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4

Quark magnetar in three-flavor Nambu–Jona-Lasinio model with vector interaction

and magnetized gluon potential

Peng-Cheng Chu,1, 2, ∗ Xin Wang,1, † Lie-Wen Chen,1, 3, ‡ and Mei Huang2, 4, §

1Department of Physics and Astronomy and Shanghai Key Laboratory for Particle Physics and Cosmology,

Shanghai Jiao Tong University, Shanghai 200240, China2Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China

3Center of Theoretical Nuclear Physics, National Laboratory of Heavy-Ion Accelerator, Lanzhou, 730000, China4Theoretical Physics Center for Science Facilities,

Chinese Academy of Sciences, Beijing 100049, China

We investigate properties of strange quark matter in the framework of SU(3) Nambu–Jona-Lasinio(NJL) model with vector interaction under strong magnetic fields. The effects of vector-isoscalar and vector-isovector interactions on the equation of state of strange quark matter areinvestigated, and it is found that the equation of state is not sensitive to the vector-isovector in-teraction, however, a repulsive interaction in the vector-isoscalar channel gives a stiffer equation ofstate for cold dense quark matter. In the presence of magnetic field, gluons will be magnetized viaquark loops, and the contribution from magnetized gluons to the equation of state is also estimated.The sound velocity square is a quantity to measure the hardness or softness of dense quark matter,and in the NJL model without vector interaction at zero magnetic field the sound velocity square isalways less than 1/3. It is found that a repulsive vector-isoscalar interaction and a positive pressurecontribution from magnetized gluons can enhance the sound velocity square, which can even reach1. To construct quark magnetars under strong magnetic fields, we consider anisotropic pressuresand use a density-dependent magnetic field profile to mimic the magnetic field distribution in aquark star. We also analyze the parameter region for the magnitude of vector-isoscalar interactionand the contribution from magnetized gluons in order to produce 2 solar mass quark magnetars.

PACS numbers: 21.65.Qr, 97.60.Jd, 26.60.Kp, 21.30.Fe,12.39.St

I. INTRODUCTION

Investigating properties of strong interaction matteris one of the main topics of quantum chromodynamics(QCD). It is believed that there will be deconfinementphase transition from hadronic matter to quark-gluonplasma (QGP) at sufficiently high temperatures and fromnuclear matter to quark matter (or color superconductor)at high baryon densities. The hot quark-gluon plasma isexpected to be created in heavy ion collisions at the Rela-tivistic Heavy Ion collider (RHIC) and the Large HadronCollider (LHC). The hot and dense quark matter mightbe created in heavy ion collisions at FAIR in GSI and theNuclotron-based Ion Collider Facility (NICA) at JINR,while the cold and dense quark matter may exist in theinner core of compact stars.

In the inner core of compact stars, the baryon densitycan reach or even be larger than about 6 times the normalnuclear matter density n0 = 0.16 fm−3, so there mightexist ”exotic” matter like hyperons [1–3], meson conden-sations [4–6] and quark matter (normal quark matter orstrange quark matter [7, 8], and color superconductor[9, 10].). Strange quark matter has been conjectured tobe the true ground state of QCD [7, 8], and many ef-

∗Electronic address: kyois@sjtu.edu.cn†Electronic address: xinwang@sjtu.edu.cn‡Electronic address: lwchen@sjtu.edu.cn§Electronic address: huangm@mail.ihep.ac.cn

forts have been put on investigating the conversion fromneutron star to quark star that consists of strange quarkmatter which is made purely by u, d, s quarks and someleptons like electron and muon due to charge neutralityand β-equilibrium [11–14]. There are also hybrid starconjectures on a transition from nuclear phase to quarkphase at such high baryon density, and several authorshave studied the phase transition in hybrid star [15–23].

The equation of state (EoS) plays a central role in in-vestigating properties of strong interaction matter, whichis one of fundamental issues in nuclear physics, astro-physics and cosmology. On the one hand, the EoS gener-ates unique mass versus radius (M-R) relations for neu-tron stars and the ultra-dense remnants of stellar evo-lution. On the other hand, the mass-radius relation ofcompact stars can put strong constraints on the EoS forstrong interaction matter at high baryon density and lowtemperature.

Recently, two heaviest neutron stars have been mea-sured with high accuracy. One is the radio pulsar J1614-2230 [24] with a mass of 1.97± 0.04M⊙, and the other isJ0348+0432 [25] with mass 2.01 ± 0.04M⊙. Even heav-ier neutron stars have been discussed in the literatures[26, 27]. It is suggested that the established existenceof two-solar-mass neutron stars would prefer hard EoSbased entirely on conventional nuclear degrees of free-dom, and many soft equations of state, including hybridstars containing significant ”exotic” proportions of hy-perons, Bose condensates or quark matter would be ruledout.

2

However, it has been pointed out that the repul-sive vector interaction in the Nambu–Jona-Lasinio (NJL)model can produce a stiff EoS for dense quark matter andthus can generate two-solar-mass compact star [28, 29].The role of the vector interaction in QCD vacuum andmedium has been discussed in many literatures [28, 30?–46]. The introduction of the vector interaction withinthe NJL model is necessary to describe vector mesons,and the coupling constant is determined by the vectorspectra [30, 33]. In order to describe vector bound stateswithin the NJL model, the vector interaction must beattractive in the space-like components, thus it is repul-sive in the time-like components, which is relevant for thenumber density in the mean field [34]. The QCD phasediagram and the critical end point is sensitive to the signof vector coupling constant as shown in [38–41].

Furthermore, the presence of external magnetic fieldcan even harden the equation of state of dense quarkmatter as shown in Ref. [47]. In recent decades, proper-ties of hot and dense quark matter under strong magneticfields have attracted lots of interests, especially in recentseveral years much progress has been made.

