
DOI: 10.22661/AAPPSBL.2019.29.6.37
Hadron Properties in a Nuclear Medium and Effective Nuclear Force from Quarks: the QuarkMeson Coupling Model
K. TSUSHIMA^{1} ^{1}LABORATÓRIO DE FÍSICA TEÓRICA E COMPUTACIONAL UNIVERSIDADE CRUZEIRO DO SUL AND UNIVERSIDADE CIDADE DE SAO PAULO, 01506000, SÃO PAOLO, SP, BRAZIL
We give a short review of the quarkmeson coupling (QMC) model, the quarkbased 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 selfconsistent exchange of Lorentzscalarisoscalar, Lorentzvectorisoscalar, and Lorentzvectorisovector 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 manybody Hamiltonian generated by the QMC model in the zerorange 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 3body and higher order Nbody forces are naturally included in the QMCgenerated effective interaction.
INTRODUCTION
This article intends to give a short review of the quarkmeson coupling (QMC) model [1], the quarkbased 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. [24].) The QMC model has been successfully applied to various studies of the properties of finite (hyper)nuclei [414], hadron properties in a nuclear medium [1520], reactions involving nuclear targets [2129], and neutron star structure [3032]. Selfconsistent exchange of Lorentzscalarisoscalar (σ), Lorentzvectorisoscalar (ω), and Lorentzvectorisovector (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 σlightquarks, ωlightquarks, and ρlightquarks 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 lightquark chiral condensates change faster than those of the strange and heavier quarks as nuclear density increases. The lightquark 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 HartreeFock 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 HartreeFock approximation in the QMC model.)
Before explaining nuclear matter in the QMC model, we start with a finite nucleus. Using the BornOppenheimer approximation, a relativistic Lagrangian density, which gives the same meanfield equations of motion for a finite (hyper)nucleus, is given [2, 3, 9] below, where the quasiparticles moving in singleparticle orbits are threequark clusters with the quantum numbers of a nucleon, strange, charm or bottom hyperon when expanded to the same order in velocity [5, 6, 9, 12, 14, 20]:
 (1)

 (2)

 (3)

For a normal nucleus, 𝓛^{Y}_{QMC} in Eq. (1), namely Eq. (3) is not needed. In the above ψ_{N} (r^{→}) and ψ_{Y} (r^{→}) are respectively the nucleon and hyperon (strange, charm or bottom baryon) fields. The meanmeson fields represented by, σ, ω and b, which directly couple to the light quarks selfconsistently, are the Lorentzscalarisoscalar, Lorentzvectorisoscalar and the third component of Lorentzvectorisovector fields, respectively, while A stands for the Coulomb field. They are defined by the mean expectations by, σ(r^{→}) = <σ(r^{→})>, ω(r^{→}) = δ^{µ}^{,0} <ω^{µ}(r^{→})>, and b(r^{→}) = δ^{µ}^{,0} δ^{i}^{,3} <ρ^{µ}^{,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 g^{Y}_{ω }= (n_{q }/3) g_{ω}, and g^{Y}_{ρ }≡ g_{ρ} = g^{q}_{ρ}, with n_{q} being the total number of valence light quarks in the hyperon Y, where g_{ω} = 3g^{q}_{ω} and g_{ρ} are the ωN and ρN coupling constants. I_{3}^{Y} and Q_{Y} are the third component of the hyperon isospin operator and its electric charge in units of the positron charge, e, respectively.
The field dependent σN and σY coupling strengths respectively for the nucleon N and hyperon Y, g_{σ}^{N}(σ) and g_{σ}^{Y} (σ), are implicitly in Eqs. (2) and (3), and defined by
 (4)

 (5)

