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Effect of isospin asymmetry in a nuclear system

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2014 J. Phys. G: Nucl. Part. Phys. 41 055201

(http://iopscience.iop.org/0954-3899/41/5/055201)

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Journal of Physics G: Nuclear and Particle Physics

J. Phys. G: Nucl. Part. Phys. 41 (2014) 055201 (19pp) doi:10.1088/0954-3899/41/5/055201

Effect of isospin asymmetry in a nuclearsystem

Shailesh K Singh, S K Biswal, M Bhuyan and S K Patra

Institute of Physics, Bhubaneswar-05, India

E-mail: shailesh@iopb.res.in

Received 19 December 2013, revised 12 February 2014Accepted for publication 25 February 2014Published 19 March 2014

AbstractThe effect of δ- and ω–ρ-meson cross couplings on asymmetry nuclear systemsis analyzed in the framework of an effective field theory motivated relativisticmean field formalism. The calculations are done on top of the G2 parameterset, where these contributions are absent. We calculate the root mean squareradius, binding energy, single particle energy (for the first and last occupiedorbits), density and spin–orbit interaction potential for some selected nucleiand evaluate the Lsym- and Esym-coefficients for nuclear matter as a functionof δ- and ω–ρ-meson coupling strengths. As expected, the influence of theseeffects is negligible for the symmetry nuclear system and these effects are veryimportant for systems with large isospin asymmetry.

Keywords: relativistic mean field theory, symmetry energy, delta meson,neutron star

(Some figures may appear in colour only in the online journal)

1. Introduction

In recent years, the effective field theory approach to quantum hadrodynamic (QHD) hasbeen studied extensively. The parameter set G2 [1, 2], obtained from the effective field theorymotivated Lagrangian (E-RMF) approach, is quite successful in reproducing the nuclear matterproperties including the structure of neutron star as well as of finite nuclei [3]. This modelwell reproduce the experimental values of binding energy (BE), root mean square (rms) radiiand other finite nuclear properties [4–6]. Similarly, the prediction of nuclear matter propertiesincluding the phase transition as well as the properties of compact star is remarkably good[7, 8]. The G2 force parameter is the largest force set available, in the relativistic mean field(RMF) model. It contains almost all interaction terms of nucleon with mesons, self and crosscoupling of mesons up to fourth order.

0954-3899/14/055201+19$33.00 © 2014 IOP Publishing Ltd Printed in the UK 1

J. Phys. G: Nucl. Part. Phys. 41 (2014) 055201 S K Singh et al

In the E-RMF model of Furnstahl et al [1, 2], the coupling of δ-meson is not takeninto account. Also, the effect of ρ- and ω-meson cross coupling was neglected. It is soonrealized that the importance of δ meson [9] and the cross coupling of ω and ρ-mesons [10]cannot be neglected while studying the nuclear and neutron matter properties. Horowitz andPiekarewicz [11] studied explicitly the importance of ρ and ω cross coupling to finite nucleias well as to the properties of neutron star structures. This coupling also influences the nuclearmatter properties, like symmetry energy Esym, slope parameters Lsym and curvature Ksym ofEsym [12].

The observation of Brown [13] and later on by Horowitz and Piekarewicz [11] make itclear that the neutron radius of heavy nuclei have a direct correlation with the equation ofstate (EOS) of compact star matter. It is shown that the collection of differences of neutronand proton radii �r = rn − rp using relativistic and non-relativistic formalisms shows twodifferent patterns. Unfortunately, the error bar in neutron radius makes no difference betweenthese two patterns. Therefore, the experimental result of JLAB [14] is much awaited. To havea better argument for all this, Horowitz and Piekarewicz [11] introduced �s and �v couplingsto take care of the skin thickness in 208Pb as well as the crust of neutron star. The symmetryenergy, and hence the neutron radius, plays an important role in the construction of asymmetricnuclear EOS. Although the new couplings �s and �v take care of the neutron radius problem,the effective mass splitting between neutron and proton is not taken care of. This effect cannotbe neglected in a highly neutron-rich dense matter system and drip-line nuclei. In addition tothis mass splitting, the rms charge radius anomaly of 40Ca and 48Ca may be resolved by thisscalar–isovector δ-meson inclusion to the E-RMF model. Our aim in this paper is to see theeffect of δ- and ρ–ω-mesons couplings in a highly asymmetric system, like asymmetry finitenuclei, neutron star and asymmetric EOS.

The paper is organized as follows: first of all, before going to a brief formalism, we outlinethe mesons and their properties in subsection 1.1. Here, we discuss the nature and their quarkconstituents along with their rest masses and spin parities. In section 2, we extend the E-RMFLagrangian by including the δ-meson and the ω–ρ cross couplings. The field equations arederived from the extended Lagrangian for finite nuclei. Then, the EOS for nuclear matter andneutron star matters is derived. The calculated results are discussed in section 3. In this section,we study the effect of δ-meson on asymmetric nuclear matter, including the neutron star. Then,we adopt the calculations for finite nuclei and see the changes in BE, radius etc. In the lastsection, the conclusions are drawn.