Strong magnetic fields with the strength of 1018 ∼1020G (equivalent to eB ∼ (0.1 − 1.0 GeV)2) can begenerated in the laboratory through non-central heavyion collisions [48, 49] at the Relativistic Heavy Ion Col-lider (RHIC) and the Large Hadron Collider (LHC).This offers a unique opportunity to study propertiesof hot strong-interaction matter under strong magneticfield. The observation of charge azimuthal correlationsat RHIC and LHC [50, 51] might indicate the anoma-lous Chiral Magnetic Effect (CME) [52–54] with local P-and CP-violation. Conventional chiral symmetry break-ing and restoration under external magnetic fields hasbeen investigated for many years. It has been recog-nized for more than 20 years that the chiral condensateincreases with B, which is called magnetic catalysis [55–57], and naturally the chiral symmetry should be restoredat a higher Tc with increasing magnetic field. However,the lattice group [58–60] has demonstrated that the tran-sition temperature decreases as a function of the externalmagnetic field, i.e. inverse magnetic catalysis around Tc,which is in contrast to the naive expectation and ma-jority of previous results. It was shown in [61, 62] thatthe chirality imbalance can explain the inverse magneticcatalysis. It was suggested that in the presence of exter-nal magnetic field, there will be vector condensation inthe QCD vacuum [63, 64], and the vector condensationwas confirmed in Refs. [65, 66]. The vector condensationin the neutron star has been discussed in Ref.[67].

Moreover, considerable efforts have also been directedto the study of the effects of intense magnetic fieldson various astrophysical phenomena. The presence ofstrong magnetic fields at the surface of conventional com-pact stars or neutron stars is 109 ∼ 1015G [68–74],which is thousand times stronger than ordinary neutronstars. These strongly magnetized objects are called mag-netars [74]. By using scalar virial theorem based on New-

tonian gravity [75], it is predicted that the magnetic fieldin the inner core of neutron stars could reach as high as1018 ∼ 1020G. Under such tremendous magnetic fields,the O(3) rotational symmetry will break and the pres-sure anisotropy of the system must be considered [76–79]. In order to mimic the spatial distribution of themagnetic field strength in magnetars, people have intro-duced a density-dependent magnetic field profile [80, 81].

Many efforts have been taken on investigating the ex-istence of quark core in neutron stars under strong mag-netic field [47, 82–88]. In this work, we will use SU(3)Nambu-Jona-Lasinio(NJL) model with vector interactionto investigate the magnetar, and we will also consider thepressure contribution from polarized gluons under mag-netic field.

It is known that the NJL model only considers thequark contribution to the pressure, therefore, the pres-sure from NJL model is smaller than that from latticecalculation at high temperature and zero chemical poten-tial [89, 90]. In order to consider the gluon contributionto the pressure, the Polyakov-loop potential was firstlyintroduced in the framework of NJL model in [92] andthen in [93], where the Polyakov-loop potential is temper-ature dependent. The PNJL model can fit the equationof state at high temperature very well with lattice QCDresults. The extension of the Polyakov-loop potential tofinite chemical potential is not trivial. In order to use thePNJL model to describe neutron stars, the Polyakov po-tential is modified to have chemical potential dependentin Refs. [94–98]. However, there is still no extension ofPolyakov-loop potential under magnetic field. Therefore,we cannot use PNJL model for our purpose in this workto investigate quark matter at high baryon density underexternal magnetic field, thus we have to find other wayto include the gluon contribution to the pressure at finitechemical potential and at magnetic fields. In this work,we will give an ansatz on the potential from magnetizedgluons hinted from hard thermal/dense loop results.

The paper is organized as follows. In Sec. II, we give ageneral description of the SU(3) NJL model with vectorinteraction under magnetic field with β equilibrium, wemake an ansatz of thermodynamical potential from mag-netized gluons, and derive the equation of state of thestrange quark matter with β-equilibrium. Our numeri-cal results are shown in Sec. III and the conclusion anddiscussion is given in Sec. IV.

II. THREE-FLAVOR NJL MODEL WITH

VECTOR INTERACTION UNDER MAGNETIC

FIELD

We study properties of three-flavor system with ex-ternal strong magnetic fields Aext

µ under β equilibriumcondition, which is described by the Lagrangian density

L = Lq + Le −1

4FµνF

µν , (1)

3

where Lq, Le are the Lagrangian densities for quarks andelectrons, respectively. Fµν = ∂µA

extν − ∂νA

extµ is the

strength tensor for external electromagnetic field. Themagnetic field B is a static magnetic field along z di-rection, and Aext

µ = δµ2x1B. In this work, we do notconsider contributions from the anomalous magnetic mo-ments [99].The electron Lagrangian density is given as

Le = e[(i∂µ − eAµext)γ

µ]e. (2)

The Lagrangian density for quarks is described by thegauged Nf = 3 NJL model with vector interaction [31,32]

Lq = ψf [γµ(i∂µ − qfA

µext)− mc]ψf + L4 + L6, (3)

where L4 indicates four-fermion interaction compatiblewith QCD symmetries SU(3)color ⊗ SU(3)L ⊗ SU(3)R,and L6 is the six-point interaction which is requiredto break the axial U(1)A symmetry. ψ = (u, d, s)T

represents a quark field with three flavors, mc =diag(mu,md,ms) is the current quark mass matrix, andqf the quark electric charge. The four-fermion inter-action includes scalar, pseudoscalar, vector, axial-vectorchannels and takes the form of

L4 = LS + LV + LI,V . (4)

The scalar part takes the form of

LS = GS

8∑

a=0

[(ψfλaψf )2 + (ψf iγ5λaψf )

2], (5)

and the vector part is given as

LV = −GV

8∑

a=0

[(ψγµλaψ)2 + (ψiγµγ5λaψ)2], (6)

where GS and GV are the coupling constants in thescalar and vector channels, respectively. λa(a = 1, · · · , 8)λa are the Gell-Mann matrices, and the generators ofthe SU(3) flavor groups, and λ0 =

√

2/3I with I the3 × 3 unit matrix. In order to describe the nonetof scalars, pseudo-scalars, vectors and axial-vectors, aconvenient representation is obtained by changing fromλ0, λ1, · · · , λ8 to the set λ0, λ±1 , λ3, λ±4 , λ±6 , λ8 with

λ±1 =

√

1

2(λ1 ± iλ2),

λ±4 =

√

1

2(λ4 ± iλ5),

λ±6 =

√

1

2(λ6 ± iλ7).

Hadrons in the u, d sector exhibit SU(2)I isospin sym-metry. Up and down quarks have isospin I = 1/2, andisospin 3-components (I3) of 1/2 and −1/2, respectively.

