Journal of the Korean Physical Society, Vol. 63, No. 2, July 2013, pp. 168∼173

Effects of the Symmetry Energy on Strongly-magnetized Neutron Stars in theDensity-dependent RMF Model

Chung-Yeol Ryu

General Education Curriculum Center, Hanyang University, Seoul 133-791, Korea

Myung-Ki Cheoun∗

Department of Physics, Soongsil University, Seoul 156-743, Korea

(Received 15 April 2013, in final form 21 June 2013)

We investigate the structure of neutron stars with strong magnetic fields by using a density-dependent relativistic mean field model. The coupling constant between nucleons and the rhomeson in the model is obtained by using the polytropic formula for the symmetry energy in theneutron star. With the density-dependent coupling constant, the effects of the symmetry energy onstrongly-magnetized neutron stars are considered in detail for particle fractions and the equationof state. The consequent changes in the masses and the radii of the neutron star are discussed byusing the roles of the symmetry energy in the neutron star.

PACS numbers: 13.75.Ev, 21.65.Ef, 26.60.KpKeywords: Hyperon-nucleon interactions, Symmetry energy, Equations of state of neutron-star matterDOI: 10.3938/jkps.63.168

I. INTRODUCTION

The density dependence of the symmetry energy in nu-clear matter is one of the main topics in nuclear physicsand astrophysics. The symmetry energy is vital to under-standing the structure of neutron-rich nuclei, the equa-tion of state (EOS) of nuclear matter, and the massesand radii of neutron stars. Because physical constraintsregarding the symmetry energy can be obtained fromthe experimental and observational data at finite nuclei,heavy-ion collisions (HIC), neutron stars and so on [1–3],various modern experiments at many accelerator facili-ties and astronomical observations have been performedto determine the constraints subject to matter density.However various ambiguities still remain owing to thelack of data, in particular, at high density.

The symmetry energy S(ρ) is defined as the energydifference between symmetric nuclear matter and neu-tron matter, E(ρ, δ) = E0(ρ, δ = 0) + S(ρ)δ2, when weignore high-order terms in δ defined as δ = (ρn − ρp)/ρ,with ρ, ρn and ρp being the nucleon, neutron and protondensities, respectively. The S(ρ) is, thus, sensitive to thematter density, and can be obtained by summing the ki-netic and the potential parts, S(ρ) = Tsym(ρ) + Vsym(ρ).

When the polytropic parametrization is employed toreproduce various data from the HIC [1–3], the potential

∗E-mail: cheoun@ssu.ac.kr

term can be written as

Vsym(ρ) =Cs,p

2

( ρ

ρ0

)γi

, (1)

where Cs,p = 35.2 MeV, as used in Ref. [1], and the sym-metry energy at saturation density is S0 = 30.1 MeV.The γi factor is usually fitted to reproduce physical ob-servables, such as the isospin diffusion and the neutron-to-proton ratios, from 124Sn+124 Sn and 112Sn+112 Sncollisions by using the improved quantum molecular dy-namics (ImQMD) transport model [1]. The γi was shownto have a range of 0.4 ≤ γi ≤ 1.05 within a 2σ uncer-tainty [1].

In this report, by employing the polytropic formulafor the density dependence in the symmetry energy, weinvestigate the effects of the symmetry energy on thestructure of neutron stars. Although the fitted γi stillhas a wide range, it would be a meaningful discussion ifwe understand how the experimental data from the HICdata will constrain the structure of the magnetar. In rel-ativistic mean field (RMF) models, the symmetry energyis described mainly by the ρ meson field. In our previouspaper [4], the density dependence of gρ is obtained fromrecent experimental data on the basis of the density-dependent RMF (DDRMF) model within a momentum-dependent interaction (MDI) scheme. In this work, weextend our previous work to a strongly-magnetized neu-tron star, in which the gρ is calculated by using the poly-tropic formula for given γi. Here, we choose three differ-ent cases, γi = 0.5, 1 and 2, by considering the physical

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Effects of the Symmetry Energy on Strongly-magnetized· · · – Chung-Yeol Ryu and Myung-Ki Cheoun -169-

range deduced from the HIC data.A recent paper [5] reported a mass of M ≈ 1.97M� for

PSR J1614-2230 from the detection of the Shapiro timedelay and the radius of a neutron star [7] was observedto be R = 12 ± 1 km. When we consider only the nu-cleonic phase composed of neutrons, protons, electronsand muons, the M ≈ 2M� can be explained, but theradii strongly depend on the symmetry energy. In ourresult, the soft (γ = 0.5) symmetry energy is shown tobe consistent with the radius observed [6,7].

