Mean field and pairing properties of nuclear matter in a quark–meson coupling model

Nuclear Physics A 697 (2002) 469–491 www.elsevier.com/locate/npe

Mean field and pairing properties of nuclear matter in a quark–meson coupling model E.F. Batista, B.V. Carlson ∗ , T. Frederico Departamento de Física, Instituto Tecnológico de Aeronáutica, 12228-900 São José dos Campos, SP, Brazil Received 22 June 2001; accepted 26 July 2001

Abstract We investigate the effects on mean-field quantities and nuclear pairing in symmetric nuclear matter of the density-dependent effective nucleon–scalar-meson coupling that is obtained in a σ –ω quark– meson coupling model. We find that the effective coupling strongly modifies mean-field quantities, such as the compressibility, the effective nucleon mass, and the spin–orbit interaction, as well as the scalar and vector components of the pairing field, but that it leaves the nonrelativistic pairing gap function almost unchanged. This model independence is due in part to the strong constraint imposed by requiring that the pairing field reproduce the two-nucleon virtual state in the vacuum, but is also a result of compensation between variations in the effective nucleon–scalar-meson coupling constant and the effective nucleon mass. We also analyze the correlation between the compressibility and the spin–orbit interaction and find that neither our model nor other quark–meson coupling models can describe the two simultaneously.  2002 Elsevier Science B.V. All rights reserved. PACS: 21.65.+f; 21.60.Jz; 21.30.Fe; 12.39.-x Keywords: Quark–meson coupling model; Nuclear matter; Pairing

1. Introduction The nuclear many-body problem has been the object of enormous theoretical and experimental study for decades. Initially, potential models were developed that took into account the experimental data on nucleon–nucleon scattering and the structure of the deuteron, but required careful accounting of the short-range part of the nucleon–nucleon interaction and three-body forces to obtain a reasonable description of the properties of nuclear matter. A significant advance was made some years ago, when models of nuclear structure were developed in which the meson degrees of freedom are explicitly treated as dynamical variables and the nucleon–nucleon interaction is mediated through * Corresponding author.

E-mail address: [email protected] (B.V. Carlson). 0375-9474/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 ( 0 1 ) 0 1 2 5 0 - 7

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the exchange of virtual mesons between Dirac nucleons [1–3]. Such models provide an excellent description of nuclear structure at the mean-field level, when adjusted effective meson masses, coupling constants and nonlinear scalar couplings, such as those of the NL3 [4] set, are used. When coupled with a Brückner approximation to the effective interaction, they provide a reasonably consistent description of both nuclear structure and the two-nucleon problem [5]. The original nucleon–meson models of nuclear structure treated both the nucleons and mesons as point-like particles. In order to study the role of nucleon structure in the nucleus, effective models were soon developed in which point-like mesons couple directly to the quarks in the nucleons [6,7]. These quark–meson coupling models were found to lead to large changes in the effective nucleon–nucleon interaction. Several basic characteristics of nuclear matter, such as the compressibility and the nucleon effective mass, show better agreement with the experimental data in these models than those obtained in a model with point-like nucleons. Since then, the Guichon bag quark–meson coupling model has been studied in great detail in a series of works by by Saito, Thomas and collaborators [8,9] and by others [10]. They have studied aspects of the structure of mesons and nucleons as well as the reflection of these on nuclear properties of nuclei and of nuclear matter, obtaining good general agreement with the experimental data. In two more recent works, the exchange term of the effective nucleon–nucleon interaction has also been examined, in an attempt to resolve the remaining discrepancies with the data [11,12]. The original objective of this study was to examine just the effects of nucleon structure on nuclear pairing, under the constraint of the empirical value of the nucleon– nucleon virtual state in the vacuum. However, upon returning to the quark–meson coupling model we had developed in Ref. [6], which is based on a confining harmonic oscillator potential rather than a bag one, we found that we could simplify its development substantially. As this simplification resulted in a different condition on its selfconsistency, we thought it relevant to reexamine the mean-field properties of the model as well. Here, we will examine both the mean-field and pairing properties within the model. An interesting aspect of our formulation of the quark–meson coupling model is that it possesses a free parameter, which we take to be the bare quark mass. By varying this mass, we observe a smooth transition from the light-quark region usually studied in quark– meson coupling models to a heavy-quark region that reproduces the results obtained with a structureless nucleon. Our model thus encompasses both quark-coupling and nucleoncoupling models. In the following, we will first develop, in Sections 2 and 3, a simpler version of the quarkcoupling model that we presented in Ref. [6], including center-of-mass corrections. In Section 4, we analyze the mean-field properties of symmetric nuclear matter in the model. The effects of the quark–meson coupling on nuclear pairing is discussed in Section 5 and we conclude in Section 6.