All other quarks have I = 0. For scalars, the couplingconstant in the scalar-isoscalar (σ) and pseudoscalar-isovector (π) interactions have to be equal, which is con-strained by chiral symmetry. However, the coupling con-stants for the vector-isoscalar (ω) and vector-isovector (ρ)interaction can be separately invariant, thus can be cho-sen independently. The ratio of the coupling constants ofthe vector-isosinglet channel ω and vector-isovector chan-nel ρ to nucleons is empirically given by gωQQ/gρQQ ≃ 3in the chirally broken phase, and gωQQ/gρQQ = 1 in thechiral symmetric phase [39, 99, 100]. To distinguish theisoscalar and isovector for vectors, we introduce an extraterm for vector-isovector channel with the form of

LIV = −GIV [(ψγµ~τψ)2 + (ψγ5γ

µ~τψ)2]. (7)

The coupling constant for vector-isoscalarGωV = GV , and

the coupling constant for vector-isovector GρV = GV +

GIV . In this work, we will investigate the role of vector-isovector interaction on the equation of state, thereforein our numerical calculations, we choose GIV as a freeparameter.The six-fermion interaction L6, i.e. the ’t Hooft term

takes the form of

Ldet = −Kdetf [ψf (1 + γ5)ψf ] + detf [ψf (1− γ5)ψf ],(8)

which is to break the U(1)A symmetry.

A. The pressure from quark contribution

The equation of state is the most important aspect forphysicists to acquire the properties of quark matter. Inorder to get the pressure and energy density of quarkmatter, we should firstly derive the thermodynamicalpotential Ωf . In procedure, we can calculate the ther-modynamical quantities by using finite temperature fieldtheory. In the mean-field approximation, the lagrangiandensity for quark part is

LM = ψf [γµ(i∂µ − qfA

µext)− M − 2GIV γ0τ3fnf ]ψf

− 2GS(σ2u + σ2

d + σ2s) + 4Kσuσdσs − 4GV γ0n

+ 2GV (n2u + n2

d + n2s) +GIV (nu − nd)

2,

(9)

where n =

nu 0 00 nd 00 0 ns

and M =

Mu 0 00 Md 00 0 Ms

.

The quark mass is determined by the gap equation of

Mi = mi − 4GSσi + 2Kσjσk, (10)

with (i, j, k) being any permutation of (u, d, s) and thechiral condensate is given as

σf = 〈ψfψf 〉 = −i∫

d4p

(2π)4tr

1

(p/ −Mf + iǫ). (11)

4

After introducing Landau quantization and severalsteps of finite temperature field theory calculations, wecan get the thermodynamical potential Ωq of quark mat-ter under magnetic fields, and the pressure density pq =−Ωq and takes the form of

pq = −2GS(σ2u + σ2

d + σ2s) + 4Kσuσdσs

+ 2GV (n2u + n2

d + n2s) +GIV (nu − nd)

2

+ (Ωuln +Ωd

ln +Ωsln) (12)

with the logarithmic contribution

Ωfln = −i

∫

d4p

(2π)4tr ln 1

T[p/− Mf + γ0µf ], (13)

here,

µf = µf − 4GV nf − 2GIV τ3f (nu − nd), (14)

where µf is the chemical potential for each flavor ofquarks and τ3f is the isospin quantum number for quarks:τ3u = 1, τ3d = −1 and τ3s = 0.Following Ref.[47], one can get the condensates and

pressure for quarks. The logarithmic contribution to thethermodynamical potential is given by

Ωfln = Ωf,vac

ln +Ωf,magln +Ωf,med

ln . (15)

The first term is the vacuum contribution

Ωf,vacln = − Nc

8π2

M4f ln

[Λ + ǫΛMf

]

− ǫΛΛ(Λ2 + ǫ2Λ)

(16)

with ǫ2Λ = Λ2 +M2f and Λ the noncovariant cutoff. The

magnetic field contribution takes the form of

Ωf,magln =

Nc

2π2(|qf |B)2

[x2f4

+ ζ′(−1, xf )

−1

2(x2f − xf ) ln (xf )

]

, (17)

where ζ(z, x) is the Riemann-Hurwitz function and

ζ′(−1, xf ) = dζ(z, x)/dz|z=−1 (18)

with xf =M2

f

2|qf |B. The medium contribution reads

Ωf,medln =

kfmax∑

k=0

αk

(|qf |BNc)

4π2

µf

√

µ2f − sf (k,B)2

− sf (k,B)2 ln[ µf +

√

µ2f − sf (k,B)2

sf (k,B)

]

,

(19)

where

sf (k,B) =√

M2 + 2|qf |Bk, (20)

and

kfmax =µ2f −M2

2|qf |B=

p2f,F2|qf |B

, (21)

is the upper Landau level with αk = 2− δk0.Then we can also give the condensates for each flavor

of quarks:

σf = σvacf + σmag

f + σmedf (22)

with

σvacf = −MfNc

2π2

Λ√

Λ2 +M2f

−M2

f

2ln[ (Λ +

√

Λ2 +M2f )

2

(M2f )

]

, (23)

σmagf = −MfNc

2π2(|qf |B)

ln[Γ(xf )]

− 1

2ln(2π) +

ln(xf )

2− xf ln(xf )

, (24)

σmedf =

kfmax∑

k=0

αk

Mf |qf |BNc

π2

ln[ µf +

√

µ2f − sf (k,B)2

sf (k,B)

]

. (25)

B. The pressure from leptons

For SQM, we assume it is neutrino-free and composedof u, d, s quarks and e− in beta-equilibrium with electriccharge neutrality. The weak beta-equilibrium conditioncan then be expressed as

µu + µe = µd = µs, (26)

where µi (i = u, d, s and e−) is the chemical potentialof the particles in SQM. Furthermore, the electric chargeneutrality condition can be written as

2

3nu =

1

3nd +

1

3ns + ne. (27)

Where

nf =

kf ,max∑

k=0

αk

|qf |BNc

2π2kF ,f (28)

is the number density for each flavor of quarks with

kF,f =√

µ2f − sf (k,B)2, and

nl =

kl,max∑

k=0

αk

|ql|B2π2

kF,l (29)

the number density of electrons.

5

We can also write the leptonic contribution to the pres-sure density, which takes the form of:

pl =

klmax∑

k=0

αk

(|ql|BNc)

4π2

µl

√

µ2l − sl(k,B)2

− sl(k,B)2 ln[µl +

√

µ2l − sl(k,B)2

sl(k,B)

]

. (30)

C. The pressure from magnetized gluon potential

It is well known that the NJL model only considersthe quark contribution to the pressure, which is smallerthan that from lattice calculation at high temperatureand zero chemical potential [89, 90]. In the presence ofstrong magnetic field, not only quarks are polarized alongthe direction of B, but also gluons will be polarized viathe quark loop. There are still few calculations of pres-sure contributed from magnetized gluon degrees of free-dom at zero temperature and finite chemical potential[60, 91].