1.1. Mesons and their properties

In this section, we discuss about the participating mesons in the NN-interaction. There arevarious mesons, which are associated with the long and short range of NN-interaction. For thecomputational and analytical point of view, one has to choose the meson–nucleon interaction,which is relevant for the nuclear system. In this context, the range Ri and the mass of themeson propagator Pi may be an important criterion for the selection. The heavier the mass ofa meson the less important it is in the interaction as Ri ∝ 1

miand Pi ∝ 1

m2i, with mi the mass

of the meson. In addition to this criterion, the coupling strength is also a deciding factor, i.e.,the moderate or smaller the coupling constant, the lesser is the importance of the meson [15].In this way, one can select the possible kind of mesons to make an optimistic meson–nucleonmodel. Also, the strange mesons are suppressed by Zweig forbidden [16] in the NN-interaction.The contribution of the meson within mass 1200–1300 MeV or more has nominal influencein the interaction. Thus, it is a reasonable criterion to have a cut off for the selection of mesonat the mass of the nucleon or slightly more to include all type of effects in the meson–nucleon

2

J. Phys. G: Nucl. Part. Phys. 41 (2014) 055201 S K Singh et al

many-body problems. The possible bosons may be π−, σ−, ω−, ρ−, δ and photon fields areimportant while considering the nucleon–meson theory.

In the RMF approximation, the pseudoscalar π -meson does not contribute to nuclear bulkproperties for finite nuclei, because of the definite spin and parity of the ground state nucleus[17–20]. The quark composition of this meson triplet is (π+ : ud), (π0 : uu−dd√

2) and (π− : du).

The masses are 139.57 and 134.9766 MeV for π± and π0, respectively and spin parity of thepion is 0−1. Therefore, in mean field level, the far long range of the nuclear potential, whichis generated by pion is adjusted by the parameters. The field corresponding to the isoscalar–scalar σ -meson, which is a broad two-pion resonance state (s wave) provides strong scalarattraction at an intermediate distance (>0.4 fm) has a mass of 400–550 MeV [21, 22], which isthe most dominating attractive part of the nuclear interaction. The quark structure of σ -mesonis uu+dd√

2and its spin parity (JP) is 0+. The nonlinearity of the σ -meson coupling included

the 3-body interaction [23, 24], which is currently noticed as an important ingredient fornuclear saturation. The isoscalar–vector ω-meson, which is a 3π -resonance state with a massof 781.94 ± 0.12 MeV, gives the strong vector repulsion at a short distance to make sure thehard core repulsion of the nuclear force. The self-coupling of the ω-meson is crucial to makethe nuclear EOS softer [3, 25–27], which has a larger consequence in determining the structureof neutron star. The quark structure of ω-meson is uu+dd√

2and JP = 1−. The isovector–vector

ρ-meson field, which is arisen due the asymmetry of the neutron–proton number density in thesystem, is a resonance of 2π -meson in p-states, contributes to the high repulsive core near thecenter and attractive behavior near the intermediate range in the even-singlet central potentialof NN-interaction [28, 29]. The ρ-meson have mass and is around 768.5 ± 0.6 MeV [22]. Thequark structure of the neutral ρ-meson is uu−dd√

2and JP = 1−. The isovector–scalar δ-meson is

the resonance state of ηπ (dominant channel) and KK (minor channel) [29, 30], which has amass of 980 ± 20 MeV [22]. Its quark structure is given as uu−dd√

2and JP = 1−. This meson

originates when there is an asymmetry in number of proton and neutron. Thus, it has a largerconsequence in highly asymmetry systems like neutron star and heavy ion collision (HIC).

It is to be noted that the bulk properties like BE and charge radius do not isolatethe contribution from the isoscalar or isovector channels. It needs an overall fitting of theparameters. That is the reason, the modern Lagrangian ignores the contribution of δ- andρ-mesons independently, i.e. once ρ-meson is included, it takes care of the properties of thenuclear system and does not need the requirement of the δ-meson [31–34]. However, theimportance of the δ-meson arises when we study the properties of a highly asymmetry systemsuch as drip-line nuclei and neutron star [9, 35–46]. In particular, at high density such asneutron star and HICs, the proton fraction of β-stable matter can increase and the splitting ofthe effective mass can affect the transfer properties. Also, at high isospin asymmetry, becausethe increase of proton fraction influences the cooling of neutron star [47–49].

2. Formalism

The relativistic treatment of the QHD models automatically includes the spin–orbit force, thefinite range and the density dependence of the nuclear interaction. The RMF or the E-RMFmodel has the advantage that, with the proper relativistic kinematics and with the mesonproperties already known or fixed from the properties of a small number of finite nuclei, it givesexcellent results for binding energies, rms radii, quadrupole and hexadecapole deformationsand other properties of spherical and deformed nuclei [50–54]. The quality of the resultsis comparable to that found in non-relativistic nuclear structure calculations with effectiveSkyrme [55] or Gogny [56] forces. The theory and the equations for finite nuclei and nuclear

3

J. Phys. G: Nucl. Part. Phys. 41 (2014) 055201 S K Singh et al

matter can be found in [1, 2, 57, 58] and we shall only outline the formalism here. We startfrom [1] where the field equations were derived from an energy density functional containingDirac baryons and classical scalar and vector mesons. Although this energy functional can beobtained from the effective Lagrangian in the Hartree approximation [2, 58], it can also beconsidered as an expansion in terms of ratios of the meson fields and their gradients to thenucleon mass. The energy density functional for finite nuclei can be written as [2, 57, 58]:

E (r) =∑

α

ϕ†α(r)

{− iα·∇ + β[M − �(r) − τ3D(r)] + W (r) + 1

2τ3R(r) + 1 + τ3

2A(r)

− iβα

2M·(

fv∇W (r) + 1

2fρτ3∇R(r)

) }ϕα(r) +

(1

2+ κ3

3!