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2. Quark–meson coupling model of nuclear matter The medium to long-range component of the nucleon–nucleon interaction can be understood as the exchange of several different types of mesons: pions, omega mesons and rho mesons being the principal ones. Yet, relatively simple σ –ω models can achieve a good description of the bulk properties of nuclear matter and of nuclei with an interaction that includes only the exchange of a repulsive effective vector meson ω and an attractive effective scalar meson σ . In a model of point-like nucleons, which we will call the nuclear σ –ω model in the following, the simplest couplings for these mesons enter the Lagrangian N gvN γµ V µ ΨN and Ψ N gsN φΨN , in which the vector and scalar coupling constants, gvN as Ψ N and gs , are adjusted to reproduce the saturation properties of nuclear matter. To replace this model with one that takes into account the nucleon substructure, we couple the same mesons directly to the quarks in the nucleons, rather than to the nucleons themselves. We thus substitute the interaction terms above with terms having forms, q gsq φΨq , where gvq and gsq are quark–meson coupling constants, which q gvq γµ V µ Ψq and Ψ Ψ will eventually be adjusted in a similar fashion to reproduce nuclear saturation. As our objective is to study the effects of the nucleon substructure on the gross nucleon properties and nuclear structure, we do not wish to enter into the details of the nature of the complex quark–gluon exchange interaction that binds the quarks in the colorless nucleon. Instead, we simply consider the quarks to be independent particles interacting through a mean field generated by the other quarks in the nucleon. This mean field should provide confinement of the quarks. To assure this and to keep our calculations a simple as possible, we use the model developed by Ferreira and Zagury [13], taking the mean field to be a spherically symmetric harmonic oscillator potential U (r) = U0 + 12 Kr 2 ,

(1)

with K an elastic constant and U0 a potential depth, which are adjusted to fit the vacuum properties of the nucleon. The potential acts only on the upper component of each Dirac quark. While the potential U (r) represents the effective color interaction between the quarks in the nucleon, the exchange of σ and ω mesons represents the long-range effective interaction resulting from the exchange of color-singlet q¯ q pairs. We assume that the mesons do not interact directly with the confining potential. Instead, they lead to an additional contribution to the mean field, which modifies the structure of the nucleon and, consequently, modifies the effective nucleon–nucleon interaction. As the mesons themselves are made of quarks, we could expect their structure to change as well in the nuclear medium [8]. We will not take into account such effects nor we will consider the expected reduction in the meson masses at finite densities due to the partial restoration of chiral symmetry [14,15]. Based on the above discussion, we take the Lagrangian density for a quark in a nucleon to be L = Lq + Lm + LI

(2)

where the three terms refer, respectively, to the quark, meson and quark–meson interaction terms in the Lagrangian. These are:

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q γµ pµ − mq − 1 (1 + γ0 )U (r) Ψq , Lq = Ψ 2   Lm = 12 ∂µ φ∂ µ φ − m2s φ 2 − 14 Fµν F µν + 12 m2v Vµ V µ ,

(3) (4)

where Fµν = ∂µ Vν − ∂ν Vµ

q gsq φΨq − Ψ q gvq γµ V µ Ψq . and LI = Ψ

(5)

A comparison between the vertices in the nucleon–meson and quark–meson coupling models yields relations between their coupling constants. Due to the coupling of the vector meson to a conserved current and to our assumption of independent quarks, we obtain q a simple relationship between the quark–vector-meson coupling constant, gv , and the effective nucleon one, gvN . We have N γµ V µ ΨN = gv N| gvN Ψ q

3 

q γµ V µ Ψq |N , q i γµ V µ Ψq i |N = 3gvq N|Ψ Ψ

(6)

i=1 q

so that gvN = 3gv . If we apply the same procedure to obtain a relation between the scalarmeson coupling constants, requiring that N φΨN = gs N| gsN Ψ q

3 

q i φΨq i |N = 3gsq N|Ψ q φΨq |N , Ψ

(7)

i=1 q q 3gs ρs ,

q

= where ρs is the scalar density of the quarks in the nucleon. The we obtain scalar-quark density is a dynamical quantity that depends on both the mean field and the binding of the quarks within the nucleon. The nucleon–scalar-meson coupling constant, gsN , is thus also a mean-field-dependent quantity. The equation of motion of the quark field is easily obtained from Lagrangian (2) as 
q q (8) γµ pµ − mq − 12 (1 + γ0 )U (r) + gs φ − gv γµ V µ Ψq = 0. gsN

In the mean-field approximation, the meson-field operators are substituted by their vacuum expectation values, φ → φ ≡ φ0

and Vµ → Vµ ≡ δµ0 V0 .

(9)

In a static, uniform system, the mean fields φ0 and V0 are independent of the coordinates xµ , while rotational invariance of the system implies that the spatial component of the vector field vanishes. Substituting into the equation for the quark field, Eq. (8), we have
   q  q  γ0 εq − gv V0 − γ · p − mq − gs φ0 − 12 (1 + γ0 )U (r) Ψq = 0. (10) We rewrite this in matrix form as     f εq − mq − U (r) − σ · p = 0, g σ · p −εq − mq

(11)

where we have made the common substitutions q

εq = εq − gv V0

q

and mq = mq − gs φ0 ,

(12)

which define the effective energy and mass of the quarks, respectively. We have also written the Dirac spinor in terms of its upper and lower Pauli spinors, f ( r ) and g( r ).

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Noting that both εq and mq are constants, we use the lower of the two coupled equations to rewrite the lower component of the Dirac spinor in terms of the upper one as g( r ) =

1 σ · p f ( r ). εq + mq

(13)

Substituting this into the upper equation permits us to write it as an equation for the upper component alone:
2  p − (εq + mq )(εq − mq − U (r)) f ( r ) = 0. (14) This equation can be recognized as that of a three-dimensional harmonic oscillator. The solution corresponding to a quark in the fundamental 1s state is 
 f0 (r) = exp −αr 2 /2 with α = 12 K(εq + mq ). (15) The total energy of the quark is given by α εq = mq + U0 + 3  . εq + mq

(16)

The complete Dirac wave function for the fundamental quark level in a nucleon can then be written as    εq + mq (17) Ψq ( r ) = N f0 (r) χν , iα σ · r where χν is a Pauli spinor and N a normalization constant, given by √ α β α 2 . N = 3/2 , where β =  3 π (εq + mq )2 1 + 2β

(18)

We construct the nucleon wave function as the symmetrized product of the wave functions of three independent quarks in the fundamental state.

