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Hadron Properties in a Nuclear Medium and Effective Nuclear Force from Quarks the Quark-Meson Coupli
K. Tsushima
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DOI: 10.22661/AAPPSBL.2019.29.6.37

Hadron Properties in a Nuclear Medium and Effective Nuclear Force from Quarks: the Quark-Meson Coupling Model

K. TSUSHIMA1
1LABORAT횙RIO DE F횒SICA TE횙RICA E COMPUTACIONAL
UNIVERSIDADE CRUZEIRO DO SUL AND
UNIVERSIDADE CIDADE DE SAO PAULO, 01506-000, S횄O PAOLO, SP, BRAZIL

We give a short review of the quark-meson coupling (QMC) model, the quark-based model of finite nuclei and hadron interactions in a nuclear medium, highlighting on the relationship with the Skyrme effective nuclear forces. The model is based on a mean field description of nonoverlapping nucleon MIT bags bound by the self-consistent exchange of Lorentz-scalar-isoscalar, Lorentz-vector-isoscalar, and Lorentz-vector-isovector meson fileds directly coupled to the light quarks up and down. In conventional nuclear physics the Skyrme effective forces are very popular, but, there is no satisfactory interpretation of the parameters appearing in the Skyrme forces. Comparing a many-body Hamiltonian generated by the QMC model in the zero-range limit with that of the Skyrme force, it is possible to obtain a remarkable agreement between the Skyrme force and the QMC effective interaction. Furthermore, it is shown that 3-body and higher order N-body forces are naturally included in the QMC-generated effective interaction.

INTRODUCTION

This article intends to give a short review of the quark-meson coupling (QMC) model [1], the quark-based model of finite nuclei and hadron properties in a nuclear medium. Aside from the model basics, we highlight on the relationship with the Skyrme effective nuclear forces. (For detailed reviews of the QMC model, see Refs. [2-4].) The QMC model has been successfully applied to various studies of the properties of finite (hyper)nuclei [4-14], hadron properties in a nuclear medium [15-20], reactions involving nuclear targets [21-29], and neutron star structure [30-32]. Self-consistent exchange of Lorentz-scalar-isoscalar (σ), Lorentz-vector-isoscalar (ω), and Lorentz-vector-isovector (ρ) mean fields directly coupled only to the light quarks up and down, is the key feature of the model for achieving the novel saturation properties of nuclear matter, despite of its simplicity. All the relevant coupling constants for the σ-light-quarks, ω-light-quarks, and ρ-light-quarks in any hadrons, are the same as those in nucleon, and they are fixed/constrained by the nuclear matter saturation properties. The physics behind this picture is the fact that the light-quark chiral condensates change faster than those of the strange and heavier quarks as nuclear density increases. The light-quark chiral condensates are the order parameters for chiral symmetry in QCD, and change in their magnitudes are one of the most important driving forces for partial restoration of chiral symmetry in a nuclear medium. This is modeled in the QMC model by the approximation that the σ, ω, and ρ fields couple directly only to the light quarks.

FINITE NUCLEUS IN THE QMC MODEL

The description below is based on Refs. [2, 3, 33]. Although a Hartree-Fock treatment is possible within the QMC model [34], the main features of the results, especially the density dependence of nuclear matter energy density, is nearly identical to that of the Hartree approximation. Then, it is suffcient to discuss the Hartree approximation. (See e.g., Ref. [31] for a neutron star structure studied by the Hartree-Fock approximation in the QMC model.)

Before explaining nuclear matter in the QMC model, we start with a finite nucleus. Using the Born-Oppenheimer approximation, a relativistic Lagrangian density, which gives the same mean-field equations of motion for a finite (hyper)nucleus, is given [2, 3, 9] below, where the quasi-particles moving in single-particle orbits are three-quark clusters with the quantum numbers of a nucleon, strange, charm or bottom hyperon when expanded to the same order in veloc-ity [5, 6, 9, 12, 14, 20]:

(1)

(2)

(3)

For a normal nucleus, 𝓛YQMC in Eq. (1), namely Eq. (3) is not needed. In the above ψN (r) and ψY (r) are respectively the nucleon and hyperon (strange, charm or bottom baryon) fields. The mean-meson fields represented by, σ, ω and b, which directly couple to the light quarks self-consistently, are the Lorentz-scalar-isoscalar, Lorentz-vector-isoscalar and the third com-ponent of Lorentz-vector-isovector fields, respectively, while A stands for the Coulomb field. They are defined by the mean expectations by, σ(r) = <σ(r)>, ω(r) = ,0 <ω(r)>, and b(r) = ,0 i,3 <ρ,i(r)>.