In order to consider the gluon contribution to the pres-sure, on the one hand, the Polyakov-loop potential wasintroduced in the framework of NJL model in [92, 93],where the Polyakov-loop potential is temperature depen-dent. The PNJL model can fit the equation of stateat high temperature very well with lattice QCD results.However, to extend the Polyakov-loop potential to finitechemical potential and strong magnetic field is nontriv-ial, though a modified version of the Polyakov potentialat finite chemical potential has been proposed in Refs.[94–98]. For our purpose in this work to describe themagnetar, we cannot use the PNJL model, but we canestimate the gluon contribution to the pressure at fi-nite chemical potential and with magnetic fields fromhard-thermal/dense-loop results. At high temperatureand zero chemical potential with zero magnetic field, thegluon contribution to the pressure in the PNJL modelshould merge with hard-thermal-loop result.

Much efforts have been put on the perturbation the-ory (PT) of hard-thermal-loop (HTL) or hard-dense-loop(HDL) calculation on the equation of state of strong in-teraction matter at high temperature and density [101–113]. At high temperature, recent progress up to three-loop HTL calculations [112] shows that the pressure den-sity, energy density and other thermodynamical proper-ties are in good agreement with available lattice data fortemperatures above approximately 300MeV. The EoSof cold quark matter is accessible through perturbativeQCD at high densities, and has been determined to orderα2s in the strong coupling constant [110].

The pressure density for ideal gas of quarks and gluons

has the form of

pSB = pSBq + pSB

g , (31)

pSBq = NcNf

(7π2T 4

180+µ2T 2

6+

µ4

12π2

)

, (32)

pSBg = (N2

c − 1)π2T 4

90. (33)

It is noticed that the pressure density for the ideal gas ofgluons is only temperature dependent and with no chem-ical potential dependent. Switching on coupling betweenquarks and gluons, the gluons will get screening mass

m2g =

1

6

[

(Nc +1

2Nf)T

2 +3

2π2

∑

f

µ2f

]

g2eff , (34)

and the perturbation theory at one-loop gives the pres-sure density of gluons as

pPTg = pSB

g −Ng

g2eff32

[5

9T 4 +

2

π2µ2T 2 +

1

π4µ4

]

, (35)

the gluon pressure becomes chemical potential dependentvia quark loop, though the coefficient in front of µ4 is only1/54 of that of T 4.Under the strong magnetic field, a massive resonance∼

g√eB is excited in the longitudinal (1+1) component of

the gluon propagator under strong magnetic fields [114].The corresponding Debye mass of the longitudinal gluonfields Aq has the screening mass

m2g(eB) =

∑

f

|qf |g2eff4π2

|eB|, (36)

at zero temperature and density. One can guess that atnonzero temperature and density, the longitudinal gluonfields take the screening mass of

m2g(T, µ, eB) = g2(aT 2 + bµ2 + ceB) (37)

where a, b, c are constants. Taking into account the trans-verse gluons, the pressure density from magnetized glu-ons can be estimated as

pg(T, µ; eB) = a0µ2eB+b0µ

4+c0T2eB+d0µ

2T 2+e0T4,

(38)with a0, b0, c0, d0, e0 being free parameters. At zero tem-perature,

pg(T = 0, µ; eB) = a0µ2eB + b0µ

4. (39)

Furthermore, for the screening mass of gluons, consid-ering the coefficients in front of eB is almost the samemagnitude as that of µ2, and we neglect the anisotropypressure density caused by the magnetic field with mag-nitude eB < 1019G, for simplicity, in this work, we usethe following ansatz of the pressure density of magnetizedgluons

pg(T = 0, µ; eB) = a0(µ2eB + µ4). (40)

6

for our numerical calculations. If we directly extend theperturbative gluon pressure Eq.(35) to nonperturbativeregion, the gluon pressure density at zero temperatureand finite density should be negative. However, in themoderate baryon density region, nonperturbative featureof gluons should still play an important role as shownin [115, 116]. The system in the moderate baryon den-sity can be regarded as compositions of quasi-quarks de-scribed by NJL model and quasi-gluons. To compen-sate the quasi-quark contribution to the pressure in theNJL model, the quasi-gluon contribution to the pressureshould be also positive. In our numerical calculation, wewill take a0 > 0 for physical case, but we will also takea0 < 0 for reference.

D. Total pressure of SQM with β-equilibrium

under magnetic field

Under strong magnetic fields, the O(3) rotational sym-metry in SQM is broken and the pressure for SQM mightbecome anisotropic, i.e., the longitudinal pressure P||

which is parallel to the magnetic field orientation is dif-ferent from the transverse pressure P⊥ which is perpen-dicular to the orientation of magnetic field. The analyticforms for longitudinal and transverse pressure densitiesof the system are given by [76]

p|| = p− 1

2B2, (41)

p⊥ = p+1

2B2 −MB, (42)

where we have defined

p = pq + pl + pg − p0, (43)

with p0 = −Ω0 = −Ω(T = 0, µ = 0, B = 0) the vacuumpressure density, which ensures p = 0 in the vacuum. Mis the system magnetization, and takes the form of

M = −∂Ω/∂B =∑

i=u,d,s,l,g

Mi. (44)

The energy density for SQM at zero temperature is givenby

ǫ = −p+∑

i=u,d,s,l

µini +1

2B2. (45)

One can find that the longitudinal pressure densityp|| satisfies the Hugenholtz-Van Hove(HVH) theorem[88, 117], while the transverse pressure density p⊥ doesnot because of the extra contributions from the mag-netic field. We can see that the magnetic energy den-sity term B2/2 contributes oppositely to the longitudi-nal and transverse pressures under magnetic fields, whichwill lead to a tremendous pressure anisotropy when themagnetic field is very strong.

