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Improved quark mass density- dependent model with quark-sigma meson and quark-omega meson couplings

Chen Wu1 , Wei-Liang Qian1? and Ru-Keng Su1,2?

1. Department of Physics, Fudan University,Shanghai 200433, P.R. China 2. CCAST(World Laboratory), P.O.Box 8730, Beijing 100080, P.R. China

arXiv:0706.1349v1 [nucl-th] 10 Jun 2007

Abstract

An improved quark mass density- dependent model with the non-linear scalar sigma ?eld and the ω-meson ?eld is presented. We show that the present model can describe saturation properties, the equation of state, the compressibility and the e?ective nuclear mass of nuclear matter under mean ?eld approximation successfully. The comparison of the present model and the quark-meson coupling model is addressed.

PACS numbers: 12.39.-x; 14.20.-c; 05.45.Yv

? ?

wlqian@fudan.edu.cn rksu@fudan.ac.cn

1

I.

INTRODUCTION

Owing to the non-perturbative nature of QCD in low energy regions, phenomenlogical models re?ecting the characteristic of the strong interaction are widely used in the studying of the properties of hadrons, nuclear matter and quark matters[1-12]. They are based on di?erent degrees of freedom, for example: nucleons and mesons, or quarks and gluons, in particular, on hybrid quarks and mesons. Some of these models have been proved to be successful. The quark mass density-dependent model(QMDD)[5] is one of such candidates. According to the QMDD model, the masses of u, d quarks and strange quarks (and the corresponding anti-quarks) are given by mq = B (i = u, d, u, d) ? ? 3nB B 3nB (1)

ms,? = ms0 + s

(2)

where nB is the baryon number density, ms0 is the current mass of the strange quark, and B is the bag constant. At zero temperature 1 nB = (nu + nd + ns ), 3 (3)

where nu , nd , ns represent the density of u quark, d quark, and s quark, respectively. The basic hypothesis Eqs.(1) and (2) corresponds to a quark con?nement mechanism because if quark goes to in?nite space, the volume of the system tends to in?nite, nB approachs to zero and the mq goes to in?nite, and the in?nite quark mass prevents the quark from going to in?nite. The con?nement mechanism is similar to that of the MIT bag model. Although the QMDD model can provide a description of con?nement and explain many dynamical properties of strange quark matter, but it is still an ideal quark gas model and cannot explain the temperature T vs. density ρ decon?nement phase diagram of QCD and the properties of nuclear matter[13-14]. To overcome this di?culty, we have introduced a coupling between quark and nonlinear scalar ?eld to improve the QMDD model in a previous paper[15]. We have found the wave functions of the ground state and the lowest one-particle excited states. By using these wave functions, we calculated many physical quantities such as root-mean-squared radius, the magnetic moment of nucleon to compare with experiments and come to a conclusion that this improved QMDD model is successful to explain the 2

properties of nucleon. In ref[16], we extended this model to ?nite temperature and studied its soliton solution by means of the ?nite temperature quantum ?eld theory. The critical temperature of quark decon?nement TC and the temperature-dependent bag constant B(T ) are found. The results of improved QMDD(IQMDD) model are qualitatively similar to that obtained from Freidberg-Lee soliton bag model[17, 18]. Instead of studying the nucleon properties, we hope to employ the IQMDD to investigate the physical properties of nuclear matter in this paper. As was shown by the Walecka model[1] and the QMC model[7-10] early, a neutral vector ?eld coupled to the conserved baryon current is very important for describing bulk properties of nuclear matter. The large neutral scalar and vector contributions have been observed empirically from NN scattering amplitude. The main qualitative features of the nucleon-nucleon interaction: a short range repulsion between baryons coming from ω-meson exchange, and a long-range attraction between baryons coming from σ-meson exchange must be included in a successful model. Obviously, if we hope to employ the IQMDD model to mimick this repulsive and attractive interactions, except the quark and σ-meson interaction, the ω meson and the qqω coupling must be added. This motivate us to introduce ω mesons and the qqω coupling in the IQMDD model in this paper. In this new IQMDD model, the nonlinear scalar ?eld coupling with quarks forms a soliton bag, and the qqω vector coupling gives the repulsion between quarks. We will prove that this model can give us a successful description of nuclear matter. The organization of this paper is as follows. In the next section, we give the main formulae of the IQMDD model under the mean ?eld approximation at zero temperature. In the third section, some numerical results are contained. The last section contains a summary and discussions.