The paper is organized as follows: In Sec. II, we in-troduce the models and formulae used in the calculation.In Sec. III, numerical results are displayed with detaileddiscussions. A summary and conclusions are presentedin Sec. IV.

II. MODEL AND SYMMETRY ENERGY

1. DDRMF Model with Strong Magnetic Fields

The DDRMF model has been developed to describethe various phenomena in nuclear physics [8]. The La-grangian of the DDRMF model for dense nuclear matterin the presence of a magnetic field can be derived by in-troducing of a vector potential Aµ [9–16]. The resultingLagrangian can generally be written in terms of baryons,leptons, and meson fields as follows;

L =∑

b

ψ̄b

[iγµ∂

µ − qbγµAµ −M∗

b (σ)

−Γωbγµωµ − Γρbγµ�τ · �ρµ − 1

2κbσµνF

µν]ψb

+∑

l

ψ̄l [iγµ∂µ − qlγµA

µ −ml]ψl

+12∂µσ∂

µσ − 12m2

σσ2 − 1

4WµνW

µν +12m2

ωwµwµ

−14RiµνR

µνi +

12m2

ρρµρµ − 1

4FµνF

µν , (2)

where the indices b and l denote the baryon octet and theleptons (e− and µ−), respectively. The effective baryonmass, M∗

b , is simply given by M∗b = Mb − Γσbσ, where

Mb is the free mass of a baryon in vacuum and Γσb isthe coupling constant between baryons and sigma mesonfields. Wµν , Riµν , and Fµν denote field tensors for ω, ρand the photon fields, respectively.

The anomalous magnetic moments (AMMs) of thebaryons interact with the external magnetic field throughκbσµνF

µν , where σµν = i2 [γµ, γν ] and κb is the strength

of the AMM for a baryon: for example, κp = 1.7928µN

for a proton with the nuclear magneton µN .For infinite nuclear matter, the equations of meson

fields in the mean-field approximation are obtained as

m2σσ =

∑b

Γσbρ̃b ,

m2ωω0 =

∑b

Γωbρb ,

m2ρρ03 =

∑b

ΓρbIb3ρb, (3)

where ρ̃b(= ψ̄bψb) and ρb(= ψ†bψb) are the scalar and the

vector densities of a baryon b, respectively, and Ib3 is itsisospin z-component. The Dirac equation of a baryon is,thus, given as

[γµ (i∂µ − Σµτ ) − (mb − Σs

τ )]ψb = 0. (4)

If we assume Γib = Γib(ρ), where ρ =∑

b ρb, the scalarand the time components of the vector self-energies areobtained as

Σsτ = Γσb σ ,

Σµ=0τ = Γωb ω0 + Γρb ρ03Ib3 + Σ0(r)

τ , (5)

with

Σ0(r)τ =

∑b

[∂Γωb

∂ρω0ρb +

∂Γρb

∂ρIb3ρbρ03 − ∂Γσb

∂ρσρ̃b

].

Note that we only need the time component of Σµτ in the

mean field approximation.Energy spectra for the baryons and the leptons are

given by

ECb =

√k2

z +(√

M∗b

2 + 2ν|qb|B − sκbB

)2

+ Γωbω0 + Ib3Γρb ρ03 + Σ0(r)τ ,

ENb =

√k2

z +(√

M∗b

2 + k2x + k2

y − sκbB

)2

+ Γωbω0 + Ib3Γρb ρ03 + Σ0(r)τ ,

El =√k2

z +m2l + 2ν|ql|B, (6)

where ECb and EN

b denote the energies of charged andneutral baryons, respectively. The Landau quantizationfor charged particles in a magnetic field is denoted asν = n+ 1/2 − sgn(q)s/2 = 0, 1, 2 · · · , where the sign ofthe electric charge is denoted as sgn(q) and s = 1(−1) isfor spin up (down).