3. Center-of-mass corrections and self-consistency When single-particle wave functions bound by a potential with fixed center are used to construct a composite particle wave function, the center-of-mass motion of the composite particle is also bound by the potential. If the composite is to be considered as a translationally invariant state, its wave function must be corrected for the effects of this spurious binding. In the previous section, we determined the wave functions of the quarks in a nucleon using such a fixed-center potential. When calculating properties of the nucleon constructed from these quarks, we must extract the contribution of the centerof-mass motion in order to obtain physically relevant results. As extracting the center-ofmass movement from the composite nucleon wave function can be an extremely difficult task, we will not do it here. However, extracting the center-of-mass effects from nucleon observables is a simpler proposition. The center-of-mass contribution to the root-meansquare radius of the nucleon can be extracted exactly. The approximate extraction of the

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center-of-mass contribution to the energy of the nucleon is a straightforward task. Here, we will extract the center-of-mass energy to first order in the difference between the fixedcenter and relative quark coordinates, using the method described in Ref. [9]. We assume that the Hamiltonian of our composite nucleon, HN , can be written as HN = Hint + Hcm ,

(19)

where Hint is the Hamiltonian corresponding to the internal degrees of freedom and Hcm is the center-of-mass Hamiltonian. Based on our development in the previous section, we can write the total nuclear Hamiltonian as HN =

3 


   q γ0 (i) γ (i) · p i + mq − gv V0 + 12 1 + γ0 (i) U (ri ) .

(20)

i=1

The internal Hamiltonian can be written in a similar fashion, but in terms of the relative rather than the fixed coordinates, as Hint =

3 


  q γ0 (i) γ (i) · π i + mq − gv V0 + 12 1 + γ0 (i) U (ρi ) ,

where

(21)

i=1

1 p i 3 3

π i = p i −

1 r i = r i − R cm . 3 3

and ρ i = r i −

i=1

(22)

i=1

The center-of-mass contribution to the Hamiltonian is then just the difference between these two: Hcm = HN − Hint =


 1 1  γ0 (i)γ (i) · p i + 1 + γ0 (i) U (ri ) − U (ρi ) . (23) 3 2 3

3

i=1

i=1

We can now estimate the center-of-mass contribution to the nucleon energy by calculating the expectation value of the center-of-mass Hamiltonian using the composite nucleon wave function obtained in the last section: (1) (2) |N + N|Hcm |N , εcm = N|Hcm |N = N|Hcm (1) |N = N|Hcm

1 N| 3

3 

where

(24)

γ0 (i)γ (i) · p i |N = N|γ0 (1)γ (1) · p 1 |N

i=1

α 1 = 3  , εq + mq 1 + 3β/2

(25)

and  1 N| (1 + γ0 (i))[U (ri ) − U (ρi )] |N 2 3

(2) N|Hcm |N =

i=1

 
K 2 N| |N (1 + γ0 (i)) 2 ri · R cm − Rcm 4 3

=

i=1

3 α 1 + 5β/6 = . 4 εq + mq (1 + 3β/2)2

(26)

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Together, the two terms yield εcm =

α 3 + 23β/6 3 . 2 εq + mq (1 + 3β/2)2

(27)

We can now define the effective nucleon mass in the medium, M  , as the center-of-mass corrected energy of the three independent quarks: M  = 3εq − εcm .

(28)

The effective nucleon mass M  is density-dependent, as both εq and εcm are. Other nucleon observables must also be corrected for center-of-mass effects. In particular, for the mean squared nucleon radius, we find 3 2 2  2

1  2 r i − R cm |N = N| r q − R cm |N = N| rq2 |N , rN = N| 3 3

(29)

i=1

where we denote by r q the coordinate vector of any one of the three quarks. Using the wavefunction of the previous section to evaluate the expectation value, we obtain 2 1 1 + 5β/2 . rN = α 1 + 3β/2

(30)

We now construct nuclear matter by assuming that our composite nucleons may be treated as Dirac nucleons with a mass of M  that may be boosted without perturbing their structure. We can then write the nuclear matter energy density, in analogy to the usual nuclear σ –ω model one, as 2 E= π

kF

dk k 2 k 2 + M 2 + 12 m2s φ02 + 12 m2v V02 ,

(31)

0

where the vector mean field is given by q

V0 = 3

gv ρB , m2v

(32)

with ρB being the baryon density. The Fermi momentum kF is determined through its relation to the baryon density, 2 3 k . (33) 3π 2 F We determine the scalar mean field φ0 by requiring that the energy density be a minimum with respect to variations in the scalar mean field. This yields the following self-consistency equation for the mean field φ0 , ρB =

∂M  1 2 φ0 = − ∂φ0 m2s π 2

kF 0

M dk k 2 √ . k 2 + M 2

When compared to the self-consistency equation for the nuclear σ –ω model,

(34)

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gN 2 φ0 = s2 2 ms π

kF 0

M dk k 2 √ , k 2 + M 2

(35)

we find the two expressions to be identical if we define the nucleon–scalar-meson coupling constant as gsN = −

∂M  . ∂φ0

(36)

This is obvious in the nuclear σ –ω model, where the relationship between M  and φ0 is a linear one: M  = M − gsN φ0

(nuclear σ −ω model).