0 1000 20000

200

400

50

100

150

200500 750 1000

Pres

sure

(MeV

/fm3 )

(MeV/fm3)

GV=GIV=0 GV=0.8 GS , GIV=0 GV=GIV=0.8 GS

B=0

FIG. 1: (Color online) The pressure density of SQM as afunction of energy density under zero magnetic field withthree cases: GV = GIV = 0, GV = 0.8GS , GIV = 0 andGV = GIV = 0.8GS .

III. NUMERICAL RESULTS AND

CONCLUSIONS

For our numerical calculations, following [47], the setof parameters we used is: Λ = 631.4MeV, mu = md =5.5MeV,ms = 135.7MeV,GΛ2 = 1.835 andKΛ5 = 9.29.

A. The effect of vector-isovector interaction and

vector-isoscalar interaction under magnetic field

Firstly, we analyze the effect of vector-isovector andvector-isoscalar interaction on the equation of state ofstrange quark matter Eq. (12) under zero magnetic fieldB = 0. For zero magnetic field, there is no anisotropy inthe system, the longitudinal pressure density is equal tothe transverse pressure density, i.e., p|| = p⊥ = p(B = 0).In Fig.1, we show the pressure density of SQM as a

function of energy density for three cases: 1) GV =GIV = 0, 2) GV = 0.8GS, GIV = 0, and 3) GV =GIV = 0.8GS . Comparing the cases of 1) GV = GIV = 0and 2) GV = 0.8GS , GIV = 0, one can find that therepulsive vector-isoscalar interaction GV gives a stifferequation of state. However, comparing the cases of 2)GV = 0.8GS , GIV = 0 and 3) GV = GIV = 0.8GS,it is observed that the equation of states for these twocases are almost the same, which indicates that thevector-isovector interaction does not affect the equationof state too much in SQM. One can find from Eq. (12)that the contribution from the vector-isovector inter-action is mainly dependent on the u-d quark isospinasymmetry(nu − nd) and the coupling constant GIV .Since the isospin asymmetry in SQM is small, the vector-isovector interaction is very tiny with the parameter setGV = GIV = 0.8GS.It can be understood like that, the repulsive vector-

7

1E18 1E19 1E20

350

400

450

500

550

600

650

700

Gv=0

Gv

=0.8GS

d,su

d,s

u

B(G)

Che

mic

al p

oten

tial(M

eV)

FIG. 2: (Color online)Chemical potentials for u, d, s quarksas functions of magnetic field with GV = 0 and GV = 0.8GS

at baryon density nb = 10n0 in SQM.

isoscalar interaction shifts the chemical potential to alarger value, which makes the equation of state stiffer.However, the interaction in the vector-isovector channelshifts the isospin chemical potential, and this effect isnegligible for the equation of state under β-equilibrium.Therefore, in the following numerical calculations, wesimply take GIV = 0.

Fig. 2 shows the chemical potentials for u, d and squarks as functions of the magnetic field B with GV = 0and GV = 0.8GS at fixed baryon number density nb =10n0 in SQM. The chemical potential for each flavor withGV = 0.8GS is enhanced magnificently comparing thecase of GV = 0, which implies that a stiffer EoS canbe generated once considering large vector-isoscalar cou-pling constant. One can also observe that, for both caseswith GV = 0 and GV = 0.8GS , the chemical poten-tial for quarks keeps a constant below the magnitude of1019G, and decreases with the constant magnetic fieldabove that.

Fig. 3 shows the constituent mass of u quark and squark as functions of baryon density in charge neutralSQM for B = 0 and B = 2 × 1019G with GV = 0 andGV = 0.8GS , respectively. In the case of zero magneticfield B = 0 and GV = 0, the constituent quark massfor u quark decreases from the vacuum mass almost lin-early to 50MeV in the region of baryon number densitybelow nb ≃ 0.35fm−3 ≃ 2n0, and then slowly decreaseswith baryon number density. The constituent quark massfor s quark also drops almost linearly from its vacuummass to around 475MeV in the region of baryon numberdensity below nb ≃ 0.35fm−3 ≃ 2n0, and almost keepsas a constant in the region of 0.35 < nb < 0.7fm−3 (2 < nb/n0 < 4), and then decreases quickly with baryonnumber density. This behavior is similar to Fig. 3.9 in[34]. When the magnetic field is turned on, it is foundthat under the magnitude of B = 2× 1019G, only a tinymagnetic catalysis effect can be observed for u quark and

0.0 0.5 1.00

50

100

150

200

250

300

350

0.0 0.5 1.0 1.5 2.0200

250

300

350

400

450

500

550

nb(fm-3)

ms(

MeV

)

B=0 GV=0 B=2e19G GV=0 B=0 GV=0.8GS

B=2e19G GV=0.8GS

mu(

MeV

)

FIG. 3: (Color online) Constituent mass for u quark (left fig-ure) and s quark (right figure) as functions of baryon numberdensity in charge neutral SQM for B = 0 and 2× 1019G withGV = 0 and 0.8Gs, respectively.

1E18 1E19 1E20300

400

500

600

Vac

uum

qua

rk m

ass (

MeV

)

B(G)

u d s

FIG. 4: (Color online) Vacuum constituent masses for u, dand s quarks as functions of the magnetic field in SQM.

s quark in the low baryon density region, and at highbaryon density region, the magnetic field with magni-tude of B = 2×1019G almost has no effect on constituentquark mass at fixed baryon density. However, it is no-ticed that the repulsive vector-isoscalar interaction cansmooth away the saturation region and make the strangequark mass decreases linearly with the baryon numberdensity.

As shown in Fig. 3, the magnetic field in the mag-nitude of B = 2 × 1019G does not essentially affect theconstituent quark mass. In Fig. 4, we investigate the vac-uum constituent mass of u, d and s quarks as functions ofmagnetic field in SQM, and one can find that the massesof three different flavors of quarks do not change so muchwhen the magnetic field is smaller than 1019G, while theu, d quark masses increase drastically when the magneticfield is bigger than 3 × 1019G, which indicates the mag-netic catalysis phenomenon. It should also be noticed

8

that the masses of d quark and s quark increase moreslowly with the magnetic field compared to the u quarkmass case.

B. The equation of state, sound velocity and

magnetar mass

As it is accepted, the magnetic field strength in theinner core region of compact stars could be much largerthan the magnetic field at the surface, then a density-dependent magnetic field distribution inside the compactstar is usually used to describe this behavior. We usethe following popular parametrization for the density-dependent magnetic field profile in QSs as Refs.[81, 118–121].