where m_{N} (m_{Y}) is the free nucleon (hyperon) mass. The dependence of these coupling strengths on the applied scalar field (σ) must be calculated selfconsistently within the quark model [1, 5, 9, 12, 13, 20]. Hence, unlike quantum hadrodynamics (QHD) [36, 37], even though g_{σ}^{Y} (σ) / g_{σ}^{N}(σ) may be 2/3 or 1/3 depending on the number of light quarks n_{q} 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 g_{σ}^{N,Y} ≡ g_{σ}^{N,Y} (σ = 0) for later convenience. Note that, we will write explicitly the σ dependence as g_{σ}^{N,Y} (σ). Therefore, without the σ dependence, g_{σ}^{N,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, ρ_{s}(r^{→}) (ρ^{Y}_{s}(r^{→})), ρ_{B}(r^{→}) = ρ_{p}(r^{→}) + ρ_{n}(r^{→}) (ρ^{Y}_{B}(r^{→})), ρ_{3}(r^{→}) = ρ_{p}(r^{→})  ρ_{n}(r^{→}), ρ_{p}(r^{→}) and ρ_{n}(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)/dσ] ≡ g_{σ}^{N} C_{N} (σ) and  [dm^{*}_{Y} (σ)/dσ] ≡ g_{σ}^{Y} C_{Y}(σ). (Recall g_{σ}^{N }= g_{σ}^{N}(σ = 0) and g_{σ}^{Y }= g_{σ}^{Y}(σ = 0).) At the hadronic level, the entire information of the quark dynamics is condensed in the effective couplings C_{N,Y}(σ)of Eq. (8), which characterize the features of the QMC model, namely, the scalar polarisability. Furthermore, when C_{N,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 C_{N,Y}(σ) ≡ S_{N,Y}(σ)/S_{N,Y}(σ = 0) is defined next, and the effective masses m^{*}_{N, Y} are related by,
 (12)

where g_{σ}^{q} is the lightquarkσ coupling constant, and ψ_{q} is the lightquark wave function in the nucleon N or hyperon Y immersed in a nuclear medium. By the above relation, we define explicitly the σN and σY coupling constants:
 (13)

Note that, the right hand side of Eq. (12) is the quark scalar charge, which is Lorentz scalar, and thus the lefthandside of Eq. (12) is Lorentz scalar, and thus m^{*}_{N} (σ) as well. Furthermore, the values of S_{N} (σ) and S_{Y} (σ) are different, because the lightquark 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 m_{ω} = 783 MeV, m_{ρ} = 770 MeV, m_{σ} = 550 MeV and e^{2}/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 𝓛^{Y}_{QMC} in Eq. (1), or correspondingly drops all the quantities with the super and subscripts 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 meanfield approximation, the nuclear (baryon) ρ_{B} and scalar ρ_{s} densities with the nucleon Fermi momentum k_{F} 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 lightquark 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, 1821],
 (16)

 (17)

 (18)

where, m^{*}_{q} = m_{q}  V_{σ} ^{q}, and the (constant) mean fields for a bag in nuclear matter are defined by V_{σ} ^{q} ≡ g_{σ} ^{q}σ, V_{ω}^{q} ≡ g_{ω}^{q}ω and V_{ρ}^{q} ≡ g_{ρ} ^{q}b, with g_{σ} ^{q}, g_{ω}^{q}, and g_{ρ} ^{q }being the corresponding quarkmeson coupling constants. We assume SU(2) symmetry, m_{u,}_{ū} = m_{d,}_{} ≡ m_{q}, thus, m^{*}_{u,}_{ū} = m^{*}_{d,}_{} = m^{*}_{q}≡ m_{q}  V_{σ} ^{q}. Since the ρmeson mean field becomes zero, V_{ρ}^{q} = 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 ρ_{3}(r^{→}) = ρ_{p}(r^{→})  ρ_{n}(r^{→}) ≠ 0.)
The same mesonmean fields σ and ω for the quarks in Eqs. (16) and (17), satisfy selfconsistently 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 C_{N} (σ).) Because of the underlying quark structure of the nucleon to calculate m^{*}_{N} (σ) in nuclear medium, C_{N} (σ) decreases as σ increases, whereas in the usual pointlike nucleonbased models it is constant, C_{N} (σ) = 1. As will be discussed later it can be parametrized in the QMC model as C_{N} (σ) = 1  a_{N} × (g_{σ}^{N} σ)(a_{N} > 0). It is this variation of C_{N} (σ) (or equivalently dependence of the scalar coupling on density, or σ as g_{σ}^{N }(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 C_{N} (σ), that induces 3body and higher order Nbody 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 pointlike nucleon model of nuclear matter), which results in the much larger value, K ≃ 500 MeV; the empirically extracted value falls in the range K = 200300 MeV. (See Ref. [39] for an extensive analysis on this issue.)
Fig. 1: Negative of binding energy per nucleon for symmetric nuclear matter
E^{tot}/Am_{N} (upper panel), and the effective lightquark mass m_{q}^{*} , and vector (V_{ω}^{q}) and scalar (V_{σ}^{q}) potentials felt by the light quarks (lower panel).
Once the selfconsistency 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, g_{σ}^{N} and g_{ω} at the nucleon level (see also Eq. (13)), by the fit to the binding energy of 15.7 MeV at the saturation density ρ_{0} = 0.15 fm^{3} for symmetric nuclear matter, as well as g_{ρ} to the symmetry energy of 35 MeV. The determined quarkmeson coupling constants, and the current quark mass values used are listed in Table I. The coupling constants at the nucleon level are (g_{σ}^{N})^{2}/4𝜋 = 3.12, g_{ω}^{2} / 4𝜋 = 5.31 and g_{ρ}^{2}/4𝜋 = 6.93. (See Eq. (13), and recall g_{ω} = 3g_{ω}^{q} and g_{ρ} = g_{ρ}^{q}.) 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), quarkmeson coupling constants and the bag pressure, B_{p}. Note that the m_{c} value is updated from Refs. [2, 3] based on Ref. [41].
m_{u,d} 
5 MeV 
g_{?q} 
5.69 
m_{s} 
250 MeV 
g_{?q} 
2.72 
m_{c} 
1270 MeV 
g_{ρq} 
9.33 
m_{b} 
4200 MeV 
B_{p1/4} 
170 MeV 
We show in Fig. 1 negative of binding energy per nucleon for symmetric nuclear matter E^{tot }/A  m_{N} (upper panel), and effective lightquark mass m^{*}_{q}, vector (V_{ω}^{q}) and scalar (V_{σ}^{q}) 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, ψ_{f} (x) = N_{f} exp^{iєft/Rh*} ψ_{f} (r^{→}), where N_{f} and ψ_{f} (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 R_{h}^{*}, 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/R_{h}* are given by,
 (22)