(37)

If our separation of the intrinsic and center-of-mass movement of the nucleon were exact, we could write, according to the Feynman–Hellman theorem: gsN = −

∂M  ∂ q q =− N|Hint |N = 3gs ρs , ∂φ0 ∂φ0

to obtain the same relation that we would expect from an inspection of the scalar-meson vertex, as given in Eq. (7). However, as this separation is not exact, we will use Eq. (36) with the effective mass given by Eq. (28). The latter relation is also used by Saito and Thomas in their development of the Guichon bag QMC model [8,9].

4. Calculation of mean-field quantities We wish to analyze the effects on several mean-field quantities of our composite model of the nucleon. However, we must first enumerate the free parameters in the model and determine conditions which fix these. In our development, we have introduced five free parameters: the quark mass mq , the two parameters of the quark potential, U0 and K, and the two quark–meson coupling constants, q q gv and gs . We will retain the quark mass mq as a free parameter. Given a value of mq , we fix the potential parameters, U0 and K, by requiring that the nucleon in the vacuum have the properties

M0 = M = 938 MeV and rN2 0 = (0.8)2 fm2 at ρb = 0. We fix the quark–meson coupling constants by fixing the Fermi momentum and binding energy per nucleon at the nuclear matter saturation point. We require these to be

 

B.E.

E

= −15.75 MeV. = − M kF,sat = 280 MeV/c and

A sat ρB sat We have performed calculations with these constraints over a large range of values of the quark mass mq . We display first, in Fig. 1, the binding energy per nucleon as a function of the Fermi momentum, for several values of the quark mass mq . We note that all curves pass through the same saturation point, as determined by the conditions above. As the quark

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Fig. 1. The nuclear matter binding energy as a function of the Fermi momentum for several values of the quark mass mq .

mass increases, the curves become narrower about the saturation point. We can quantify this narrowing in terms of the compressibility,

2

−1 2 d (B.E./A) , (38) KV = kF

2 dkF kF,sat which measures the stiffness of nuclear matter at the saturation point. In Fig. 2, we plot the compressibility as a function of the quark mass mq . Its value increases monotonically with the quark mass. The compressibility saturates at high values of the mass, approaching the value obtained in the nuclear σ –ω model, which is a reasonable result. As the quark mass increases, the quarks move less and less relative to the nucleon center-of-mass and finally

Fig. 2. The nuclear matter compressibility as a function of the quark mass mq .

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stop when their masses become infinite. The model then reduces to the original σ –ω model of a structureless nucleon. The physical value of the compressibility is usually taken to be about 200 MeV. We see in Fig. 2 that we obtain a similar value with a quark mass of about 100 MeV. Saito and Thomas obtain a compressibility of about 200 to 300 MeV in their bag QMC calculations [8]. Typical Hartree parameter sets that have been fit to nuclear properties over the mass table, such as the NL3 one [4], possess compressibilities closer to 300 MeV, which would correspond to a quark mass of about 200 MeV in our model. One of the fundamental derived quantities of the model is the effective nucleon mass M  , shown in Fig. 3 as a function of the Fermi momentum for several values of the quark mass. In all cases, the effective mass decreases as the baryon density increases and then saturates at high baryon densities. The density dependence grows stronger as the bare quark mass becomes larger. At extremely high values of the quark mass, the behavior of the effective mass as a function of the density approaches that of the nuclear σ –ω model. The effective nucleon mass of the nuclear σ –ω model at saturation density is about 550 MeV, a value considered to be extremely small. A value of about 700 to 750 MeV is usually obtained in nonrelativistic calculations and considered to be more consistent with the observed value of the density of states near the Fermi surface. As can be seen in Fig. 3, an effective mass in this range results when a quark mass mq of about 300 to 400 MeV is used. The bag QMC calculations of Saito and Thomas yield a density dependence of the effective mass similar to that seen in the curve corresponding to a quark mass of 200 MeV. The other fundamental derived quantity of the model is the nucleon–scalar-meson coupling constant, gsN , shown as a function of the Fermi momentum for several values of the quark mass in Fig. 4. Like the effective nucleon mass, the nucleon–scalar-meson coupling constant decreases as the baryon density increases. The coupling constant, however, displays its strongest density dependence at low values of the quark mass mq .

Fig. 3. The effective nucleon mass M  as a function of the Fermi momentum for several values of the quark mass mq .

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Fig. 4. The nucleon–scalar-meson coupling constant gsN as a function of the Fermi momentum for several values of the quark mass mq .

When the quark mass becomes extremely large, the coupling constant tends to a constant, independent of the baryon density, just as it is in the nuclear σ –ω model. The vacuum values of the quark–meson and nucleon–meson coupling constants are shown in Fig. 5. The quark–meson coupling constants have been determined by fitting the saturation point of the nuclear binding energy. The nucleon–vector-meson coupling constant is three times the quark one, as discussed above. The vacuum nucleon–scalarmeson coupling constant has the value detemined by Eq. (36) at zero baryon density. It reflects the behavior of the quark scalar density of a nucleon in the vacuum. At high values of the quark mass, the nucleon–meson coupling constants tend to the values of the nuclear

Fig. 5. The vacuum values of the quark and nucleon coupling constants as a function of the quark mass mq .