II.

FORMULAE OF THE IMPROVED QMDD MODEL

The Lagrangian density of the IQMDD model is : 1 1 1 q q ? L = ψ[iγ ? ?? ? mq + gσ σ ? gω γ ? ω? ]ψ + ?? σ? ? σ ? U(σ) ? F?ν F ?ν + m2 ω? ω ? 2 4 2 ω where F?ν = ?? ων ? ?ν ω? (5) (4)

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and the quark mass mq is given by Eqs.(1) and (2), mσ and mω are the masses of σ and ω

q q mesons, gσ and gω are the couplings constant between quark-σ meson and quark-ω meson

respectively. And 1 1 1 U(σ) = m2 σ 2 + bσ 3 + cσ 4 + B σ 2 3 4 ?B = m2 2 b 3 c 4 σ σ + σ + σ . 2 v 3 v 4 v (6)

(7)

where σv is the absolute minimum of U(σ), U(σv ) = 0 and U(0) = B. It can easily show the equation of motion for quark ?eld in the whole space is

q q [γ ? (i?? + gω ω? ) ? (mq ? gσ σ)]ψ = 0

(8)

Under mean ?eld approximation, the e?ective quark mass m? is given by: q

q m? = mq ? gσ σ ? q

(9)

In nuclear matter, three quarks constitute a bag, and the e?ective nucleon mass is obtained from the bag energy and reads: γq 4 ? MN = Σq Eq = Σq πR3 3 (2π)3

q KF

0

m? 2 + k 2 ( q

dNq )dk dk

(10)

q where quark degeneracy γq =6, KF is the Fermi energy of quarks. dNq /dk is the density of

states for various quarks in a spherical cavity. It is given by[19]: N(k) = A(kR)3 + B(KR)2 + C(KR) where A= 2γq . 9π (12) (11)

B(

mq γq mq k mq π )= {[1 + ( )2 ]arctan( ) ? ? }. k 2π k mq k 2

(13)

C(

mq γq mq ? mq . ) = C( ) + ( )1.45 mq k k k 3.42( k ? 6.5)2 + 100

(14)

γq 1 mq k k πk ? mq C( ) = { +( + )arctan( ) ? }. k 2π 3 k mq mq 2mq 4

(15)

Eqs. (12) and (13) are in good agreement with those given by multire?ection theory[21, 22] and the Eqs. (14) and (15) are given by a best ?t of numerical calculation for the MIT bag ? model. The curvature term C cannot be evaluated by this theory except for two limiting cases mq → 0 and mq → ∞. Madsen[20] proposed the Eq. (15), but as was pointed out by Ref. [19], the best ?t of numerical data is given by Eq. (14). This density of state has been employed by Refs. [13,14] to study the strangelets.

q The Fermi energy KF of quarks is given by

4 3 = πR3 nB 3 where nB satis?es γq nB = Σq (2π)3

q KF

(16)

(

0

dNq )dk dk

(17)

The bag radius R is determined by the equilibrium condition for the nucleon bag:

? δMN =0 δR

(18)

In nuclear matter, the total energy density is given by εmatter γN = (2π)3

N KF

0

? MN 2

+

p2 dp3

2 1 1 1 gω 2 ρ + m2 σ 2 + b? 3 + c? 4 σ σ + σ? 2 B 2mω 2 3 4

(19)

N where γN = 4 is degeneracy of nucleon, KF is fermi energy of nucleon and ρB is the density

of nuclear matter ρB = γN (2π)3

N KF

d3 k

(20)

0

In Eqs. (19), gω is the coupling constant between the nucleon and the ω meson and it

q satis?es gω = 3gω . As that of the QMC model[7], the σ is yielded by the equation: ?

N KF

m2 σ + b? 2 + c? 3 = ? σ σ σ?

γN (2π)3

? MN ? MN 2 + p2

d3 p(

0

? ?MN )R ?σ ?

(21)

Eqs. (9)-(21) form a complete set of equations and we can solve them numerically. Our numerical results will be shown in the next section.

III.

NUMERICAL RESULT

Before numerical calculation, let us consider the parameters in IQMDD model. As that of Ref.[1, 23], the masses of ω-meson and σ-meson are ?xed as mω = 783 MeV, mω = 509 5

MeV respectively. We choose the bag constant B = 174 MeV fm?3 to ?t the mass of nucleon MN = 939 MeV. When B is determined, the parameters b and c are not independent because of Eq. (7). we choose the b is free parameter. There are still three parameters, namely,

q q gω , gσ , b are needed to be ?xed in IQMDD model.