 (23)

 (24)

The hadron mass in a nuclear medium, m^{*}_{h} (free mass is
hdenoted by m_{h}), is calculated for a given baryon density
together with the mass stability condition,
 (25)

 (26)

where Ω^{*}_{q} = Ω^{*}_{q—} = [x_{q}^{2} + (R^{*}_{h}m^{*}_{q})^{2}]^{1/2} (q = u, d), with m^{*}_{q} = m_{q}  g_{σ}^{q}σ = m_{q}  V_{σ}^{q}, Ω^{*}_{Q} = Ω^{*}_{Q—} = [x_{Q}^{2}+ (R^{*}_{h}m_{Q})^{2}]^{1/2} (Q = s, c, b), and x_{q, Q} are the lowest mode bag eigenvalues. B_{p} is the bag pressure (constant), n_{q} (n_{q—}) and n_{Q} (n_{Q—}) are the lowest mode valence quark (antiquark) numbers for the quark flavors q and Q in the hadron h, respectively, while z_{h} parametrizes the sum of the centerofmass and gluon fluctuation effects, which are assumed to be density independent [5]. The bag pressure B_{p} = (170 MeV)^{4} (density independent) is determined by the free nucleon mass m_{N} = 939 MeV with the bag radius in vacuum R_{N} = 0.8 fm and m_{q} = 5 MeV as inputs (this yields S_{N} (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 lightquark 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 quarkmeson coupling constants, g_{σ}^{q}, g_{ω}^{q} and g_{ρ}^{q}, have already been determined by the nuclear matter saturation properties. Exactly the same coupling constants, g_{σ}^{q}, g_{ω}^{q} and g_{ρ}^{q}, 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 C_{B}(σ) (B = N, Λ, Σ, Ξ, Λ_{c}, Σ_{c}, Ξ_{c}, Λ_{b}, Σ_{b}, Ξ_{b}) (see Eq. (12)), can be parameterized as a linear form in the σ field, g_{σ}^{N}σ, for a practical use [5, 6, 9, 33]:
 (27)

The values obtained for a_{B} are listed in Table II. This parameterization works well up to about three times the normal nuclear matter density 3ρ_{0}. Then, the effective mass of baryons B in nuclear matter is also well approximated up to 3ρ_{0} by:
 (28)