�(r)

M+ κ4

4!

�2(r)

M2

)

×m2s

g2s

�2(r) − ζ0

4!

1

g2v

W 4(r) + 1

2g2s

(1 + α1

�(r)

M

)(∇�(r))2

− 1

2g2v

(1 + α2

�(r)

M

)(∇W (r))2 − 1

2

(1 + η1

�(r)

M+ η2

2

�2(r)

M2

)m2

v

g2v

W 2(r)

− 1

2e2(∇A(r))2 − 1

2g2ρ

(∇R(r))2 − 1

2

(1 + ηρ

�(r)

M

)m2

ρ

g2ρ

R2(r)

−�v(R2(r) × W 2(r)) + 1

2g2δ

(∇D(r))2 − 1

2

mδ2

g2δ

(D2(r)), (1)

where �, W , R, D and A are the fields for σ, ω, ρ, δ and photon and gσ , gω, gρ , gδ and e2

4πare

their coupling constant, respectively. The masses of the mesons are mσ , mω, mρ and mδ for�0, V0, b0 and δ0, respectively. In the energy functional, the nonlinearity as well as the crosscoupling up to a maximum of fourth order is taken into account. This is restricted due to thecondition 1 > field

M (M = nucleon mass) and a non-significant contribution of the higher order[4]. We fit the coupling constants using the naive dimensional analysis without destroying thenaturalness. The higher nonlinear coupling of ρ- and δ-meson fields is not taken in the energyfunctional, because the expectation values of the ρ- and δ-fields are an order of magnitude lessthan that of ω-field and they have only marginal contribution to finite nuclei. For example,in calculations of the high-density EOS, Muller and Serot [57] found the effects of a quarticρ-meson coupling (R4) to be appreciable only in stars made of pure neutron matter. A surfacecontribution −α3� (∇R)2/(2g2

ρM) was tested in [59] and it was found to have absolutelynegligible effects. We should note, nevertheless, that very recently it has been shown thatcouplings of the type �2R2 and W 2R2 are useful to modify the neutron radius in heavy nucleiwhile making very small changes to the proton radius and the BE [11].

The Dirac equation corresponding to the energy density equation (1) becomes{− iα·∇ + β[M − �(r) − τ3D(r)] + W (r) + 1

2τ3R(r) + 1 + τ3

2A(r)

− iβα

2M·[

fv∇W (r) + 1

2fρτ3∇R(r)

] }ϕα(r) = εα ϕα(r). (2)

The mean field equations for �, W , R, D and A are given by

− ��(r) + m2s �(r) = g2

sρs(r) − m2s

M�2(r)

(κ3

2+ κ4

3!

�(r)

M

)

+ g2s

2M

(η1 + η2

�(r)

M

)m2

v

g2v

W 2(r) + ηρ

2M

g2s

gρ2

m2ρR2(r)

+ α1

2M[(∇�(r))2 + 2�(r)��(r)] + α2

2M

g2s

g2v

(∇W (r))2, (3)

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J. Phys. G: Nucl. Part. Phys. 41 (2014) 055201 S K Singh et al

− �W (r) + m2vW (r) = g2

v

(ρ(r) + fv

2ρT(r)

)−

(η1 + η2

2

�(r)

M

)�(r)

Mm2

vW (r)

− 1

3!ζ0W

3(r) + α2

M[∇�(r) · ∇W (r) + �(r)�W (r)]

−2 �vgv2R2(r)W (r), (4)

− �R(r) + m2ρR(r) = 1

2g2

ρ

(ρ3(r) + 1

2fρρT,3(r)

)− ηρ

�(r)

Mm2

ρR(r) − 2�vgρ2R(r)W 2(r),

(5)

− �A(r) = e2ρp(r), (6)

− �D(r) + mδ2D(r) = g2

δρs3, (7)

where the baryon, scalar, isovector, proton and tensor densities are

ρ(r) =∑

α

ϕ†α(r)ϕα(r), (8)

ρs(r) =∑

α

ϕ†α(r)βϕα(r), (9)

ρ3(r) =∑

α

ϕ†α(r)τ3ϕα(r), (10)

ρp(r) =∑

α

ϕ†α(r)

(1 + τ3

2

)ϕα(r), (11)

ρT(r) =∑

α

i

M∇ ·[ϕ†

α(r)βαϕα(r)], (12)

ρT,3(r) =∑

α

i

M∇ · [ϕ†

α(r)βατ3ϕα(r)], (13)

ρs3(r) =∑

α

ϕ†α(r)τ3βϕα(r), (14)

where ρs3 = ρsp − ρsn, ρsp and ρsn are scalar densities for proton and neutron, respectively.The scalar density ρs is expressed as the sum of proton (p) and neutron (n) scalar densitiesρs = 〈ψψ〉 = ρsp+ρsn, which are given by

ρsi = 2

(2π)3

∫ ki

0d3k

M∗i(

k2 + M∗2i

) 12

, i = p, n (15)

ki is the nucleon’s Fermi momentum and M∗p, M∗

n are the proton and neutron effective masses,respectively and can be written as

M∗p = M − gsφ0 − gδδ, (16)

M∗n = M − gsφ0 + gδδ. (17)

Thus, the δ field splits the nucleon effective masses. The baryon density is given by

ρB = 〈ψγ 0ψ〉 = γ

∫ kF

0

d3k

(2π)3, (18)