Chemical potentials for the baryons and the leptonsare given by

µb = Ebf + Γωbω0 + Ib3Γρb ρ03 + Σ0(r)

τ , (7)

µl =√k2

f +m2l + 2ν|ql|B, (8)

where Ebf is the baryon Fermi energy and kf is the lepton

Fermi momentum. For charged particles, Ebf is written

as

Ebf

2= kb

f

2+ (

√m∗

b2 + 2ν|qb|B − sκbB)2, (9)

-170- Journal of the Korean Physical Society, Vol. 63, No. 2, July 2013

where kbf is the baryon Fermi momentum and ν = 0 for

neutral baryons.We apply three constraints for calculating the proper-

ties of a neutron star: 1) baryon number conservation,2) charge neutrality and 3) chemical equilibrium. Themeson field equations are solved along with the chemi-cal potentials for the baryons and the leptons subject tothese three constraints. The total energy density is thengiven by εtot = εm +εf , where the energy density for thematter fields is given as

εm =∑

b

εb +∑

l

εl +12m2

σσ2 +

12m2

ωω2

+12m2

ρρ2, (10)

and the energy density due to the magnetic field is givenby εf = B2/2. The total pressure is then

Ptot = Pm +12B2, (11)

where the pressure due to the matter fields is ob-tained from Pm =

∑i µiρ

iv − εm. The relation between

the mass and the radius for a static and spherically-symmetric neutron star is generated from the solution tothe Tolman-Oppenheimer-Volkoff (TOV) equations ob-tained by using the equation of state (EOS) describedabove.

2. Coupling Constants in DDRMF and Mag-netic Fields

In this work, we employ the density-dependent cou-pling constants for σ and ω mesons proposed by Typel& Wolter (TW99) [8]. For i = σ, ω, we can write thedensity-dependent coupling constants in Eq. (3) as

Γib(ρ) = gibfi(n), (12)

where n = ρ/ρ0 with the saturation density ρ0. Weassume that fi(1) = 1, so gib denotes the coupling con-stant at the saturation density. The density-dependentpart fi(n) is given by

fi(n) = ai1 + bi(n+ di)2

1 + ci(n+ di)2. (13)

The details for the procedure to fix the parameters infi(n) can be found in Ref. [8]. Here, we simply tab-ulate the values of the parameters in Table 1, and theresulting saturation properties in the caption. The den-sity dependence and the coupling constants of ρ mesonwill be discussed in the next subsection.

Since the magnetic fields may also depend on the den-sity, we take the density-dependent magnetic fields usedin Refs. [13] and [14]:

B (ρ/ρ0) = Bsurf +B0

[1 − exp{−α (ρ/ρ0)

β}], (14)

Meson(i) giN ai bi ci di

σ 10.87854 1.365469 0.226061 0.409704 0.901995

ω 13.29015 1.402488 0.172577 0.344293 0.983955

Table 1. Parameters of the density-dependent couplingconstants in Typel & Wolter [8] fitted to the saturationdensity ρ0 = 0.153 fm−3, the binding energy per nucleon16.247 MeV, and the compression modulus K0 = 240 MeV.The masses of the mesons used were mσ = 550 MeV andmω = 783 MeV.

where Bsurf is the magnetic field at the surface of aneutron star, which is taken as 1015 G from observations,and B0 represents the magnetic field saturated at highdensities. In the present work, we use the set α = 0.02and β = 3 for varying magnetic fields. The magneticfields are usually written with the unit of the electroncritical magnetic fields Be = 4.414 × 1013, so that B0 =B∗Be, where B∗ is regarded as a free parameter.