 (29)

with n_{q} being the valence lightquark number in the baryon B. See Eqs.(4) and (5) to compare with g^{N,Y}(σ) and the above expression. The obtained values of the "slope parameter" a_{B} for various baryons are listed in Table II.
Table II. Slope parameter values a_{B} obtained for various baryons [33].
Note that the tiny differences in values of a_{B} from those in Refs. [2, 3], are due to the differences in the number of data points for evaluating a_{B}, 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 manybody problem in a nonrelativistic framework. This allows us to take the limit corresponding to a zerorange 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 V_{s.o}. is the spinorbit interaction.
To get the dynamical mass m_{N}^{*}(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 n_{q} = 3 and d = a_{N} and g_{σ} ≡ g_{σ}^{N} hereafter,
 (31)

where d of the MIT bag model gives d = 0.22R_{N} (in MeV^{1}) with the nucleon bag radius R_{N} (fm) corresponding to Table II with R_{N} = 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 E_{N} (r^{→}) was approximated by neglecting the velocity dependent terms (∇^{→}σ)^{2},
 (34)

where we define the classical densities as
ρ^{cl} (r^{→}) = ∑_{i} δ(r^{→}  r^{→}_{i}) and ρ_{s}^{cl} (r^{→}) = ∑_{i} (1  p^{→}_{i}^{2} / 2m_{N}^{2})δ(r^{→}  r^{→}_{i}).
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, δE_{tot} / δσ(r^{→}) = δE_{tot} / δω(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 ρ_{s}^{cl} and ρ^{cl} except in the leading term. When inserted into Eq. (34), the series for the meson fields generates Nbody 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 G_{i} = g_{i}^{2 }/ m_{i}^{2}(i = σ, ω, ρ) and A_{ij} = G_{σ} + (2µ_{s}  1) G_{ω} + (2µ_{v}  1)G_{ρ}τ^{→}_{i} · τ^{→}_{j} /4, with µ_{s} and µ_{v} being respectively, the nucleon isoscalar and isovector magnetic moments. Here r^{→}_{ij} = r^{→}_{i} r^{→} _{j} and ∇^{→}_{i} is the gradient with respect to r^{→}_{i} acting on the delta function. In Eq. (35) we we have used the notation δ^{2} (ijk) for δ (r^{→}_{ij}) δ (r^{→}_{jk}) and analogously for δ^{3} (ijkl). Furthermore, we have dropped the contact interactions involving more than 4bodies because their matrix elements vanish for antisymmetrized states.
To fix the free parameters, G_{i}, the volume and symmetry coeffcients of the binding energy per nucleon of infinite nuclear matter, E_{B }/A = a_{1} + a_{4} (N  Z)^{2}/A^{2}, are calculated and fitted so as to produce the experimental values. Using the bag model with the radius R_{N} = 0.8 fm and the physical masses for the mesons and m_{σ} = 600 MeV, one gets, in fm^{2}, G_{σ} = 11.97, G_{ω} = 8.1 and G_{ρ} = 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 4body 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 4body 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 singleparticle states with j = l + 1/2 and j = l  1/2. Then, by comparing the HartreeFock Hamiltonian obtained from H_{QMC} 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 3t_{1} + 5t_{2}, which controls the effective mass, and 5t_{2}  9t_{1}, 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 spinorbit strength W_{0} comes out with approximately the correct value. The middle column (N = 3) shows the results when we switch off the 4body force. The main change is expected to decrease of the predicted 3body 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 4body 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 m_{σ} = 600 MeV) [35] compared with the Skyrme force [43].

QMC 
QMC(N=3) 
SkIII 
t_{0}_{ }(MeV fm^{3}) 
1082 
1047 
1129 
x_{0} 
0.59 
0.61 
0.45 
t_{3}_{ }(MeV fm^{6}) 
14926 
12513 
14000 
3t_{1}_{ }+ 5t_{2}_{ }(MeV fm^{5}) 
475 
451 
710 
5t_{2}_{ } 9t_{1}_{ }(MeV fm^{5}) 
4330 
4036 
4030 
W_{0}_{ }(MeV fm^{5}) 
97 
91 
120 
K (MeV) 
327 
364 
355 
SUMMARY
We have given a short review on the basics of the quarkmeson coupling (QMC) model, a quarkbased 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 zerorange 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 QMCgenerated effective interaction automatically contains the 3body and higher order Nbody 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 inmedium 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 Científico e Tecnológico (CNPq) Process, No. 313063/20184, and No. 426150/20180, and Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Process, No. 2019/007630, and was also part of the projects, Instituto Nacional de Ciência e Tecnologia  Nuclear Physicsand Applications (INCTFNA), Brazil, Process. No. 464898/20145, and FAPESP Temático, Brazil, Process, No. 2017/056600.
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