B = Bsurf +B0[1− exp (−β0(nb/n0)γ)], (46)

where Bsurf is the magnetic field strength at the surfaceof compact stars and its value is fixed at Bsurf = 1015Gin this work, n0 = 0.16fm−3 is the normal nuclear mat-ter density, B0 is the constant magnetic field, which is aparameter with dimension of B, and β0 and γ are twodimensionless parameters that control how exactly themagnetic field strength decays from the center to the sur-face. In order to reproduce the magnetic field which isweak below the nuclear saturation point while gettingstronger at higher density, we take B0 = 4× 1018G, β =0.003 and γ = 3 as the set of parameters in the follow-ing calculations, and this magnetic field distribution isalready proved to be a gentle magnetic field distributionfor SQM inside QSs, which can lead to a small pressureanisotropy and small maximum mass splitting for QSs ina density-dependent quark model[88].The anisotropic pressure densities in Eq.(41) and

Eq.(42) are calculated in cases of B0 = 0 and B0 =4 × 1018G with a0 = −0.01, 0, 0.01 and GV = 0, GV =0.4GS, GV = 0.8GS and GV = 1.1GS, respectively. Wecalculate the transverse pressure density as a function ofenergy density for SQM in Fig. 5 while the longitudinalpressure case in Fig. 6.One can see from Fig. 5 that: 1) With fixed GV

and fixed a0, the transverse pressure density for B0 =4 × 1018G is higher than that for B0 = 0; 2) Withfixed GV , the positive magnetized gluon pressure density(a0 = 0.01) and the case B0 = 4 × 1018G always givesthe hardest equation of state, while the negative magne-tized gluon pressure (a0 = −0.01) and the case B0 = 0always gives the softest equation of state; 3) For the caseof negative magnetized gluon pressure (a0 = −0.01), itis found that the equations of state for both B0 = 0 andB0 = 4×1018G are not sensitive to the value ofGV . How-ever, for the case of positive magnetized gluon pressure(a0 = 0.01), the EoS for both B0 = 0 and B0 = 4×1018Gare very sensitive to the value of GV ; 4)When GV = 0,the magnetic field contribution to equation of state is im-portant, while when GV increases, the contribution from

0 1E30

500

1000

Gv=0

P (M

eV/fm

3 )

0 1E3

Gv=0.4GS

(MeV/fm3)

SQM

0 1E3

Gv=0.8GS

B0=4e18G a

0=0.01

B0=4e18G a

0=0

B0=4e18G a

0=-0.01

B0=0 a

0=0.01

B0=0 a

0=0

B0=0 a

0=-0.01

0 1E3

Gv=1.1GS

FIG. 5: (Color online) Transverse pressure density as a func-tion of energy density for SQM in cases of B0 = 0 andB0 = 4 × 1018G with a0 = 0, 0.01,−0.01 and GV = 0,GV = 0.4GS , GV = 0.8GS and GV = 1.1GS , respectively.

0 10000

500

1000

Gv=0

P || (MeV

/fm3 )

0 1000

SQM

Gv=0.4GS

(MeV/fm3)0 1000

Gv=0.8GS

B0=4e18G a0=0.01 B0=4e18G a0=0 B0=4e18G a0=-0.01 B0=0 a0=0.01 B0=0 a0=0 B0=0 a0=-0.01

0 1000

Gv=1.1GS

FIG. 6: (Color online)Longitudinal pressure density as afunction of energy density SQM in cases of B0 = 0 andB0 = 4 × 1018G with a0 = 0, 0.01,−0.01 and GV = 0,GV = 0.4GS , GV = 0.8GS and GV = 1.1GS , respectively.

magnetized gluon pressure becomes more and more im-portant to the equation of state.Shown in Fig. 6 is the longitudinal pressure as a func-

tion of energy density of SQM in cases of B0 = 0 andB0 = 4 × 1018G with a0 = 0, 0.01,−0.01 and GV = 0,GV = 0.4GS , GV = 0.8GS and GV = 1.1GS , re-spectively. From this figure one can find that: 1) ForGV = 0 and fixed a0, the longitudinal pressure densityfor B0 = 4×1018G is a little smaller than that at B0 = 0,which is opposite to the case of transverse pressure; 2)With fixed GV , the positive magnetized gluon pressure(a0 = 0.01) and the case B0 = 0 always gives the hardestequation of state, while the negative magnetized gluonpressure (a0 = −0.01) and the case B0 = 4 × 1018G al-

9

ways gives the softest equation of state; 3) For the caseof negative magnetized gluon pressure (a0 = −0.01), itis found that the equations of state for both B0 = 0 andB0 = 4 × 1018G are not sensitive to the value of GV .However, for the case of positive magnetized gluon pres-sure (a0 = 0.01), the equations of state for both B0 = 0and B0 = 4 × 1018G are very sensitive to the value ofGV . Compared to Fig 5, we can find that the pres-sure anisotropy for longitudinal and transverse pressureis small when GV is as big as GV = 0.8GS, GV = 1.1GS

for a0 = 0.01, while the pressure anisotropy gets largeras the decrement of GV for a0 = −0.01. Compared tothe result from Fig. 5, we find the magnetized gluon pres-sure contribution is more important to stiffen the EoS forSQM than the contribution from magnetic field, and byusing this contribution from magnetized gluon one candescribe a heavy QS (about 2M⊙) with small pressureanisotropy (like GV = 0.8GS, GV = 1.1GS for a0 = 0.01cases) under a reasonable magnetic field distribution in-side QSs.It was pointed out in the work [122] that to construct

a hybrid star with mass heavier than 2M⊙, a large soundvelocity square for quark matter, say larger than 1/3 ispreferred. It is known that the sound velocity square forideal gas or for strongly coupled conformal theory can be1/3. However, for strongly interacting liquid, the soundvelocity square is normally smaller than 1/3. Therefore,it is interesting to investigate the sound velocity in ourcurrent model, and the results of sound velocity

c2s =dp

dǫ(47)

directly derived from equations of state in Fig. 5 andFig. 6 are shown in Fig.7 and Fig.8. Because the mainpurpose of this work is to explore the properties of SQMunder strong magnetic field by considering the magne-tized gluon contribution, we choose B0 = 4× 1018G witha0 = 0, 0.01,−0.01 and B0 = 0 with a = 0 by consideringthe vector-isoscalar interaction as GV = 0, GV = 0.8GS

and GV = 1.1GS in the following parts.The sound velocity square of dense quark matter for

transverse pressure density in Fig 7 in the NJL modelwithout vector interaction (GV = 0) and at B0 = 0 isalways less than 1/3, which is in agreement with com-mon sense. It is also understood that when the system ismore strongly coupled, the sound velocity square is get-ting smaller, and reach the smallest value at the phasetransition point, where the system is regarded as moststrongly coupled [123].However, it is observed that under magnetic field with

magnitude of B0 = 4×1018G, even at GV = 0, the soundvelocity square for transverse pressure density case canincrease to 0.6 at most. Also in the case of B0 = 0, if oneswitches on a repulsive interaction in the vector-isoscalarchannel, the sound velocity square for transverse pressuredensity case also increases and can become bigger than1/3. The stronger the GV is, the larger sound velocitysquare will be.