3. Symmetry Energy Beyond the Nuclear Mat-ter Density

The density-dependent ΓρN in DDRMF model can beextracted from a phenomenological formula [4], and here,we briefly review the method. The energy per baryon ininfinite nuclear matter is conventionally defined as

E(ρ, δ) = E(ρ, 0) + S(ρ) δ2 +O(δ4), (15)

where ρ = ρn + ρp is the nucleon number density andδ = (ρn−ρp)/ρ. From the above equation, we can definethe symmetry energy as

S(ρ)δ=1 = E(ρ, 1) − E(ρ, 0). (16)

In the DDRMF model described above, the symmetryenergy density is given as

εsym = ε(ρ, δ = 1) − ε(ρ, δ = 0)

= ∆Ekin +Γ2

ρN

8m2ρ

ρ2. (17)

If we divide the symmetry energy into kinetic and po-tential terms as

S(ρ)δ=1 = Tsym + Vsym, (18)

the kinetic term reads

Tsym =∆Ekin

ρ=

(kNF )2

6√

(kNF )2 + (m

N )2, (19)

where kNF is the Fermi momentum of the nucleon in sym-

metric nuclear matter. The potential part is then writtenas

Vsym =Γ2

ρN

8m2ρ

ρ. (20)

Effects of the Symmetry Energy on Strongly-magnetized· · · – Chung-Yeol Ryu and Myung-Ki Cheoun -171-

0

10

20

30

40

0 0.5 1 1.5 2

Vsy

m (

MeV

)

ρ / ρ0

γ = 0.51.02.0

Fig. 1. (Color online) Potential part in the symmetry en-ergy by using the DDRMF.

As mentioned in the Introduction, the polytropic for-mula of the potential term is written as

Vsym(ρ) =Cs,p

2

( ρ

ρ0

)γi

, (21)

where Cs,p = 35.2 MeV and the symmetry energy atsaturation density is S0 = 30.1 MeV [1]. Therefore, ΓρN

can be obtained from this formula for given γi. In thiswork, we test γi = 0.5, 1 and 2.

III. RESULTS AND DISCUSSION

In Fig. 1, we show the potential term of the symmetryenergy for three given γi values. In this work, we excludethe super-soft model suggested by the MDI scheme [4] inthe polytropic formula considered here. The γ = 0.5,γ = 1 and γ = 2 give us soft, middle and stiff symmetryenergies, respectively. As mentioned in the Introduction,the range of γi = 0.4 < γi < 1.05 was fitted to reproducethe constraints from heavy-ion collision data. Above thesaturation density ρ0, the stiffer symmetric energy dueto the larger γi favors more symmetric matter while thesofter one favors more asymmetric matter. The situationis reversed below ρ0.

In Figs. 2 and 3, the populations of particles, the EOSsand the mass-radius relations are displayed for given γi

values. First, particle profiles are presented in Fig. 2to understand the effect of the symmetry energy on thepopulations of particles. With increasing γi, the neutronfraction decreases with increasing proton fraction. Be-cause symmetry energy is the energy difference betweenneutron matter and symmetric matter, the increase inthe symmetry energy with increasing γi factor causesproton fractions to be increased in a neutron star.

Because the EOS can be easily affected by the particlefractions, we show the change in the EOS in Fig. 3. Be-cause the stiffer symmetry energy favors more symmetric

0.001

0.01

0.1

1

0 2 4 6 8 10

Yi

ρ/ρ0

p

n

e-

µ-

γi = 0.5

0.001

0.01

0.1

1

0 2 4 6 8 10

Yi

ρ/ρ0

p

n

e-

µ-

γi = 1.0

0.001

0.01

0.1

1

0 2 4 6 8 10

Yi

ρ/ρ0

pn

e-

µ-

γi = 2.0

Fig. 2. (Color online) Populations of particles for given γi

values. B∗ = 1 × 104 is used for all calculations.

matter, the pressure should be decreased because an in-crease in the number of protons leads to a decrease inthe chemical potentials of all nucleons, as shown in Fig.2. However in Fig. 3, one can see an opposite behaviorin which the stiff symmetry energy also gives a stiff EOS.This can be explained by the ρ meson’s contribution tothe competition between the kinetic and the potentialterms, Eqs. (19) and (20), in the symmetry energy.