In the approximation that the σ, ω and ρ fields couple only to the u and d light quarks, the coupling constants for the hyperon appearing in Eq. (3) are obtained/identified as gYω = (nq /3) gω, and gYρ gρ = gqρ, with nq being the total number of valence light quarks in the hyperon Y, where gω = 3gqω and gρ are the ω-N and ρ-N coupling constants. I3Y and QY are the third component of the hyperon isospin operator and its electric charge in units of the positron charge, e, respectively.

The field dependent σ-N and σ-Y coupling strengths respectively for the nucleon N and hyperon Y, gσN(σ) and gσY (σ), are implicitly in Eqs. (2) and (3), and defined by

(4)

(5)

where mN (mY) is the free nucleon (hyperon) mass. The dependence of these coupling strengths on the applied scalar field (σ) must be calculated self-consistently within the quark model [1, 5, 9, 12, 13, 20]. Hence, unlike quantum hadrodynamics (QHD) [36, 37], even though gσY (σ) / gσN(σ) may be 2/3 or 1/3 depending on the number of light quarks nq in the hyperon in free space, σ = 0 (even this is true only when their bag radii in free space are exactly equal in the QMC model using the MIT bag), this will not necessarily be the case in a nuclear medium. We define gσN,YgσN,Y (σ = 0) for later convenience. Note that, we will write explicitly the σ dependence as gσN,Y (σ). Therefore, without the σ dependence, gσN,Y are the coupling constants when σ = 0 in this article. (The explicit expression will be given by Eq. (13).)

The Lagrangian density Eq. (1) [or Eqs. (2) and (3)] leads [lead] to a set of equations of motion for the finite (hyper)nuclear system:

(6)

(7)

(8)

(9)

(10)

(11)

where, ρs(r) (ρYs(r)), ρB(r) = ρp(r) + ρn(r) (ρYB(r)), ρ3(r) = ρp(r) - ρn(r), ρp(r) and ρn(r) are the nucleon (hyperon) scalar, nucleon (hyperon) baryon, third component of isovector, proton and neutron densities at the position r in the (hyper)nucleus. Notice that the terms on the right hand side of Eq. (8), -[dm*N (σ)/] ≡ gσN CN (σ) and - [dm*Y (σ)/] ≡ gσY CY(σ). (Recall gσN = gσN(σ = 0) and gσY = gσY(σ = 0).) At the hadronic level, the entire information of the quark dynamics is condensed in the effective couplings CN,Y(σ)of Eq. (8), which characterize the features of the QMC model, namely, the scalar polarisability. Furthermore, when CN,Y(σ)= 1, which corresponds to a structureless nucleon or hyperon, the equations of motion given by Eqs. (6)-(11) can be identified with those derived from naive QHD [36, 37].

The effective mass of hadron h (in the present case nucleon and hyperon), will be calculated by Eq. (25). The explicit expressions for CN,Y(σ) ≡ SN,Y(σ)/SN,Y(σ = 0) is defined next, and the effective masses m*N, Y are related by,

(12)

where gσq is the light-quark-σ coupling constant, and ψq is the light-quark wave function in the nucleon N or hyperon Y immersed in a nuclear medium. By the above relation, we define explicitly the σ-N and σ-Y coupling constants:

(13)

Note that, the right hand side of Eq. (12) is the quark scalar charge, which is Lorentz scalar, and thus the left-hand-side of Eq. (12) is Lorentz scalar, and thus m*N (σ) as well. Furthermore, the values of SN (σ) and SY (σ) are different, because the light-quark wave functions in the nucleon N and hyperon Y are different in vacuum as well as in medium, because the bag radii of the N and Y are different in each case. Since the light quarks in the other hadrons feel the same scalar and vector mean fields as those in the nucleon, we can systematically study the hadron properties in medium without introducing any new coupling constants for the σ, ω, and ρ fields for different hadrons.

