

√M∗2

N + −→k

2. The quantity νN

is related to the usual chemical potential, µN , byνN = µN − gN

ω ω − gNρ ρ. The energy per unit

volume and the pressure of the system are re-spectively ε = Ω − 1

T∂Ω∂T + νNρN and p = −Ω,

where ρN is the baryon density. The mean-fieldequation for meson φi is obtained by the formula∂Ω/∂φi = 0.

Let us now discuss the liquid-gas phase transi-tion. For asymmetric nuclear matter, the system

Figure 1. The effective nucleon mass M∗N/MN

versus nuclear density ρN . The solid and dashedlines are for linear and square root treatments ofeffective baryon mass, respectively.

will be stable against separation into two phases ifthe free energy of a single phase is lower than thefree energy in all two-phase configurations. Thisrequirement can be formulated as [14]

F (T, ρ) < (1 − λ)F (T, ρ′) + λF (T, ρ′′), (12)

where

ρ = (1 − λ)ρ′ + λρ′′, 0 < λ < 1, (13)

and F is the Helmholtz free energy per unit vol-ume. The two phases are denoted by a prime anda double prime. If the stability condition is vio-lated, a system with two phases is energeticallyfavorable. The phase coexistence is governed bythe Gibbs conditions:

µ′j(T, ρ′) = µ′′

j (T, ρ′′) (j = p, n), (14)

p′(T, ρ′) = p′′(T, ρ′′), (15)

where the temperature is the same in the twophases. For finite nuclei, compared with thecase of infinite nuclear matter, the size effect andCoulomb interaction are important. When theCoulomb interaction is considered, the chemicalpotential for the protons will have an additionalterm [15,16].

P. Wang et al. / Nuclear Physics B (Proc. Suppl.) 141 (2005) 273–278 275

Figure 2. The energy per nucleon E/A versus nu-clear density ρN . The solid and dashed lines arefor linear and square root treatments of effectivebaryon mass, respectively.

4. NUMERICAL RESULTS

The parameters in this model are determinedby the meson masses in vacuum and the proper-ties of nuclear matter which have been discussedin the earlier papers [8,17]. We first consider in-finite nuclear matter. In Fig. 1, we show the ef-fective nucleon mass divided by the free nucleonmass, M∗

N/MN , versus nuclear density. The ef-fective mass for the square root ansatz decreasesfaster than that for the linear definition and, asa consequence, at some critical density the effec-tive mass drops to zero. However, in the case ofthe linear definition, the effective mass decreasesslowly at high density and there is no phase tran-sition to a state of chiral symmetry restoration.We plot the energy per nucleon versus nucleardensity in Fig. 2. The behavior of the solid lineand dashed lines are close when the density issmall, say ρB < 0.2 fm−3. Both curves passthrough the saturation point of nuclear matter.For the square root ansatz for effective mass, E/Achanges discontinuously at the critical density.

For finite systems, we do not adjust the param-eters, rather they remain the same as in infinitehadronic matter. The charge density of 208Pbversus the radius is shown in Fig. 3. The two

Figure 3. The charge density versus radius for208Pb. The dashed and dotted lines are calcu-lated with linear and square root treatments ofeffective baryon mass, respectively. The solid lineis from the experimental data.

definitions give similar results. This arises natu-rally since the nucleon density is around satura-tion density in the center of finite nuclei and bothtreatments yield the correct saturation propertiesof nuclear matter. We plot the proton energy lev-els of 208Pb in Fig. 4. The energy levels are qual-itatively reproduced. For the square root ansatzfor baryon mass, the spin-orbit splitting is quiteclose to experiment. For example, the splittingof 1g7/2 and 1g9/2 ( 2d3/2 and 2d5/2 ) of 208Pb isabout 4.2 MeV ( 1.6 MeV ) which is close to theexperimental value 3.9 MeV ( 1.5 MeV ). For thelinear definition of baryon mass, the spin-orbitsplitting is smaller. This is because at saturationdensity, the effective nucleon mass is higher inthis definition and the spin-orbit splitting is pro-portional to the decrease of the effective nucleonmass. However, the smaller spin-orbit splittingcan be improved by going beyond the mean-fieldapproximation [18].

We now discuss the liquid-gas phase transitionof asymmetric nuclear systems. The α depen-dence of the critical temperature for infinite nu-clear matter is shown in Fig. 5 where α is definedas (ρn−ρp)/(ρn +ρp). The solid and dashed linescorrespond to the linear definition and the square

P. Wang et al. / Nuclear Physics B (Proc. Suppl.) 141 (2005) 273–278276

Figure 4. The proton energy levels for 208Pb. Thefirst and second columns are calculated with lin-ear and square root definition of effective baryonmass, respectively. The third column is from theexperimental data.

root ansatz of effective nucleon mass, respectively.The critical temperature, Tc, decreases with in-creasing α. For the square root case, when α isless than 0.2, the decrease of Tc is very small.When α is larger than 0.6, Tc decreases very fast.If α is larger than 0.88, the system can only bein the gas phase at any temperature. In the lin-ear case, the critical temperature is 2 MeV largerthan that in the square root case. The liquid-gas phase transition can occur for nuclear matterwith any α if the temperature is lower than thecritical temperature.