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σ –ω model. In this limit, the nucleon–scalar-meson coupling constant is also three times the quark–scalar-meson one, consistent with the diminishing motion of the quarks in the nucleon, since the scalar density has the same value as the vector density for motionless quarks. For small values of the quark mass, the nucleon–scalar-meson coupling constant approaches the quark–scalar-meson one, reflecting the substantial difference between the two densities in this limit. The vector-meson coupling constants necessary to fit the saturation point decrease as the quark mass decreases and reach a value of zero for a quark mass of about 32 MeV. For quark masses below this value, we cannot fit the nuclear matter saturation point. The effects of the nuclear medium also appear in the root-mean-square nucleon radius, shown in Fig. 6, as a function of the Fermi momentum for several values of the quark mass mq . The nucleon radius increases with the nucleon density for all values of the quark mass. The rate of increase tends to be larger for smaller values of the quark mass. The nucleon radius at the saturation density is approximately 0.86 fm for values of the quark mass below about 500 MeV. For larger values of the quark mass, the value of the radius drops slowly with increasing mass, eventually reaching the constant nuclear σ –ω value of 0.80 fm as the quark mass tends to infinity. Of the nucleon properties studied, the nucleon radius is the slowest to respond to the infinite quark mass limit. With a quark mass of 5 GeV, the nucleon radius is still greater than 0.81 fm at the nuclear matter saturation density. We may also compare the effective potential strengths extracted from our model with their phenomenological values. To make this comparison, we write the Dirac equation for a nucleon in its two-component Hamiltonian form as    u E − Vv − M  − σ · p = 0, (39) v − σ · p E − Vv + M  and reduce this to an equation for the upper component u alone. After some manipulation, we can rewrite the latter as

Fig. 6. The root-mean-square radius of the nucleon as a function of the Fermi momentum for several values of the quark mass mq .

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 p2 E2 − M 2 + Vc (r) + Vso (r) u = Enr u, σ · , + Vd (r) u = 2M 2M

481

(40)

where we have assumed for the potentials a dependence on the radial coordinate r, as would be the case for a sperically symmetric nucleus. On the right-hand side, we have defined a nonrelativistic energy as Enr because of its similarity in form with the usual kinetic one. The central and spin–orbit potentials have the form E Vs (r)2 − Vv (r)2 Vv (r) − Vs (r) + and (41) M 2M  h¯ 2 1 d Vv (r) + Vs (r) . Vso (r) = (42) 2M(E + M − Vv (r) − Vs (r)) r dr The Darwin potential, Vd (r), can be incorporated in a redefinition of the wave function u and will be neglected here. To make contact with our nuclear matter calculations, we take our nuclear matter potentials, Vc (r) =

Vs = M − M 

and Vv =

gvN2 ρB , m2v

(43)

as potential strengths that multiply a common form factor: Vs (r) = Vs f (r)

and Vv (r) = Vv f (r),

(44)

which we take to have a Wood–Saxon form: 1 . f (r) = 1 + exp[(r − R)/a]

(45)

In a standard phenomenological analysis, the potentials are parametrized as Vc (r) = −Vc f (r) and Vso (r) = Cso Vso

1 df , r dr

(46)

where Cso ≈ 2 fm2 . A comparison between these expressions and those of Eqs. (41) and (42) permit us to make the approximate associations E V 2 − Vs2 Vv + v and M 2M h¯ 2 (Vv + Vs ) , ≈ 2Cso M(E + M − (Vv + Vs )/2)

Vc ≈ Vs − Vso

(47) (48)

where we have evaluated the form factors in the denominator of the spin–orbit potential at r = R, the point at which the derivative in the numerator has its maximum value. We observe that the central potential strength is determined by the difference between the two large Dirac potential strengths, Vs and Vv , while the spin–orbit potential strength is determined by their sum. To compare the potential strengths with the phenomenological values obtained from fits to low-energy nucleon–nucleus scattering, we evaluate them at E = M. We display the spin–orbit potential strength Vso obtained from our calculations in Fig. 7. It possesses values close to zero for small values of the quark mass, rises smoothly with

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Fig. 7. The spin–orbit potential strength as a function of the quark mass mq .

increasing quark mass and saturates at about 5 MeV, for values of the quark mass larger than about 1 GeV. Phenomenological values of the spin–orbit strength are in the range from 5 to 7 MeV, which puts them at the extreme upper limit of values we have calculated, corresponding to quark masses larger than 1 GeV. For quark masses in the range of 200 to 400 MeV, where most of our calculated mean-field properties show reasonable to good agreement with observation, the strength of the spin–orbit potential is about one third of the phenomenological value. The central potential strength, which is not shown here, displays a dependence on the quark mass similar to that of the spin orbit strength. It varies from a value close to 58 MeV at small values of mq to about 38 MeV for values of the quark mass above 1 GeV. Phenomenological studies put its value in the range from 45 to 55 MeV, which is consistent with our calculations for quark masses from about 200 to 400 MeV. With the exception of the spin–orbit potential strength, the mean-field quantities that we can compare with phenomenological values show reasonable agreement for values of the quark mass of about 200 to 300 MeV. The compressibility, in particular, yields reasonable values in this range but is much too large at the large values of the quark mass for which the spin–orbit potential strength is reasonable. We illustrate this correlation between the the two values in Fig. 8, where we plot the spin–orbit strength as a function of the compressibility obtained for the same value of the quark mass mq . Reasonable values for the pair are given by the NL3 nucleon–meson coupling result (X in Fig. 8), in the upper left-hand side of the figure, and very far from our curve [4]. This discrepancy, however, seems to be common to other quark–meson coupling models as well. We have taken the liberty of applying our procedure for estimating the spin–orbit potential strength to the Hartree scalar and vector potential strengths given in Ref. [12], plotting the resulting spin– orbit potential strengths as triangles in Fig. 8. We find the results to be very similar to ours. In this reference and an earlier one [11], the authors suggest that the Fock exchange terms could increase the spin–orbit strength without altering the compressibility. Using