0 1000 20000.0

0.5

1.0

0 1000 2000 0 1000 2000 0 1000 2000

(Sou

nd v

eloc

ity)2

(MeV/fm3)

Gv=0

B0=4e18G a0=0.01 B0=4e18G a0=0 B0=4e18G a0=-0.01 B0=0 a0=0

0.33

P

Gv=0.4Gs Gv=0.8Gs

SQM

Gv=1.1Gs

FIG. 7: (Color online) The sound velocity square for trans-verse pressure as a function of energy density for SQM in casesof B0 = 0 and B0 = 4 × 1018G with a0 = 0, 0.01,−0.01 andGV = 0, GV = 0.8GS and GV = 1.1GS , respectively.

0 1000 20000.0

0.5

1.0

0 1000 2000 0 1000 2000 0 1000 2000

(Sou

nd v

eloc

ity)2

(MeV/fm3)

Gv=0

B0=4e18G a0=0.01 B0=4e18G a0=0 B0=4e18G a0=-0.01 B0=0 a0=0

0.33

SQM

Gv=0.4Gs Gv=0.8Gs Gv=1.1Gs

P||

FIG. 8: (Color online) The sound velocity square for longi-tudinal pressure as a function of energy density for SQM incases of B0 = 0 and B0 = 4× 1018G with a0 = 0, 0.01,−0.01and GV = 0, GV = 0.8GS and GV = 1.1GS , respectively.

Another factor to increase the sound velocity is fromthe magnetized gluons. With a positive contributionfrom magnetized gluon pressure, and also taken intoaccount the repulsive interaction in the vector-isoscalarchannel, the sound velocity for transverse pressure casecan be even larger than 1(for GV = 1.1GS, B0 =4× 1018G and a0 = 0.01 case), i.e., larger than the speedof light, which is of course not physical. So we can usethe condition c2s < 1 to constrain the EoS.

For the sound velocity square corresponding to longi-tudinal pressure from Fig.8, we can find that the soundvelocity square under B0 = 4× 1018G with different GV

and a0 are all smaller than those C2s corresponding to

10

4 6 8 100.0

0.4

0.8

1.2

1.6

2.0

2.4

R (km)

M/M

GV=0

PSR J0348+0432

4 6 8 10

GV=0.4G

S

4 6 8 10

for P|| B0=4e18G

a0=0.01

B0=0

a0=0

for P B0=4e18G

a0=0.01

for P B0=4e18G

a0=0

for P B0=4e18G

a0=-0.01

GV=0.8G

S

4 6 8 101214

GV=1.1G

S

FIG. 9: (Color online)Maximum mass - radius relation forquark star for transverse and longitudinal pressure caseswithin B0 = 0 and B0 = 4 × 1018G with a0 = 0, 0.01,−0.01and GV = 0, GV = 0.4GS , GV = 0.8GS and GV = 1.1GS ,respectively.

the transverse pressure case, which is self-consisted withFig. 5 and Fig. 6. One can also find that all the sound ve-locity square corresponding to longitudinal pressure aresmaller than 1.Since we have calculated that the pressure anisotropy

for SQM with the contribution from magnetized gluonpressure under the magnetic field distribution inside theQSs within B0 = 4× 1018G, β0 = 0.003 and γ = 3 is notbig, we can approximately use the EoS from longitudinalor transverse pressure case to construct the magnetar un-der density dependent magnetic field. We should firstlyintroduce the TOV equations [124] which can give thequark star with isotropic pressure:

dM

dr= 4πr2ǫ(r), (48)

dp

dr= −Gǫ(r)M(r)

r2[1 +

p(r)

ǫ(r)]

[1 +4πp(r)r3

M(r)][1− 2GM(r)

r]−1. (49)

We give the maximum mass of quark star in Figure 9by using transverse pressure and longitudinal pressurerespectively, and we use B0 = 4 × 1018G with a0 =0, 0.01,−0.01 and GV = 0, GV = 0.4GS, GV = 0.8GS

and GV = 1.1GS for transverse pressure case whileB0 = 4 × 1018G, a0 = 0.01 with GV = 0.8GS andGV = 1.1GS for longitudinal pressure case. We canread the following information from Figure 9: 1) AtB0 = 0 and GV = 0, the three-flavor NJL model givethe maximum mass of quark star is about 1.4M⊙; 2)At B0 = 0 but increases the repulsive interaction GV

in the vector-isoscalar channel, the the maximum massof quark star can reach 1.75M⊙ for GV = 1.1GS ; 3) In

the case of GV = 0, when put quark matter under themagnetic field, the maximum mass of quark magnetarfor transverse pressure case can be as heavy as 1.65M⊙,and if one takes into account a positive pressure densitycontributed from magnetized quasi-gluons, the mass ofquark magnetar can reach 1.8M⊙; 4) For the magnitudeof B0 = 4 × 1018G, the sound velocity square for trans-verse pressure case reaches almost 0.9 for GV = 0.8GS

and a0 = 0.01, and the magnetar mass is 2.17M⊙, whilethe maximum mass of quark star is 2.01M⊙ for longitu-dinal pressure within GV = 0.8GS and a0 = 0.01 underB0 = 4×1018G, which is consistent with the recently dis-covered large mass pulsar J0348+0432 (2.01± 0.04)M⊙.In order to investigate the difference for the maximum

mass of QSs by using longitudinal pressure and transversepressure, we define a mass difference parameter