The parameters appearing at the nucleon, hyperon and meson Lagrangian level are mω = 783 MeV, mρ = 770 MeV, mσ = 550 MeV and e2/4𝜋 = 1/137.036 [5, 6]. (See Ref. [6] for a discussion on the parameter fixing in the QMC model, in treating finite nuclei.)

BARYON PROPERTIES IN A NUCLEAR MEDIUM

We consider the rest frame of infinitely large, symmetric nuclear matter, a spin and isospin saturated system with only strong interaction (Coulomb force is dropped as usual). One first keeps only 𝓛YQMC in Eq. (1), or correspondingly drops all the quantities with the super- and sub-scripts Y, and sets the Coulomb field A(r) = 0 in Eqs. (6)-(11). Next one sets all the terms with any derivatives of the fields to be zero. Then, within the Hartree mean-field approximation, the nuclear (baryon) ρB and scalar ρs densities with the nucleon Fermi momentum kF are respectively given by,

(14)

(15)

Here, m*N (σ) is the value (constant) of the effective nucleon mass at a given nuclear density. In the standard QMC model [1], the MIT bag model is used for describing nucleons and hyperons (hadrons). The use of this quark model is an essential ingredient for the QMC model, namely the use of the relativistic, confined quarks.

The Dirac equations for the quarks and antiquarks with the effective light-quark masses m*q (to be defined below) in nuclear matter in a bag of a hadron h, with q = u or d, and Q = s, c or b, neglecting the Coulomb force are given by [16, 18-21],

(16)

(17)

(18)

where, m*q = mq - Vσ q, and the (constant) mean fields for a bag in nuclear matter are defined by Vσ q gσ qσ, Vωq gωqω and Vρq gρ qb, with gσ q, gωq, and gρ q being the corresponding quark-meson coupling constants. We assume SU(2) symmetry, mu,ū = md,mq, thus, m*u,ū = m*d, = m*qmq - Vσ q. Since the ρ-meson mean field becomes zero, Vρq = 0 in Eqs. (16) and (17) in symmetric nuclear matter in the Hartree approximation, we will ignore it. (This is not true in a finite nucleus with equal and more than two protons even with equal numbers of protons and neutrons, since the Coulomb interactions among the protons induce an asymmetry between the proton and neutron density distributions to give ρ3(r) = ρp(r) - ρn(r) ≠ 0.)

The same meson-mean fields σ and ω for the quarks in Eqs. (16) and (17), satisfy self-consistently the following equations at the nucleon level, together with the effective nucleon mass m*N (σ) of Eq. (4) to be calculated by Eq. (25):

(19)

(20)

(See Eq. (12) for CN (σ).) Because of the underlying quark structure of the nucleon to calculate m*N (σ) in nuclear medium, CN (σ) decreases as σ increases, whereas in the usual point-like nucleon-based models it is constant, CN (σ) = 1. As will be discussed later it can be parametrized in the QMC model as CN (σ) = 1 - aN 횞 (gσN σ)(aN > 0). It is this variation of CN (σ) (or equivalently dependence of the scalar coupling on density, or σ as gσN (σ)) that yields a novel saturation mechanism for nuclear matter in the QMC model, and contains the important dynamics originating from the quark structure of nucleons and hadrons. It is also the variation of this CN (σ), that induces 3-body and higher order N-body forces [35]. (This issue will be discussed separately in the next section.) As a consequence of the derived, nonlinear couplings of the meson fields in the Lagrangian density at the nucleon (hyperon) and meson level, the standard QMC model yields the nuclear incompressibility of K ≃ 280 MeV.

This is in contrast to a naive version of QHD [36, 37] (the point-like nucleon model of nuclear matter), which results in the much larger value, K ≃ 500 MeV; the empirically extracted value falls in the range K = 200-300 MeV. (See Ref. [39] for an extensive analysis on this issue.)

 

Fig. 1: Negative of binding energy per nucleon for symmetric nuclear matter Etot/A-mN (upper panel), and the effective light-quark mass mq* , and vector (Vωq) and scalar (-Vσq) potentials felt by the light quarks (lower panel).