To study the physical properties of nuclear matter, we investigate the nuclear saturation, the equation of state and the compressibility. The pressure of nuclear matter P is given by P = ρ2 B ? εmatter ?ρB ρB (22)

where ρB is the baryon density. The compressibility for nuclear matter reads: K=9 ? P ?ρB (23)

at saturation point, the binding energy per particle E/A = ?15 MeV, and the saturation density ρ0 = 0.15 fm?3 . Our numerical results are shown in Fig. 1-Fig. 4. In Fig. 1, we choose ω-meson and σ-meson satisfy σ = 0, ω = 0 and depict the bag energy as a function of bag radius at zero ? ? temperature. We ?nd the stable radius of a ”free” nucleon R = 0.85 fm. In Figs. 2-4 we show the e?ective mass M ? of nucleon, the saturation curve and the equations of state of nuclear matter at zero temperature for IQMDD model respectively, where we ?x the parameter b=-1460 (MeV), gσ = 4.67 and gω = 2.44 respectively, and ?nd E/A = ?15 MeV and ρ0 = 0.15 fm?3 and K(ρ0 ) = 210 MeV. We ?nd our model can explain the properties of nuclear matter successfully. To illustrate our results more transparently, we show the dependence of the properties of

q q nuclear matter on the parameters b, gσ , gω in Table.1 for ?xing binding energy E/A = ?15

MeV and ρ0 = 0.15 fm?3 . We ?nd that the compressibility K(ρ0 ) and e?ective nucleon mass

? q q MN (ρ0 ) at saturation point all decrease when gσ , gω increase and b decreases. At was shown ? in Table.1, the variational regions for K(ρ0 ) and MN (ρ0 ) are small, and the decreasements ? of K(ρ0 ) and MN (ρ0 ) are slowly.

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TABLE 1. Variation of the nuclear matter properties to b.

q b(MeV) gσ q gω

K(ρ0 )(MeV) 218.8 215.5 213.6 211.2 208.1 205.7

? MN (ρ0 )(MeV)

-800 -1000 -1200 -1400 -1600 -1800

4.59 4.61 4.64 4.66 4.69 4.71

2.35 2.38 2.40 2.43 2.46 2.48

782.9 781.2 778.5 776.8 774.3 772.6

It was pointed in the Refs. [24] early, adding a nonlinear scalar ?eld in the model will cause unphysical behavior under mean ?eld approximation in nuclear matter. This can easily be seen from Eq.(21) because the left hand side of Eq. (21) is a cubic order function of σ , and σ = 0 is one of its solutions. There are two solutions in low-density regions. In ? ?

? Fig. 5 and Fig. 6, these two solutions are shown explicitly for σ vs. ρB curve and for MN ?

vs. ρB curve respectively where the parameters are ?xing as b = ?3655 (MeV). Noting that the term of non-linear scalar ?eld is essential to form a soliton bag, we conclude that the unphysical branch cannot be avoided for the soliton solution under mean ?eld approximation. Fortunately, the lower branch cannot be ended at the point(MN = 939 MeV, ρB = 0), and give us a experimental value of nucleon mass, we will give up this unphysical branch in our calculation. Finally, It is of interest to compare the properties of nuclear matter for IQMDD model and for the QMC model. Our results are shown in Table. 2. we ?nd their results are very similar. But as was pointed in our previous paper[15], the ?rst advantage of the IQMDD model is that the MIT bag boundary constraint has been given up because it mimicks to a Friedberg-lee soliton bag model[13-16]. The second advantage of the IQMDD model is that the interactions between qqσ and qqω are extended to the whole free space. We can easily write down the propagators of quarks, σ-meson and ω-meson respectively and do the many-body calculations beyond mean ?eld approximation. But for QMC model, the propagators of quarks, σ-meson and ω-meson cannot be written down easily because one must consider the multire?ection by the MIT bag boundary as well as the e?ect of the interactions limited into a nucleon space only. The IQMDD model provides a good substitute of the QMC model which is more suitable for the study of nuclear matter beyond

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mean ?eld. TABLE 2. Comparison of properties for IQMDD and QMC model. R(fm) QMC IQMDD(b=0) IQMDD(b=-1460) QHD-1 0.80 0.85 0.85

q gσ q gω ? K(ρ0 )(MeV) MN (ρ0 )(MeV)

5.53 4.54 4.67 gσ = 9.57

1.26 2.21 2.44 gω = 11.6

200 227 210 540

851 798 775 522

IV.