5

J. Phys. G: Nucl. Part. Phys. 41 (2014) 055201 S K Singh et al

where γ is spin or isospin multiplicity (γ = 4 for symmetric nuclear matter (SNM) and γ = 2for pure neutron matter). The proton and neutron Fermi momentum will also split, while theyhave to fulfil the following condition:

ρB = ρp + ρn

= 2

(2π)3

∫ kp

0d3k + 2

(2π)3

∫ kn

0d3k. (19)

Because of the uniformity of the nuclear system for infinite nuclear matter, all of the gradientsof the fields in equations (1)–(7) vanish and only the κ3, κ4, η1, η2 and ζ0 nonlinear couplingsremain. Due to the fact that the solution of SNM in the mean field depends on the ratios g2

s/m2s

and g2v/m2

v [20], we have seven unknown parameters. By imposing the values of the saturationdensity, total energy, incompressibility modulus and effective mass, we still have three freeparameters (the value of g2

ρ/m2ρ is fixed from the bulk symmetry energy coefficient J). The

energy density and the pressure of nuclear matter are given by

ε = 2

(2π)3

∫d3kE∗

i (k) + ρ(r)W (r) + m2s �

2

g2s

(1

2+ κ3

3!

�(r)

M+ κ4

4!

�2(r)

M2

)

−1

2m2

v

W 2(r)

g2v

(1 + η1

�

M+ η2

2

�2

M2

)− 1

4!

ζ0W 4(r)

g2v

+ 1

2ρ3(r)R(r)

−1

2

(1 + ηρ�(r)

M

)m2

ρ

g2ρ

R2(r) − �vR2(r) × W 2(r)

+1

2

m2δ

g2δ

(D2(r)), (20)

P = 2

3(2π)3

∫d3k

k2

E∗i (k)

− m2s �

2

g2s

(1

2+ κ3

3!

�(r)

M+ κ4

4!

�2(r)

M2

)

+1

2m2

v

W 2(r)

g2v

(1 + η1

�

M+ η2

2

�2

M2

)+ 1

4!

ζ0W 4(r)

g2v

+1

2

(1 + ηρ�(r)

M

)m2

ρ

g2ρ

R2(r) + �vR2(r) × W 2(r)

−1

2

m2δ

g2δ

(D2(r)), (21)

where E∗i (k) =

√k2 + M∗

i2 (i = p, n). In the context of density functional theory, it is

possible to parametrize the exchange and correlation effects through local potentials (Kohn–Sham potentials), as long as those contributions be small enough [60]. The Hartree values arethe ones that control the dynamics in the relativistic Dirac–Bruckner–Hartree–Fock (DBHF)calculations. Therefore, the local meson fields in the RMF formalism can be interpreted asKohn–Sham potentials and in this sense equations (3)–(7) include effects beyond the Hartreeapproach through the nonlinear couplings [1, 2, 58].

3. Results and discussions

Our calculated results are shown in figures 1–10 and table 1 for both finite nuclei and infinitenuclear matter systems. The effect of δ-meson and the crossed coupling constant �v of ω − ρ

fields on some selected nuclei like 48Ca and 208Pb is demonstrated in figures 1–4 and thenuclear matter outcomes are displayed in the rest of the figures and table. In one of our recent

6

J. Phys. G: Nucl. Part. Phys. 41 (2014) 055201 S K Singh et al

-460

-440

-420B

E (

MeV

)

3.4

3.5

3.6rmsrn

rp

rch

0 0.5 1gδ

0

15

30

45

0 0.05 0.1Λ

v

1s(n)1s(p)1f(n)2s(p)

Rad

ius

(fm

)

48Ca

(a)

(b)

(c)

(d)

(e)

(f)

ε n,p (

MeV

)

Figure 1. Binding energy (BE), rms radius and first (1sn,p) and last (1 f n, 2sp) occupiedorbits for 48Ca as a function of gδ and �v .

-2000

-1900

-1800

-1700

-1600

5.4

5.5

5.6

5.7 rrms

rn

rp

rch

0 0.5 10

20

40

601s(n)3p(n)1s(p)3s(p)

0 0.05 0.1 0.15gδ Λ

v

ε n,p (

MeV

)R

adiu

s (f

m)

BE

(M

eV)

208Pb Energy Variation

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2. Same as figure 1 for 208Pb.

publication [12], the explicit dependence of �v(ω − ρ) on nuclear matter properties is shownand it is found that it has significant implication on various physical properties, like mass andradius of neutron star and Esym asymmetry energy and its slope parameter Lsym for an infinite

7

J. Phys. G: Nucl. Part. Phys. 41 (2014) 055201 S K Singh et al

0

0.05

0.1

0.15

0.2

0.01.3

0.000.16

2 4 6-420

-280

-140

2 4 6

48Ca

r(fm)

E(s

o) (

MeV

)ρ

(fm

-3)

Total Density

Neutron Density

Proton Density

Total Density

Neutron Density

Proton Density

gδ Λv

(a)

(b)

(c)

(d)

Figure 3. The neutron, proton and total density with radial coordinate r( f m) at differentvalues of gδ (a) and �v (c). The variations of spin–orbit potential for proton and neutronare shown in (b) and (d) by keeping the same gδ and �v as (a) and (c), respectively.

Table 1 The symmetry energy Esym (MeV), slope co-efficient Lsym (MeV) and Ksym

(MeV) at different values of gδ .