Once the self-consistency equation for the σ field Eq. (20) is solved, one can evaluate the total energy of symmetric nuclear matter per nucleon:

(21)

We then determine the coupling constants, gσN and gω at the nucleon level (see also Eq. (13)), by the fit to the binding energy of 15.7 MeV at the saturation density ρ0 = 0.15 fm-3 for symmetric nuclear matter, as well as gρ to the symmetry energy of 35 MeV. The determined quark-meson coupling constants, and the current quark mass values used are listed in Table I. The coupling constants at the nucleon level are (gσN)2/4𝜋 = 3.12, gω2 / 4𝜋 = 5.31 and gρ2/4𝜋 = 6.93. (See Eq. (13), and recall gω = 3gωq and gρ = gρq.) These values are ω determined with the standard QMC model inputs at the quark level which will be given later.

 

Table I. Current quark mass values (inputs), quark-meson coupling constants and the bag pressure, Bp. Note that the mc value is updated from Refs. [2, 3] based on Ref. [41].

mu,d

5 MeV

g?q

5.69

ms

250 MeV

g?q

2.72

mc

1270 MeV

gρq

9.33

mb

4200 MeV

Bp1/4

170 MeV

We show in Fig. 1 negative of binding energy per nucleon for symmetric nuclear matter Etot /A - mN (upper panel), and effective light-quark mass m*q, vector (Vωq) and scalar (-Vσq) potentialsfelt by the light quarks (lower panel).

Let us consider the situation that a hadron h is immersed in nuclear matter. The normalized, static solution for the ground state quarks or antiquarks with flavor f in the hadron h may be written, ψf (x) = Nf exp-iєft/Rh* ψf (r), where Nf and ψf (r) are the normalization factor and corresponding spin and spatial part of the wave function. The bag radius in medium for the hadron h, denoted by Rh*, is determined through the stability condition for the mass of the hadron against the variation of the bag radius [1,10] (see Eq.(26)). The eigenenergies in units of 1/Rh* are given by,

(22)

(23)

(24)

The hadron mass in a nuclear medium, m*h (free mass is hdenoted by mh), is calculated for a given baryon density together with the mass stability condition,

(25)

(26)

where 廓*q = 廓*q = [xq2 + (R*hm*q)2]1/2 (q = u, d), with m*q = mq - gσqσ = mq - Vσq, 廓*Q = *Q = [xQ2+ (R*hmQ)2]1/2 (Q = s, c, b), and xq, Q are the lowest mode bag eigenvalues. Bp is the bag pressure (constant), nq (nq) and nQ (nQ) are the lowest mode valence quark (antiquark) numbers for the quark flavors q and Q in the hadron h, respectively, while zh parametrizes the sum of the center-of-mass and gluon fluctuation effects, which are assumed to be density independent [5]. The bag pressure Bp = (170 MeV)4 (density independent) is determined by the free nucleon mass mN = 939 MeV with the bag radius in vacuum RN = 0.8 fm and mq = 5 MeV as inputs (this yields SN (0) = 0.48265 for Eq. (13)), which are considered to be standard values in the QMC model [2]. (See also Table I.) Concerning the effective light-quark mass m*q in nuclear medium, it reflects nothing but the strength of the attractive scalar potential as in Eqs. (16) and (17), and thus naive interpretation of the mass for a (physical) particle, which is positive, should not be applied. The model parameters are determined to reproduce the corresponding masses in free space. The quark-meson coupling constants, gσq, gωq and gρq, have already been determined by the nuclear matter saturation properties. Exactly the same coupling constants, gσq, gωq and gρq, are used for the light quarks in all the hadrons as in the nucleon.

We show in Fig. 2 the scalar potentials of baryons and mesons, [m* - m] (MeV), calculated in the QMC model [20]. (See Eq. (25) for m*.) One can notice that the scalar potentials of hadrons are well proportional to the light quark numbers of the corresponding hadrons.

 



Fig. 2: Baryon and meson scalar potentials, [m* - m] (MeV) [20].