SUMMARY AND DISCUSSION

In summary, we present an Improved quark mass density dependent model which has the non-linear σ meson ?eld, and the ω meson ?eld. The qqσ coupling and the qqω coupling are introduced to mimick the repulsive and the attractive interactions between quarks in this model. It is shown that the present model is successful for describing the saturation properties, the equation of state and compressibility of nuclear matter. The e?ective nucleon mass decreases with baryon density in this model more rapidly than that of QMC model. After comparing the IQMDD model and the QMC model, we come to a conclusion that the IQMDD model is a good substitute for QMC model.

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[9] X. Jin and B. K. Jennings, Phys. Lett. B 374, 13(1996); Phys. Rev. C 54,1427 (1996). [10] H. Q. Song and R. K. Su, Phys. Lett. B 358, 179(1995); J. Phys. G 22, 1025(1996). [11] P. Wang, R. K. Su, H. Q. Song and L.L. Zhang, Nucl. Phys. A 653, 166(1999). [12] P. Wang, Z. Y. Zhang, Y. M. Yu, R.K. Su and H.Q. Song, Nucl. Phys. A 688, 791(2001). [13] Y. Zhang and R. K. Su, Phys. Rev. C65, 035202(2002) , Phys. Rev. C67, 015202(2003). [14] Y. Zhang and R. K. Su, S. Q. Ying and P. Wang, Europhys. Lett. 53,361(2001). [15] C. Wu, W. L. Qian and R. K. Su, Phys. Rev. C 72, 035205(2005). [16] H. Mao, R. K. Su and W. Q. Zhao, Phys. Rev. C 74, 055204(2006). [17] M. Li, M. C. Birse and L. Wilets, J. Phys. G 13 1(1987); R. Gold?am and L. Wilets, Phys. Rev. D 25, 1951 (1982). [18] S. Gao, E. K. Wang and J. R. Li, Phys. Rev. D 46, 3211 (1992). [19] Y. Zhang, W.L. Qian, S.Q. Ying and R.K. Su, J. Phys. G 27, 2241 (2001). [20] J. Madsen, Phys. Rev. D47, 5156(1993); Phys. Rev. D50, 3328(1994). [21] R. Balian and C. Bloch, Ann. Phys.(N. Y). 60, 401(1970). [22] T. H. Hansson and R. L. Ja?e, Phys. Rev. D 35, 213 (1987). [23] R. J. Furnstahl, B. D. Serot and H. B. Tang, Nucl. Phys. A 615, 441(1997). [24] B. M. Waldhauser, J. A. Maruhn, H. St¨cker and W. Greiner, Phys. Rev. C 38, 1003(1988). o

9

1020

1000

Ebag(MeV)

980

960

940

0.80

0.85

0.90

0.95

1.00

R (fm)

Fig.1

FIG. 1: The bag energy as a function of bag radius at zero temperature for ω = 0, ω = 0. ? ?

10

950

M (MeV)

900

*

850

800

750

700 0.00

0.05

0.10

0.15

-3

0.20

ρB (fm )

Fig.2

FIG. 2: E?ective nucleon mass vs. baryon density at zero temperature where the parameters

q q gσ = 4.67, gω = 2.44, b = ?1460 (MeV).

11

-10

E/V-939 (MeV)

-12

-14

-16 0.10

0.15

0.20

0.25

ρB (fm )

-3

Fig.3

FIG. 3: Saturation curve of nuclear matter at zero temperature. the parameters is same as that of Fig. 2.

12

4

3

P (MeV fm )

-3

2

1

0

-1

-2 0.00

0.05

0.10

-3

0.15

0.20

0.25

ρB (fm )

Fig.4

FIG. 4: Pressure of nuclear matter as a function of ρB . the parameters is same as that of Fig. 2.

13

100 90 80 70 60 50 40 30 20 10 0 0.00 0.05 0.10 0.15

ρB (fm )

-3

Fig.5

q q FIG. 5: the σ ?eld vs. baryon density for b=-3655 (MeV), gσ = 5.23, gω = 3.12. ?

14

950 900 850 800 750 700 650 600 550 0.00 0.05 0.10 0.15

M (MeV)

*

ρB (fm )

-3

Fig.6

FIG. 6: E?ective nucleon mass M ? vs. baryon density. the parameters is same as that of Fig. 5.

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