δm =M⊥ −M||

(M⊥ +M||)/2, (50)

where M⊥ (M||) represents the maximum mass of QSswith transverse (longitudinal) orientation pressure, re-spectively. We calculate the mass difference for the trans-verse and longitudinal pressure within GV = 0.8GS anda0 = 0.01 under B0 = 4 × 1018G is δm = 7.65%, whichimplies that the pressure anisotropy in this case is verysmall due to the tiny mass asymmetry, and this is thereason why we can use the isotropic TOV equation tocalculate the properties of QSs approximately. We canalso find the similar results from GV = 1.1GS, a0 = 0.01and B0 = 4× 1018G case that the maximum mass of QSfor transverse pressure case is 2.30M⊙ while 2.20M⊙ forlongitudinal pressure case, and the mass difference forthis case is δm = 4.44%, which implies a smaller pressureanisotropy than GV = 0.8GS case. Therefore, our resultsindicate that we can get stiffer EoS by considering thecontributions from density dependent magnetic field, therepulsive interaction in the vector-isoscalar channel andmagnetized gluon pressure. Since the pressure anisotropyfrom density-dependent magnetic field is not big, we cancalculate the properties of QSs under magnetic field byusing isotropic TOV equation approximately, and we findthe mass difference of magnetars by using transverse andlongitudinal pressure is also very small.In Fig. 10, we show the parameter region for a0 andGV

to produce 2 solar mass QSs under B0 = 4× 1018G andby using longitudinal pressure and transverse pressure,respectively. The parameter sets of a0 and GV on thistwo lines can describe 2 solar mass QSs. One can see thatin order to produce 2 solar mass QSs, if the repulsive vec-tor interaction is stronger, the needed contribution frommagnetized gluons is smaller, or vice versa. It is alsoobserved that larger a0 and GV parameters are neededto produce 2 solar mass QSs if one use the longitudi-nal pressure density. However, when the magnitude ofrepulsive vector interaction increases, it is noticed thatthe difference between using longitudinal pressure andtransverse pressure to produce 2 solar mass QSs becomessmaller, because the pressure anisotropy decreases when

11

0.0 0.2 0.4 0.6 0.8 1.00.00

0.02

0.04

with P

GV/G

S

a 0B 0= 4e18G

0 = 0.003

= 3with P

FIG. 10: (Color online)The parameter region for a0 and GV

to produce 2 solar mass QSs under B0 = 4 × 1018G and byusing longitudinal pressure and transverse pressure, respec-tively. The parameter sets of a0 and GV on this two lines canproduce 2 solar mass QSs.

large contribution from the repulsive interaction in thevector-isoscalar channel is considered, which can be readfrom Fig. 5 and Fig. 6.

IV. CONCLUSION AND DISCUSSION

In this work, we construct quark magnetars in theframework of SU(3) NJL model with vector interactionunder strong magnetic field.We investigate the effect of vector-isoscalar and vector-

isovector interaction on the equation of state, and it isfound that the equation of state is not sensitive to thevector-isovector interaction, however, a repulsive interac-tion in the vector-isoscalar channel gives a stiffer equationof state for cold dense quark matter. The result is reason-able because the interaction in the vector-isovector chan-nel shifts the isospin chemical potential, and this effect isnegligible on the equation of state under β-equilibrium,while the repulsive vector-isoscalar interaction shifts thechemical potential to a larger value, which makes theequation of state stiffer.In the presence of magnetic field, the pressure of the

system is shown to be anisotropic along and perpendicu-lar to the magnetic field direction with the former beinggenerally larger than the latter. Gluons will be magne-tized via quark loops, and we also estimate the pressurecontributed from magnetized gluons. Normally the NJLmodel only considers the contribution from quark degreesof freedom on the pressure, which is always underesti-mated. We estimate the pressure density contributedfrom magnetized gluons, which should be positive in or-der to compensate the pressure density of quasi-quarksdescribed by the NJL model. It is found that magnetized

quarks and gluons also give a stiffer equation of state.The sound velocity square is one of fundamental prop-

erties of hot/dense matter, which measures the hardnessor softness of the equation of state. It is known thathot and dense quark matter in the NJL model withoutvector interaction at zero magnetic field is always lessthan 1/3. It is also understood that when the system ismore strongly coupled, the sound velocity square is get-ting smaller, and reach the smallest value at the phasetransition point, where the system is regarded as moststrongly coupled. However, it is found that the soundvelocity square can be larger than 1/3 when a strongrepulsive interaction is introduced, and the sound veloc-ity square can change a lot when strong magnetic fieldis added. Furthermore, the sound velocity square corre-sponding to the transverse pressure density case can evenreach 1 when the magnetized gluon contribution is takeninto account.

Since the pressure anisotropy from density-dependentmagnetic field is small, we can calculate the properties ofQSs under magnetic field by using isotropic TOV equa-tion approximately, and we find the magnetar mass differ-ence by using transverse and longitudinal pressure casesare very small and the maximum mass of quark starcan be enhanced by the contribution from the density-dependent magnetic field, the repulsive interaction in thevector-isovector channel and the magnetized gluon pres-sure. We also give the parameter region for a0 and GV

which can describe the 2 solar mass quark star by us-ing longitudinal pressure and transverse pressure respec-tively.

Acknowledgement.— We thank valuable discus-sions with M. Alford and A. Schmitt on sound veloc-ity of quark matter, with J. Schaffner-Bielich and A.Sedrakian on magnetars, and with I. Shovkovy and A.Vuorinen on pressure from magnetized gluon potential.M. Huang thanks the hospitality of Frankfurt Univer-sity, Tours University and TU Viena, where the finalstage of this work is performed. This work is sup-ported by the National Basic Research Program of China(973 Program) under Contract No. 2015CB856904 and2013CB834405, the NSFC under Grant Nos. 11275213,11275125, 11135011 and 11261130311(CRC 110 by DFGand NSFC), also supported by CAS key project KJCX2-EW-N01, and Youth Innovation Promotion Associationof CAS, the China-France collaboration Project ”CaiYuanpei 2013”, the Shanghai Rising-Star Program un-der grant No. 11QH1401100, the “Shu Guang” projectsupported by Shanghai Municipal Education Commis-sion, and Shanghai Education Development Foundation,the Program for Professor of Special Appointment (East-ern Scholar) at Shanghai Institutions of Higher Learning,and the Science and Technology Commission of ShanghaiMunicipality (11DZ2260700).

12

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