In connection with the effective baryon masses, it is found that the function CB(σ) (B = N, Λ, 誇, Ξ, Λc, c, Ξc, Λb, b, Ξb) (see Eq. (12)), can be parameterized as a linear form in the σ field, gσNσ, for a practical use [5, 6, 9, 33]:

(27)

The values obtained for aB are listed in Table II. This parameterization works well up to about three times the normal nuclear matter density 3ρ0. Then, the effective mass of baryons B in nuclear matter is also well approximated up to 3ρ0 by:

(28)

(29)

with nq being the valence light-quark number in the baryon B. See Eqs.(4) and (5) to compare with gN,Y(σ) and the above expression. The obtained values of the "slope parameter" aB for various baryons are listed in Table II.

 

Table II. Slope parameter values aB obtained for various baryons [33]. Note that the tiny differences in values of aB from those in Refs. [2, 3], are due to the differences in the number of data points for evaluating aB, but such differences give negligible effects.

THE QMC MODEL AND CONVENTIONAL NUCLEAR MODELS

In this section we discuss the relationship between the QMC model and a conventional Skyrme effective nuclear force according to Ref. [35]. (For a review including further developments, see Refs. [4, 8, 38, 40].) The QMC model description was reformulated to describe a nucleus as a many-body problem in a nonrelativistic framework. This allows us to take the limit corresponding to a zero-range force which can be compared with the Skyrme effective forces in conventional nuclear physics [35]. The classical energy of a nucleon with position (r) and momentum (p) is given by [35],

(30)

where Vs.o. is the spin-orbit interaction.

To get the dynamical mass mN*(r) one has to solve a quark model of the nucleon (in the present case the MIT bag model) in the field σ(r). For the present purpose, it is suffcient to use the approximated relation Eq. (29) with nq = 3 and d = aN and gσgσN hereafter,

(31)

where d of the MIT bag model gives d = 0.22RN (in MeV-1) with the nucleon bag radius RN (fm) corresponding to Table II with RN = 0.8 fm. The last term, which represents the response of the nucleon to the applied scalar field - the scalar polarizability - is an essential element of the QMC model. From the numerical studies we know that the approximation Eq. (31) is quite accurate at moderate nuclear densities.

The energy (30) is for one particular nucleon moving classically in the nuclear meson fields. The total energy of the system is then given by the sum of the energy of each nucleon and the energy carried by the fields [2]:

(32)

(33)

The expression of EN (r) was approximated by neglecting the velocity dependent terms (σ)2,

(34)

where we define the classical densities as ρcl (r) = ∑i (r - ri) and ρscl (r) = ∑i (1 - pi2 / 2mN2)(r - ri).

This will be the starting point for the many body formulation of the QMC model.

To eliminate the meson fields from the energy, we use the equations, 灌Etot / 灌σ(r) = 灌Etot / 灌ω(r) = 0, and leave a system whose dynamics depends only on the nucleon coordinates. Roughly speaking, since the meson fields should follow the matter density, the typical scale for the operator acting on σ or ω is the thickness of the nuclear surface, that is about 1 fm. Therefore, it seems reasonable that we can consider the second derivative terms acting on the meson fields as perturbations. Then, starting from the lowest order approximation, we solve the equations for the meson fields iteratively, and neglect a small difference between ρscl and ρcl except in the leading term. When inserted into Eq. (34), the series for the meson fields generates N-body forces in the Hamiltonian. To complete the effective Hamiltonian, we now include the effect of the isovector ρ meson as well.

The quantum effective Hamiltonian finally takes the form

(35)

where Gi = gi2 / mi2(i = σ, ω, ρ) and Aij = Gσ + (2s - 1) Gω + (2v - 1)Gρτiτj /4, with s and v being respectively, the nucleon isoscalar and isovector magnetic moments. Here rij = ri -r j and i is the gradient with respect to ri acting on the delta function. In Eq. (35) we we have used the notation 2 (ijk) for (rij) (rjk) and analogously for 3 (ijkl). Furthermore, we have dropped the contact interactions involving more than 4-bodies because their matrix elements vanish for antisymmetrized states. To fix the free parameters, Gi, the volume and symmetry coeffcients of the binding energy per nucleon of infinite nuclear matter, EB /A = a1 + a4 (N - Z)2/A2, are calculated and fitted so as to produce the experimental values. Using the bag model with the radius RN = 0.8 fm and the physical masses for the mesons and mσ = 600 MeV, one gets, in fm2, Gσ = 11.97, Gω = 8.1 and Gρ = 6.46.