We show in Fig. 6 the mass number dependenceof the limiting temperature, Tlim, for nuclei alongthe line of β-stability:

Z = 0.5A − 0.3 × 10−2A5/3. (16)

The solid and dashed lines correspond to thelinear definition and the square root ansatz ofthe effective nucleon mass, respectively. The ex-perimental values obtained recently by Natowitz

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

α

Tc (

MeV

)

Figure 5. The critical temperature, Tc, versusasymmetry parameter, α. The solid and dashedlines are for the linear and square root cases, re-spectively.

[19] are also plotted in the figure for compari-son. The calculated results in this model arein good agreement with the experimental data.The limiting temperature decreases with increas-ing mass number. This means that when thetemperature is higher than the limiting tempera-ture, the heavy nuclei will fragment to light nu-clei. Two useful parameterizations of Tlim/Tc,valid for 10 ≤ A ≤ 208, are (Tlim/TC) = 0.611 −0.00193A+ 3.32× 10−6A2 (square root case) and(Tlim/TC) = 0.591 − 0.00203A + 3.80 × 10−6A2

(linear case), which are comparable with thatgiven in Ref. [19].

In summary, we have used an improved treat-ment of the c.m. motion in calculating the effec-tive, in-medium, baryon mass in an investigationof infinite hadronic matter and finite nuclei withinthe chiral SU(3) quark mean-field model. Theresults are compared with earlier results whichused the square root ansatz for the effective mass.Both treatments fit the saturation properties ofnuclear matter and therefore, for densities lowerthan the saturation density, these two treatmentsgive reasonably similar results. For high baryondensity, the predictions of these two treatmentsare quite different. There is a phase transition

P. Wang et al. / Nuclear Physics B (Proc. Suppl.) 141 (2005) 273–278 277

0 100 200 300 4000

2

4

6

8

10

12

14

Tlim

( M

eV )

A

Figure 6. The limiting temperature, Tlim, versusmass number A of finite nuclei. The solid anddashed lines are for the linear and square rootcases, respectively. The points with error barsare from Ref. [19].

of chiral symmetry restoration in the case of thesquare-root ansatz for the baryon mass. Physi-cal quantities, such as effective baryon mass andenergy per baryon change discontinuously at thecritical density. In the linear case, no such tran-sition occurs. Rather, the effective baryon massdecreases slowly at high density. These qualita-tively different behaviors lead to significant dif-ferences in the predicted high-density physics.

ACKNOWLEDGEMENTS

This work was supported by the Australian Re-search Council and by DOE contract DE-AC05-84ER40150, under which SURA operates Jeffer-son Laboratory.

REFERENCES

1. J. D. Walecka, Ann. Phys. 83 (1974) 491;S. A. Chin and J. D. Walecka, Phys. Lett.B52 (1974) 24.

2. J. Zimanyi and S. A. Moszkowski, Phys. Rev.C42 (1990) 1416.

3. A. R. Bodmer, Nucl. Phys. A526 (1991) 703.4. P. A. M. Guichon, Phys. Lett. B200 (1988)

235; S. Fleck, W. Bentz, K. Shimizu and K.Yazaki, Nucl. Phys. A510 (1990) 731; K.Saito and A. W. Thomas, Phys. Lett. B327(1994) 9.

5. A. W. Thomas, S. Theberge and G. A. Miller,Phys. Rev. D24 (1981) 216; A. W. Thomas,Adv. Nucl. Phys. 13 (1984) 1; G. A. Miller,A. W. Thomas and S. Theberge, Phys. Lett.B91 (1980) 192.

6. H. Toki, U. Meyer, A. Faessler and R. Brock-mann, Phys. Rev. C58 (1998) 3749.

7. W. Bentz and A. W. Thomas, Nucl.Phys. A696 (2001) 138. M. Buballa, hep-ph/0402234.

8. P. Wang, Z. Y. Zhang, Y. W. Yu, R. K. Suand H. Q. Song, Nucl. Phys. A688 (2001)791.

9. P. Wang, H. Guo, Z. Y. Zhang, Y. W. Yu,R. K. Su and H. Q. Song, Nucl. Phys. A705(2002) 455.

10. P. Wang, Z. Y. Zhang, Y. W. Yu, Commun.Theor. Phys. 36 (2001) 71.

11. P. Wang, V. E. Lyubovitskij, Th. Gutscheand Amand Faessler, Phys. Rev. C67 (2003)015210.

12. P. Wang, H. Guo, Y. B. Dong, Z. Y. Zhangand Y. W. Yu, J. Phys. G28 (2002) 2265.

13. P. A. M. Guichon, K. Saito, E. Rodionov andA. W. Thomas, Nucl. Phys. A601 (1996) 349.

14. H. Muller and B. D. Serot, Phys. Rev. C52(1995) 2072.

15. A. L. Goodman, J. I. Kapusta and A. Z.Mekjian, Phys. Rev. C30 (1984) 851.

16. P. Wang, D. B. Leinweber, A.W. Thomas andA. G. Williams, nucl-th/0407057.

17. P. Wang, D. B. Leinweber, A.W. Thomas andA. G. Williams, nucl-th/0404079.

18. T. S. Biro and J. Zimanyi, Phys. Lett. B391(1997) 1; G. Krein, A. W. Thomas and K.Tsushima, Nucl. Phys. A650 (1999) 313; P.A. M. Guichon and A. W. Thomas, nucl-th/0402064.

19. J. B. Natowitz et al., Phys. Rev. Lett. 89(2002) 212701.

P. Wang et al. / Nuclear Physics B (Proc. Suppl.) 141 (2005) 273–278278