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−1 Fig. 8. The spin–orbit potential strength as a function of the compressibility KV for several different models. Our QCM results are given as a line. The values obtained using the QCM Hartree and Hartree–Fock potential strengths of Ref. [12] are shown as triangles and squares, respectively. The value obtained using the nuclear NL3 parameter set of Ref. [4] is shown as an ×.

their Hartree–Fock potential strengths, we do indeed obtain spin–orbit potential strengths (squares in Fig. 8) about 25% higher than the Hartree ones, but still very far from the phenomenological value. Lacking in the QMC models, but contained in the NL3 nucleon parameter set used to obtain the point in the upper left-hand side of the figure, are the scalarmeson self-interaction terms that appear to provide one means of obtaining simultaneously reasonable values of the compressibility and the spin–orbit interaction.

5. Nuclear pairing W.H. Dickhoff has carefully analyzed the intimate relationship between pairing and a bound-state pole in the Brückner G-matrix [16]. We will loosely follow his analysis in the development of the approximation we use here. In the ladder approximation, the Brückner G-matrix Γ (E, K) describing the scattering between two particles in the nuclear medium satisfies a Bethe–Salpeter equation of the form Γ (E, K) = V + V G(E, K)Γ (E, K),

(49)

which we represent diagrammatically in Fig. 9. In this expression, V is the full two-body scattering kernel and G(E, K) is the propagator for two noninteracting particles, with total energy E and center-of-mass momentum K, that takes into account the one-body propagation in the nuclear medium of each of them, as well as the blocking of occupied states. When the G-matrix possesses a bound state at εb (K), it can be decomposed into a separable term and a remainder as

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Fig. 9. A diagrammatic representation of the Bethe–Salpeter equation for the Brückner G-matrix. The propagator is represented by the arrows on the particle lines while the interaction V is represented by the zigzag lines.

Fig. 10. A diagrammatic representation of the decomposition of a G-matrix containing a bound state into a factorizable bound-state term and the rest of the G-matrix.

Fig. 11. A diagrammatic representation of the Bethe–Salpeter equation for the vertex function of a bound state. The propagator is represented by the arrows on the particle lines while the interaction V is represented by the zigzag line.

Fig. 12. A diagrammatic representation of the association between a two-body bound-state wave function and the anomalous HFB propagator.

Γ (E, K) = |dK

1 dK | + Γc (E, K), E − εb (K)

and as shown in Fig. 10, where |dK is the vertex function of the bound state and Γc (E, K) is the remainder of the G-matrix. Substitution into the Bethe–Salpeter equation (49), yields the following equation for the vertex function: |dK = V G(εb , K)|dK ,

(50)

represented diagrammatically in Fig. 11. Cooper originally formulated pairing in terms of such an equation for the pair vertex function [17]. The integral equation for the vertex function can be recast in the more usual form of a wavefunction equation by defining |ΨK = G(εb , K) |dK , shown diagrammatically in Fig. 12, so that G−1 (εb , K)|ΨK = V |ΨK

⇒

(εb − H0 (K) − V )|ΨK = 0

in the vacuum. Rather than analyze pairing as the two-body-among-many problem that it is, we will follow Bogoliubov [18] and Gorkov [19], treating it approximately in a simpler one-body

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Fig. 13. A diagrammatic representation of the components of the HFB propagator.

formalism. To do this, we associate the pair wavefunction |ΨK=0 with an anomalous propagator F (k), in which one of the particles leaves the vertex going forward in time with momentum k while the other leaves it going backward in time with momentum −k, as shown in Fig. 12. This is a good approximation when the bound pair state is limited to a small region about center-of-mass momentum K = 0 [16], which we will assume to be the case here. To construct a complete description of the propagation of the system,  and the time-reversed we must also include the usual propagator G, its time reverse G, , in which one particle enters the vertex going forward in time anomalous propagator F while the second enters it going backwards in time. The four terms may combined in an extended propagator, which is shown together with its diagrammatic representation in Fig. 13. The Hartree–Fock equations for the extended propagator are known as the Hartree– Fock–Bogoliubov (HFB) equations. Dirac HFB equations were first derived in Ref. [20]. A nonrelativistic reduction of these was applied to nuclear matter in Ref. [21]. Complete self-consistent Dirac HFB equations in nuclear matter were derived and studied in Ref. [22]. These consist of the usual Hartree– Fock self-energy equation,  
 0 + d4 q αβ Γj α (0)Dj (0) Tr Γjβ (0)G(q) eiq 0 Σ(k) = −i 4 (2π) j   d4 q αβ +i Γj α (q)Dj (q)G(k − q)Γjβ (−q), (51) (2π)4 j

and an equation for the pairing field,   d4 q αβ T Γj α (q)Dj (q)F (k − q)BΓjβ (−q)B † , ∆(k) = i (2π)4