4πgδ Esym Lsym Ksym

0.0 45.09 120.60 −29.281.0 44.58 119.37 −27.612.0 43.07 115.67 −22.873.0 40.55 109.41 −15.874.0 37.00 100.44 −7.725.0 32.40 88.61 0.436.0 26.76 73.77 7.617.0 20.04 55.78 13.27

nuclear matter system at high densities. Here, only the influence of �v on finite nuclei andthat of gδ on both finite and infinite nuclear systems is studied.

3.1. Finite nuclei

In this section, we analyzed the effects of δ-meson and �v coupling in finite nuclei. For this,we calculate the BE, rms radii (rn, rp, rch, rrms) and energy of first and last filled orbitals of48Ca and 208Pb with gδ and �v . The finite size of the nucleon is taken into account for thecharge radius using the relation rch =

√r2

p + 0.64. The results are shown in figures 1 and 2.

8

J. Phys. G: Nucl. Part. Phys. 41 (2014) 055201 S K Singh et al

0

0.05

0.1

0.15

0.01.3

0.000.17

2 4 6 8 10

-50

0

50

100

2 4 6 8 10r(fm)

E(s

o) (

MeV

)ρ

(fm

-3)

208Pb

gδΛ

v

Total Density

Neutron Density

Proton Density

Total Density

Neutron Density

Proton Density

(a)

(b)

(c)

(d)

Figure 4. Same as figure 3 for 208Pb.

In our calculations, while analyzing the effect of gδ , we keep �v = 0 and vice versa. Fromthe figures, it is evident that the BE, radii and single particle levels εn,p affected drasticallywith gδ contrary to the effect of �v . A careful inspection shows a slight decrease of rn withthe increase of �v consistent with the analysis of [61]. Again, it is found that the BE increaseswith the increase of the coupling strength up to gδ ∼ 1.5 and no convergence solution isavailable beyond this value. Similar to the gδ limit, there is a limit for �v also, beyond whichno solution exists. From the anatomy of gδ on rn and rp, we find their opposite trend in size.That means, the value of rn decreases and rp increases with gδ for both 48Ca and 208Pb. It sohappens that both the radii meet at a point near gδ = 1.0 (figures 1 and 2) and again showreverse character on increasing gδ , i.e., the neutron skin thickness (rn − rp) changes its signwith gδ . These interesting results may help us to settle the charge radius anomaly of 40Ca and48Ca.

In figure 1(c), we have shown the first (1sn,p) and last (1 f n and 2sp) filled orbitals for 48Caas a function of gδ and �v . The effect of �v is marginal, i.e., almost negligible on εn,p orbitals.However, this is significance with the increasing value of gδ . The top most filled orbital evencrosses each other at gδ ∼ 1, although initially, it is well separated. On the other hand, thefirst filled orbital 1s both for proton and neutron gets separated more and more with gδ , whichhas almost same single particle energy εn,p at gδ = 0. We get similar trend for 208Pb, which isshown in figure 2(c). In both the representative cases, we notice orbital flipping only for thelast filled levels.

The nucleon density distribution (proton ρp and neutron ρn) and spin–orbit interactionpotential Eso of finite nuclei are shown in figures 3 and 4. The calculations are done with twodifferent values of gδ and �v as shown in the figures. Here, the solid line is drawn for initialand dotted one is for the limiting values. In figure 3(a), we have depicted the neutron, protonand total density distribution for 48Ca at values of gδ = 0.0 and 1.3. Comparing figures 3(a)

9

J. Phys. G: Nucl. Part. Phys. 41 (2014) 055201 S K Singh et al

and 3(c), one can see that the sensitivity of gδ is more than �v on density distribution. Thespin–orbit potential Eso of 48Ca with different values of gδ is shown in figure 3(b) and for �v

in figure 3(d). Similarly, we have given these observables for 208Pb in figure 4. In general, fora light mass region both coupling constants gδ and �v are less effective in density distributionand spin–orbit potential. It is clear from this analysis that the coupling strength of δ-meson ismore influential than the isoscalar–vector and isovector–vector cross coupling. This effect ismostly confined to the central region of the nucleus.

3.2. Nuclear matter

In this section, we perform calculation for nuclear matter properties like energy and pressuredensities, symmetry energy, radii and mass of the neutron star using ω–ρ and δ couplings ontop of G2 parametrization. Recently, it has been reported [12] that the ω–ρ cross couplingplays a vital role for the nuclear matter system on important physical observables like EOS,symmetry energy coefficient, Lsym coefficient, etc. A detailed account is available in [12] forω–ρ coupling on the nuclear matter system. The main aim of this section is to take δ-mesonas an additional degree of freedom in our calculations and elaborate the effect on the nuclearmatter system within G2 parameter set. In a highly asymmetric system like neutron star andsupernova explosion, the contribution of δ-meson is important. This is because of the highasymmetry due to the isospin as well as the difference in neutron and proton masses. Here inthe calculations, the β-equilibrium and charge neutrality conditions are not considered. Weonly varies the neutron and proton components with an asymmetry parameter α, defined asα = ρn−ρp

ρn+ρp. The splitting in nucleon masses is evident from equations (16) and (17) due to

the inclusion of isovector scalar δ-meson. For α = 0.0, the nuclear matter system is purelysymmetrical and for other non-zero value of α, the system gets more and more asymmetry.For α = 1.0, it is a case of pure neutron matter.