If we compare the kinetic terms in both the soft andthe stiff symmetry energies, the stiffer symmetry energyleads to a softer EOS because of the decrease in the Fermimomenta of the nucleons. However a stiff symmetry en-ergy causes the ρ meson’s contribution to become largebecause the ρ meson is proportional to the ρ meson cou-

-172- Journal of the Korean Physical Society, Vol. 63, No. 2, July 2013

0

50

100

150

200

250

300

0 1 2 3 4 5

Pres

sure

(M

eV f

m-3

)

ρ (fm-3)

γi = 0.51.02.0

0.5

1

1.5

2

2.5

9 10 11 12 13 14 15 16

M /

Mso

lar

Radius (km)

γi = 0.51.02.0

Fig. 3. (Color online) Equations of state and mass-radiusrelations for various γi. B∗ = 1 × 104 is used for all calcula-tions.

plings. Therefore, the ρ meson term in pressure can ex-ceed the contribution by the Fermi momentum, so thata stiff symmetry energy can give rise to a stiffer EOS inthe upper panel of Fig. 3.

In the lower panel of Fig. 3, we show the mass-radiusrelations of neutron stars. To calculate the radii of neu-tron stars, we use the Baym-Pethick-Sutherland (BPS)[17] and Baym-Bethe-Pethick (BBP) [18] EOSs for thecrust of the neutron star and our EOS in the core re-gion. In Ref. [7], radii of neutron stars were reported asR = 12± 1 km. If we accept that value, the soft symme-try energy (γi = 0.5) in our model is more compatiblewith the data. In particular, the stiff symmetry energy(γi > 1.05) can be ruled out in the radius of the neutronstar and in the heavy-ion collision data.

In order to evaluate the effect of the magnetic fields,we show the EOS and the mass-radius relation for somegiven B∗ values in the case of γi = 0.69 in Fig. 4. Be-cause we use the density-dependent magnetic fields, in-creasing magnetic fields mean that the magnetic fieldsstrength in the core region increases with increasing B∗.Therefore, the magnetic field effect may not contributeto the radius, but does contribute to the maximum massof the neutron star, as shown in lower panel.

0

50

100

150

200

250

300

0 1 2 3 4 5

Pres

sure

(M

eV f

m-3

)

ρ (fm-3)

B* = 1 x 104

5 x 104

1 x 105

0.5

1

1.5

2

2.5

9 10 11 12 13 14 15

M /

Mso

lar

Radius (km)

B* = 1 x 104

5 x 104

1 x 105

Fig. 4. (Color online) Equations of state and mass-radiusrelations for strongly-magnetized neutron stars. γi = 0.69 isused.

IV. SUMMARY

In summary, 1) we calculated the symmetry energy ofnuclear matter by the density-dependence RMF, whichtakes recent heavy ion scattering data into account by us-ing the polytropic formula. In particular, the potentialpart of the symmetry energy adjusted by the polytropicparameter is shown to be consistent with the soft po-tential rather than the super-soft or hard potential. 2)Although the proton fractions increase with increasingpotential parameter, the stiff symmetry energy leads toa bit stiffer equation of state of magnetized neutron starbecause of the competition of the role of the ρ meson inthe kinetic and potential terms of the symmetry energy.3) In the mass-radius relation, the polytropic parameters(0.4 < γi < 1.05) constrained by the heavy-ion scatter-ing data turn out to be consistent with the observed radiiof magnetized neutron stars.

Finally, the radius of a neutron star depends verylargely on the EOS at densities below the saturation den-sity, where the EOS is sensitive to the matter state in theouter region of a neutron star, i.e., the lattice structure ofthe crust, the phase transition between the crust and theouter core, etc. However, even though various ambigui-ties from the low density EOS, the symmetry energy withγ > 1 might be ruled out based on the observed radii of

Effects of the Symmetry Energy on Strongly-magnetized· · · – Chung-Yeol Ryu and Myung-Ki Cheoun -173-

neutron stars and the heavy-ion collision data. Espe-cially, γ ≈ 0.5 is favored in order to explain R = 11− 13km. It means that the EOS below saturation density be-cause of the smaller γ factor in the symmetry energy isfavored for the observed mass-radius relations.

Because the main topic in this paper is to exploit thepolytropic parameter in the density-dependent RMF tomagnetized neutron star, our present calculation is lim-ited to a nucleonic neutron star. Results for the hyper-onic neutron star will be discussed elsewhere in the nearfuture.

ACKNOWLEDGMENT

This work was supported by the research fund ofHanyang University (HY-2012-G).

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