(52)

j

in which the sum over j represents the sum over the different meson exchanges, with the αβ Dj being the meson propagators and the Γj α the respective nucleon–meson vertices. The matrix B is defined as B = τ2 ⊗ γ5 C, in which the Pauli matrix τ2 acts in the isospin space and C is the charge conjugation matrix. The components of the nucleon propagator satisfy the HFB equation,    G(k) F (k) ∆(k) γν k ν − M − Σ(k) + µγ0 = 1, (53) (k) G(k)  ¯ F ∆(k) γν k ν + M + ΣT (k) − µγ0 where ΣT (k) = BΣ T (−k)B † . The pairing self-consistency equation, Eq. (52), is the direct analogue of the two-body vertex function equation (50). In the vacuum, the pairing

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equation reduces to the ladder approximation to a bound-state Bethe–Salpeter equation with center-of-mass momentum zero. The HFB self-consistency equations use the same interaction to determine the selfenergy and pairing fields. This is not the case in Dickhoff’s more general analysis. There it is found that the interaction determining the self-energy field is a dressed one, the Brückner G-matrix Γ , while the interaction determining the pairing field is the bare one. We will take this to mean that we have a certain liberty in the choice of interaction in the self-consistency equations for the two fields. We will study 1 S0 pairing in symmetric nuclear matter here. The self-energy has the usual Dirac structure: V (k), Σ(k) = ΣS (k) − γ0 Σ0 (k) + γ · kΣ while the 1 S0 pairing field takes the form 
¯ ˆ ∆(k) = ∆(k) = ∆S (k) − γ0 ∆0 (k) − iγ0 γ · k ∆T (k) τ · n,

(54)

(55)

where the orientation in isospin, n, ˆ is arbitrary. The Dirac pairing field can be reduced in a manner similar to that of the potentials of the self-energy field to yield a nonrelativistic pairing gap function. The reduction yields ˆ ∆ = ∆g τ · n, where the gap function ∆g is given by M k ∆0 − ∆s − i ∆T . E E The gap function is the quantity that corresponds most closely to the nonrelativistic pairing gap. Like the nonrelativistic central potential strength of Eq. (47), it is a difference between two larger relativistic quantities. A comparison between Hartree, Hartree–Fock and Brückner calculations in nuclear matter shows that they generally yield very similar results if the coupling constants are adjusted so that the calculations fit the nuclear matter saturation point. We have seen this explicitly for the case of the spin–orbit potential strengths obtained in Ref. [12], which differ by about 25% (a large difference when compared to that found for most quantities). We thus feel justified in neglecting the Fock term in the HFB self-energy equation (51) and treating the equation in the Hartree approximation. If we were then to use the effective nucleon–meson coupling constant of Eq. (36) in the HFB mean-field equation (51) as well, we would obtain an equation very similar to the mean-field equation of the quark–mesoncoupling model, Eq. (34). The only difference would be the partial occupations of states near the Fermi level, due the pairing. As the effect of the partial occupations on the mean field are extremely small, we can neglect them. Based on these arguments, we use the mean field obtained from Eq. (34) as the self-energy field in our HFB calculations. An important feature of the pairing self-consistency equation is its reduction to a Bethe– Salpeter equation in the vacuum. We have shown in Ref. [23] that the solution to this Bethe–Salpeter equation dominates the behavior of the pairing gap function at low densities. Since this solution is observable — the two-nucleon system possesses a virtual ∆g =

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Table 1 Value of constant gpr used in the pairing self-consistency equation for each value of the quark mass mq mq (MeV) gpr

100 0.664

200 0.994

1000 1.027

5000 1.012

state at kv ≈ −0.05i fm–1 in the 1 S0 channel — we should constrain our calculations so as to be consistent with it. In Ref. [23], we found that, with such a constraint, many different sets of Dirac–Hartree–Fock coupling constants and meson masses (all of which were adjusted to reproduce the nuclear saturation point) yield very similar pairing gap functions. In that work, the constraint was imposed by varying the cutoff Λ in the momentum integrals. Such a procedure has the advantage of not modifying the coupling constants. However, it does modify the effective ranges of the exchanged mesons. Here, we wish to use a simpler procedure. We assume that the effective nucleon–nucleon interaction that produces the pairing is the same, up to a constant factor, as that which produces the mean field. We thus fix the momentum cutoff, at Λ = 1.5 GeV, in the pairingfield equation and take its coupling constants to be the scalar and vector coupling constants, gvN and gsN , multiplied by an overall factor gpr . The value of gpr necessary to fix the virtual state is different for each value of the quark mass mq . The values that we use are given in Table 1. Given our earlier remarks about the difference in effective interactions in the selfenergy and pairing equations, our use of the same density-dependence of the scalar coupling coefficient gsN in the two equations could be questioned. It has been shown that a Brückner description of the nuclear matter self-energy can be reproduced within the Hartree approximation through the use of density-dependent coupling coefficients [24]. However, the density dependence of the scalar coupling coefficient found here is not a Brückner correction but a medium correction due to the composite nature of the nucleon in the model. If we were to attempt a Brückner calculation within the model, we might expect an additional density dependence of the coupling constants relevant to the calculation of the self-energy. But, as found in Ref. [24], such an additional density dependence would be very small at the low densities for which the pairing field is nonzero. We thus feel justified in using the same density-dependent coupling coefficient, up to an overall normalization, in both the self-energy and the pairing equations. We display the two principal components of the Dirac pairing field, ∆s and ∆0 , as a function of the Fermi momentum for several values of the quark mass mq in Fig. 14. We refer to Refs. [22,23] for details of the calculations. The tensor component ∆T is two orders of magnitude smaller than the others and is not shown. The magnitude of the components of the pairing field vary greatly for the values of the quark mass less than 1 GeV. This variation saturates for large quark masses, as the pairing-field approach the nuclear σ –ω values. The Fermi-momentum dependence of the pairing-field components is about the same for values of the quark mass of 200 MeV or more. The peaks of the components shift in momentum for smaller values of the quark mass, with the component ∆s becoming extremely small