In figure 5(a), the effective masses of proton and neutron are given as a function of gδ . Aswe have mentioned, δ-meson is responsible for the splitting of effective masses (equations (16)and (17)), this splitting increases continuously with the coupling strength gδ . In figure 5, thesplitting is shown for few representative cases at α = 0.0, 0.75 and 1.0. The solid line is forα = 0.0 and α = 0.75, 1.0 are shown by dotted and dashed line, respectively. From the figure,it is clear that the effective mass is unaffected for symmetric matter. The proton effective massM∗

p is above the reference line with α = 0 and the neutron effective mass always lies below it.The effect of gδ on BE per nucleon is shown in figure 5(b) and pressure density in figure 5(c).One can easily see the effect of δ-meson interaction on the energy and pressure density of thenuclear system. The energy and pressure density show an opposite trend to each other withthe increased function of gδ .

3.3. Energy and pressure density

We analyze the BE per nucleon and pressure density including the contribution of δ-meson inthe G2 Lagrangian as a function of density. As it is mentioned earlier, the addition of δ-mesonis done due to its importance on asymmetry nuclear matter as well as to make a full-fledgedE-RMF model. This is tested by calculating the observables at different values of δ-mesoncoupling strength gδ . In figure 6, the calculated BE/A and P for pure neutron matter withbaryonic density for different gδ are shown. Unlike the small value of gδ up to 1.5 in finitenuclei, the instability arises at gδ = 7.0 in nuclear matter. Of course, this limiting value of gδ

depends on the asymmetry of the system.

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J. Phys. G: Nucl. Part. Phys. 41 (2014) 055201 S K Singh et al

Figure 5. Variation of nucleonic effective masses, binding energy per particle (BE/A)and pressure density as a function of gδ on top of G2 parameter set for nuclear matter.

Figure 6. Energy per particle and pressure density with respect to density with variousgδ .

In figure 6(a), we have given BE/A for different values of gδ . It is seen from figure 6(a), thebinding increases with gδ in the lower density region and maximum value of BE is ∼ 7 MeVfor gδ = 7.0. On the other hand, in higher density region, the BE curve for finite gδ crossesthe one with gδ = 0.0. That means, the EOS with δ-meson is stiffer than the one with pureG2 parametrization. As a result, one gets a heavier mass of the neutrons star, which suitedwith the present experimental finding [62]. For comparing the data at lower density (dilute

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J. Phys. G: Nucl. Part. Phys. 41 (2014) 055201 S K Singh et al

system, 0 < ρ/ρ0 < 0.16), the zoomed version of the region is shown as an inset figure(c) inside figure 6(a). From the zoomed inset portion, it is clearly seen that the curves withvarious gδ at α = 1.0 (pure neutron matter) deviate from other theoretical predictions, such asBaldo–Maieron [63], DBHF [64], Friedman [65], auxiliary-field diffusion Monte Carlo [66]and Skyrme interaction [67]. This is an inherited problem in the RMF or E-RMF formalisms,which needs more theoretical attention. Similarly, the pressure density for different valuesof gδ with G2 parameter set is given in figure 6(b). At high density, we can easily see thatthe curve becomes more stiffer with the coupling strength gδ . The experimental constraint ofEOS obtained from the heavy ion flow data for both stiff and soft EOS is also displayed forcomparison in the region 2 < ρ/ρ0 < 4.6 [68]. Our results match with the stiff EOS data of[68].

3.4. Symmetry energy

The symmetric energy Esym is important in infinite nuclear matter and finite nuclei, becauseof isospin dependence in the interaction. The isospin asymmetry arises due to the differencein densities and masses of the neutron and proton, respectively. The density type isospinasymmetry is taken care by ρ-meson (isovector–vector meson) and mass asymmetry byδ-meson (isovector–scalar meson). The expression of symmetry energy Esym is a combinedexpression of ρ- and δ-mesons, which is defined as [4, 9, 69, 70]

Esym(ρ) = Ekinsym(ρ) + Eρ

sym(ρ) + Eδsym(ρ), (22)

with

Ekinsym(ρ) = k2

F

6E∗F

; Eρsym(ρ) = g2

ρρ

8m∗2ρ

(23)

and

Eδsym(ρ) = −1

2ρ

g2δ

m2δ

(M∗

EF

)2

uδ (ρ, M∗). (24)

The last function uδ is from the discreteness of the Fermi momentum. This momentum is quitelarge in the nuclear matter system and can be treated as a continuum and continuous system.The function uδ is defined as

uδ (ρ, M∗) = 1

1 + 3 g2δ

m2δ

(ρs

M∗ − ρ

EF

). (25)

In the limit of continuum, the function uδ ≈ 1. The whole symmetry energy (Ekinsym + Epot

sym)arisen from ρ- and δ-mesons is given as

Esym(ρ) = k2F

6E∗F

+ g2ρρ

8m∗2ρ

− 1

2ρ

g2δ

m2δ

(M∗

EF

)2

uδ (ρ, M∗), (26)

where the effective energy E∗F =

√(k2

F + M∗2), kF is the Fermi momentum and the effectivemass M∗ = M − gsφ0 ± gδδ0. The effective mass of the ρ-meson modified, because of crosscoupling of ρ–ω and is given by

m∗2ρ =

(1 + ηρ

gσ σ

M

)m2

ρ + 2g2ρ (�vg2

vω20). (27)

The cross coupling of isoscalar–isovector mesons (�v) modified the density dependent ofEsym without affecting the saturation properties of the SNM. This is explained explicitly in[12] and does not need special attention here. In the E-RMF model with pure G2 set, the

12

J. Phys. G: Nucl. Part. Phys. 41 (2014) 055201 S K Singh et al

0 0.5 1 1.5 20

10

20

30

40

50

HIC data0.01.02.03.04.05.06.07.0SHF(GSkII)SHF(Skxs20)

ρ/ρ0

Esy

m(M

eV)

Neutron Matter (α=1.0)

4π δ=0

.0

4πg δ=7.0

G2+δ, Λ=0.0

}4πgδ

Figure 7. Symmetry energy Esym (MeV) of neutron matter with respect to the differentvalue of gδ on top of G2 parameter set. The heavy ion collision (HIC) experimentaldata [71] (shaded region) and non-relativistic Skyrme GSkII [72], and Skxs20 [73]predictions are also given. �v = 0.0 is taken.