It is now possible to compare the present Hamiltonian with the Skyrme effective interaction. Since, in our formulation, the medium effects are summarized in the 3- and 4-body forces, we consider Skyrme forces of the same type, that is, without density dependent interactions. They are defined by a potential energy of the form

(36)

with ∇ij = ∇i -∇j. There is no 4-body force in Eq. (36).

Comparison of Eq. (36) with the QMC Hamiltonian, Eq. (35), allows one to identify

(37)

Furthermore, we restrict our considerations to doubly closed shell nuclei, and assume that one can neglect the difference between the radial wave functions of the single-particle states with j = l + 1/2 and j = l - 1/2. Then, by comparing the Hartree-Fock Hamiltonian obtained from HQMC and that of Ref. [42] corresponding to the Skyrme force, we obtain the relations

(38)

(39)

(40)

We compare in Table III the results with the parameters of the force SkIII [43], which is considered a good representative of density independent effective interactions. We show the combinations 3t1 + 5t2, which controls the effective mass, and 5t2 - 9t1, which controls the shape of the nuclear surface [42]. From the Table III, one sees that the level of agreement with SkIII is very impressive. An important point is that the spin-orbit strength W0 comes out with approximately the correct value. The middle column (N = 3) shows the results when we switch off the 4-body force. The main change is expected to decrease of the predicted 3-body force. However, this is not the case. If we look at the incompressibility of nuclear matter, K, this decreases by as much as 37 MeV when we restore this 4-body force.

Now one can recognize a remarkable agreement between the phenomenologically successful Skyrme force SkIII) and the effective interaction corresponding to the QMC model - a result which suggests that the response of nucleon internal structure to the nuclear medium (scalar polarizability) indeed plays a vital role in nuclear structure.

 

Table III. QMC predictions (with mσ = 600 MeV) [35] compared with the Skyrme force [43].

QMC

QMC(N=3)

SkIII

t0 (MeV fm3)

-1082

-1047

-1129

x0

0.59

0.61

0.45

t3 (MeV fm6)

14926

12513

14000

3t1 + 5t2 (MeV fm5)

475

451

710

5t2 - 9t1 (MeV fm5)

-4330

-4036

-4030

W0 (MeV fm5)

97

91

120

K (MeV)

327

364

355

SUMMARY

We have given a short review on the basics of the quark-meson coupling (QMC) model, a quark-based model of finite nuclei and hadron properties in a nuclear medium. The highlight was on the relationship between the QMC model and a conventional Skyrmeeffective nuclear force, by reformulating the QMC model in nonrelativistic form and taking the zero-range interaction limit. It was shown that the derived, effective QMC interaction has a remarkable agreement with a successful Skyrme force. Furthermore, it was shown that the QMC-generated effective interaction automatically contains the 3-body and higher order N-body forces. Since the QMC model is based on the quark degrees of freedom, the model enables us to study the properties of finite nuclei and in-medium hadron properties in a very systematic manner.

Acknowledgments: The author would like to thank P. A. M. Guichon, K. Saito, and A. W. Thomas for exciting collaborations, and the Asia Pacific Center for Theoretical Physics (APCTP), Kyungpook National University, and Yongseok Oh for warm hospitality and supports during his visit. This work was supported by the Conselho Nacional de Desenvolvimento Cient챠fico e Tecnol처gico (CNPq) Process, No. 313063/2018-4, and No. 426150/2018-0, and Funda챌찾o de Amparo 횪 Pesquisa do Estado de S찾o Paulo (FAPESP) Process, No. 2019/00763-0, and was also part of the projects, Instituto Nacional de Ci챗ncia e Tecnologia - Nuclear Physicsand Applications (INCT-FNA), Brazil, Process. No. 464898/2014-5, and FAPESP Tem찼tico, Brazil, Process, No. 2017/05660-0.

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