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Fig. 14. The strength of the scalar and vector pairing fields, ∆s and ∆0 , as a function of the Fermi momentum for several values of the quark mass mq . The vector field ∆0 is always the larger of the two.

and shifting to higher values of the Fermi momentum while the component ∆0 shifts to lower values. Note that we display the pairing fields at a smallest value of mq = 100 MeV rather than the value of 40 MeV that we used in the mean-field calculations. At values of the quark mass below about 100 MeV, we encountered convergence problems in our calculations, related to the fact that the component ∆s of the Dirac pairing field vanishes for smaller values of the quark mass. We display in Fig. 15 the pairing gap function as a function of the Fermi momentum for several values of the quark mass. The variation in the gap function at low densities is fixed

Fig. 15. The nonrelativistic pairing gap as a function of the Fermi momentum for several values of the quark mass mq .

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by the virtual state and is the same for all values of the quark mass. Surprisingly, we see very little difference at high densities between the curves corresponding to quark masses of 200 MeV or more, despite the large variations in the component fields ∆s and ∆0 . Due to the faster decrease with density of the effective scalar coupling constant, gsN , for smaller values of mq , as seen in Fig. 4, we might expect to see variations similar to those seen in the previous figure. However, the gap function depends on both the effective coupling constant gsN and the effective mass M  (explicitly and through the density of states in the integrals which determine ∆s and ∆0 ). The variations with the quark mass of gsN and M  tend to compensate one another to yield a pairing gap function which is almost invariant for quark masses above about 200 MeV. As the quark mass decreases below 200 MeV, the variation in the coupling constant begins to dominate and the peak of the gap function tends to lower values of the Fermi momentum and diminishes in magnitude. Finally we wish to examine the correlation length of a nucleon pair. To do this, we calculate the pair wavefunction, which is the frequency sum over the anomalous propagator:  ω)eiω0+ . = 1 dω F (k, χ(k) 2πi Using this wavefunction, we can define the pair correlation length as the root-mean-square radius of the pair wave function [25]:   d3 k χ † χ, rc2 = d3 k ∇k χ † · ∇k χ which we display in Fig. 16 as a function of the Fermi momentum for several values of the quark mass. The calculations in all cases are very similar. The correlation length is slightly smaller for a quark mass of 200 MeV, suggesting that the pair is slightly more bound at this value of the quark mass. The correlation length for a quark mass of 100 MeV rises more

Fig. 16. The pair correlation length as a function of the Fermi momentum for several values of the quark mass mq .

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quickly at higher densities, reflecting the decrease in the corresponding scalar coupling constant gsN at these densities.

6. Conclusions We have analyzed nuclear mean-field properties and nuclear pairing in a quark–meson coupling model of nuclear matter. We have found the mean-field properties — the nucleon effective mass and radius, the effective nucleon–scalar-meson coupling constant, and the equivalent nonrelativistic potentials — to depend strongly on the quark mass used in the calculations. For extremely large values of the quark mass, the mean-field properties approach those of the nucleon–meson coupling model. The best general agreement with experimental data is obtained with a quark mass in the range from 200 to 400 MeV. However, discrepancies between the calculations and observables remain. We find a strong correlation between the values of the compressibility and the spin–orbit interaction strength. As we have shown, this is also the case in the bag quark-coupling meson model, with and without exchange contributions. Thus, within the hypothesis of linear coupling of the meson fields with the quark fields, we find it impossible to adjust the compressibility and the spin– orbit interaction strength simultaneously. Nucleon–meson models that include higher-order scalar-meson self-interactions are capable of describing these two observables and suggest a manner in which the quark–meson coupling models might be corrected. We have found that, like the mean-field quantities, the Dirac components of the pairing field depend strongly on the quark mass. However, the nonrelativistic gap function and the pair correlation radius are quite insensitive to variations in the quark mass. This is due in part to the strong constraint imposed by requiring that the pairing field reproduce the twonucleon virtual state in the vacuum, but, at higher values of the density, it is also a result of compensation between variations in the effective nucleon–scalar-meson coupling constant gsN and the effective nucleon mass M  . In the future, we intend to include in the model the nonlinear scalar-meson interaction terms that should permit the simultaneous description of most mean-field quantities. We also plan to study the effects of variations in the meson masses, such as those due to the partial restoration of chiral symmetry. [14]

Acknowledgements E.F.B. acknowledges the support of FAPESP. B.V.C. and T.F. acknowledge partial support from FAPESP and the CNPq.

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