SNM saturates at ρ0 = 0.153 fm−3, BE/A = 16.07 MeV, compressibility K0 = 215 MeV andsymmetry energy of Esym = 36.42 MeV [1, 2].

In the numerical calculation, the coefficient of symmetry energy Esym is obtained bythe energy difference of symmetry and pure neutron matter at saturation and it is definedby equation (26) for a quantitative description at various densities. Our results for Esym arecompared in figure 7 with the experimental HIC data [71] and other theoretical predictionsof the non-relativistic Skyrme–Hartree–Fock model. The calculation is done for pure neutronmatter with different values of gδ , which are compared with two selective force parametersets GSkII [72] and Skxs20 [73]. For more discussion one can see [67], where 240 differentSkyrme parametrization are used. Here in our calculation, as usual �v = 0 to see the effect ofδ-meson coupling on Esym. In this figure, shaded region represents the HIC data [71] within0.3 < ρ/ρ0 < 1.0 region and the symbols square and circle represent the SHF results forGSkII and Skxs20, respectively. Analyzing figure 7, Esym of G2 matches with the shaded regionin the low density region, however as the density increases, the value of Esym moves away.Again, the symmetry energy becomes softer by increasing the value of coupling strength gδ .For higher value of gδ , again the curve moves far from the empirical shaded area. In this way,we can fix the limiting constraint on the coupling strength of δ-meson and nucleon. Similarto the finite nuclear case, the nuclear matter system becomes unstable for excessive value ofgδ (>7.0). This constrained may help to improve the G2+gδ parameter set for both finite andinfinite nuclear systems.

The symmetry energy of a nuclear system is a function of baryonic density ρ, hence canbe expanded in a Taylor series around the saturation density ρ0 as (26):

Esym(ρ) = E0 + LsymY + 12 KsymY2 + O[Y3], (28)

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J. Phys. G: Nucl. Part. Phys. 41 (2014) 055201 S K Singh et al

0 1 2 3 4 5 6 7

0

50

100L

sym

Esym

Ksym

Esy

m, L

sym

, Ksy

m (

MeV

)

4π gδ

Figure 8. Symmetry energy Esym (MeV), slope coefficients Lsym (MeV) and Ksym (MeV)at different gδ with �v = 0.0.

where E0 = Esym(ρ = ρ0), Y = ρ−ρ0

3ρ0and the coefficients Lsym and Ksym are defined as:

Lsym = 3ρ

(∂Esym

∂ρ

)ρ=ρ0

, Ksym = 9ρ2

(∂2Esym

∂ρ2

)ρ=ρ0

. (29)

Here, Lsym is the slope parameter defined as the slope of Esym at saturation. The quantity Ksym

represents the curvature of Esym with respect to density. A large number of investigations havebeen made to fix the value of Esym, Lsym and Ksym [12, 67, 71, 74–77]. In figure 9, we havegiven the symmetry energy with its first derivative at saturation density with different valuesof coupling strength starting from gδ = 0.0–7.0. The variation of Esym, Lsym and Ksym with gδ

is listed in table 1. The variation in symmetry energy takes place from 45.09 to 20.04 MeV,Lsym from 120.60 to 55.78 MeV and Ksym from −29.28 to 13.27 MeV at saturation densitycorresponding to 0.0 < gδ < 7.0. From this investigation, one can see that G2 set is notsufficient to predict this constrained on Esym and Lsym. It is suggestive to introduce the δ-meson as an extra degree of freedom into the model to bring the data within the prediction ofexperimental and other theoretical constraints.

The above tabulated results are also depicted in figure 8 to get a graphical representationof Esym, Lsym and Ksym. The values of Esym are marginally effective with the δ-meson couplingstrength. However, in the same time Lsym and Ksym vary substantially as shown in the figure.The slope parameter Lsym decreases almost exponentially opposite to the similar exponentialincrease of Ksym. At large value of gδ all the three quantities almost emerge very closely to thesimilar region.

3.5. Neutron star

In this section, we study the effect of δ-meson on mass and radius of neutron star. Recently,an experimental observation has predicted the constraint on mass of neutron star and its radius[62]. This observation suggests that the theoretical models should predict the star mass andradius as M � (1.97 ± 0.04)M and 11 < R(km) < 15. Keeping this point in mind, wecalculate the mass and radius of neutron star and analyzed their variation with gδ .

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J. Phys. G: Nucl. Part. Phys. 41 (2014) 055201 S K Singh et al

Figure 9. Constraints on Esym with its first derivative, i.e., Lsym at saturation density forneutron matter. The experimental results of HIC [71], PDR [78, 79] and IAS [80] aregiven. The theoretical prediction of finite range droplet model (FRDM) and Skyrmeparametrization is also given [81], SHF [67].

In the interior part of neutron star, the neutron chemical potential exceeds the combinedmass of the proton and electron. Therefore, asymmetric matter with an admixture of electronsrather than pure neutron matter is a more probable composition. The concentrations of neutrons,protons and electrons can be determined from the condition of β-equilibrium. This equilibriumcan be given by the combination of two equations, known as the direct URCA process:

n → p + e + ν; p + e → n + ν, (30)

assuming that neutrinos are non-degenerate due to the charge neutrality condition. Here, n, p,e, ν have usual meaning as neutron, proton, electron and neutrino. The direct URCA processis governed by the concentration of the proton, which has a critical value after which theprocess is possible. However, the modified URCA process takes place in cooling the neutronstar below the critical value of proton abundance, which is given as [47–49]

(n, p) + p + e → (n, p) + n + ν,

(n, p) + n → (n, p) + p + e + ν. (31)

The concentration of the proton is determined by the symmetry energy Esym of the system.Thus, constraining the symmetry energy is a crucial physical quantity for the URCA processand can be controlled by the isospin dependence of the density with the help of the isospinchannel.

The β-equilibrium condition for neutron star is given by μn = μp + μe, where μi

represents the chemical potential of the ith system. It is given as μn = E∗f n +gωV0 − 1

2 gρb0 and

μp = E∗f p + gωV0 + 1

2 gρb0 with E∗f n =

√(k2

f n + M∗2n) and E∗

f p =√

(k2f p + M∗2

p), where E∗f n,

E∗f p are the Fermi energy, and k f n and k f p are the Fermi momentum for neutron and proton,

respectively. Imposing this conditions, in the expressions of E and P (equations (20)–(21)),

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J. Phys. G: Nucl. Part. Phys. 41 (2014) 055201 S K Singh et al

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

0.01.02.03.04.05.06.0

9.5 10.0 10.5 11.0 11.5R (km)

10.0 20.0 30.0 40.0ρ

c(gm/cm

3)

M/M

O.

(a) (b)

G2 + gδ{4πgδ Λv=0.0

Figure 10. The mass and radius of neutron star at different values of gδ . (a) M/M withneutron star density (gm cm−3), (b) M/M with neutron star radius (km).

we evaluate E and P as a function of density. To calculate the star structure, we use theTolman–Oppenheimer–Volkoff (TOV) equations for the structure of a relativistic sphericaland static star composed of a perfect fluid derived from Einstein’s equations [82], where thepressure and energy densities obtained from equations (20) and (21) are the inputs. The TOVequation is given by [82]

dPdr

= −G

r

[E + P][M + 4πr3P]

(r − 2GM),

dM

dr= 4πr2E, (32)

with G as the gravitational constant and M(r) as the enclosed gravitational mass. We haveused c = 1. Given the P and E , these equations can be integrated from the origin as an initialvalue problem for a given choice of central energy density, (εc). The value of r(= R), wherethe pressure vanishes defines the surface of the star.

The results of mass and radius with various δ-meson coupling strength gδ are shown infigure 10. In the left panel, the neutron star mass with density (gm cm−3) is given, where wecan see the effect of the newly introduced extra degree of freedom δ-meson into the system. Onthe right side of the figure, (figure 10), M/M is depicted with respect to radius (km), whereM is the mass of the star and M is the solar mass. The gδ coupling changes the star massby ∼5.41% and radius by 5.39% with a variation of gδ from 0 to 6.0. From this observation,we can say that δ-meson is important not only for the asymmetry system normal density, butalso substantially effective in the high-density system. If we compare these results with theprevious results [12], i.e., with the effects of cross coupling of ω–ρ on mass and radius ofneutron star, the effects are opposite to each other. That means the star masses decrease with�v , whereas they increase with gδ . Thus a finer tuning in mass and radius of neutron star ispossible by a suitable adjustment on gδ value in the extended parametrization of G2+�v +gδ

to keep the star properties within the recent experimental observations [62].

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J. Phys. G: Nucl. Part. Phys. 41 (2014) 055201 S K Singh et al

4. Summary and conclusions

In summary, we rigorously discussed the effects of cross coupling of ω–ρ-mesons in finitenuclei on top of the pure G2 parameter set. The variation of binding energy (BE), rms radiiand energy levels of protons and neutrons are analyzed with increasing values of �v . Thechange in neutron distribution radius rn with �v is found to be substantially compared tothe less effectiveness of BE and proton distribution radius for the two representative nuclei48Ca and 208Pb. Thus, to fix the neutrons distribution radius depending on the outcome ofPREX experimental [14] result, the inclusion of �v coupling strength is crucial. As it isdiscussed widely by various authors [12], the role of ω–ρ-mesons in the nuclear matter systemis important on nuclear EOSs.

We emphasized strongly the importance of the effect of the extra degree of freedom, i.e.,δ-meson coupling into the standard RMF or E-RMF model, where generally it is ignored. Wehave seen the effect of this coupling strength of δ-meson with nucleon in finite and neutronmatter is substantial and very different in nature, which may be extremely helpful to fix variousexperimental constraints. For example, with the help of gδ , it is possible to modify the BE,charge radius and flipping of the orbits in asymmetry finite nuclei systems. The nuclear EOScan be made stiffer with the inclusion of δ-meson coupling. On the other hand, softening ofsymmetry energy is also possible with the help of this extra degree of freedom. In a compactsystem, it is possible to fix the limiting values of gδ and �v by testing the effect on availableconstraints on symmetry energy and its first derivative with respect to the matter density. Thiscoupling may be extremely useful to fix the mass and radius of neutron star keeping in viewthe recent observation [62].

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