Hadronic and hybrid stars subject to density dependent magnetic fields
aa r X i v : . [ a s t r o - ph . S R ] N ov Hadronic and hybrid stars subject to density dependent magneticfields
R. Casali, ∗ L. B. Castro, † and D. P. Menezes ‡ Departamento de F´ısica CFM, Universidade Federal de Santa Catarina,Florian´opolis, SC CP. 476, CEP 88.040-900, Brazil
Abstract
In the light of the very massive neutron stars recently detected and the new possible constraintsfor the radii of these compact objects, we revisit some equations of state obtained for hadronicand hybrid stars under the influence of strong magnetic fields. We present our results for hadronicmatter taking into account the effects of the inclusion of anomalous magnetic moment. Addi-tionally, the case of hybrid stars under the influence of strong magnetic fields is considered. Westudy the structure of hybrid stars based on the Maxwell condition (without a mixed phase), wherethe hadron phase is described by the non-linear Walecka model (NLW) and the quark phase bythe Nambu-Jona-Lasinio model (NJL). The mass-radius relation for each case are calculated anddiscussed. We show that both hadronic and hybrid stars can bear very high masses and radiicompatible with the recently observed high mass neutron stars.
PACS numbers: 12.39.Ki, 14.20.Jn, 26.60.Kp ∗ rcasali@fisica.ufsc.br † [email protected] ‡ [email protected] . INTRODUCTION The study of neutron stars provides an excellent laboratory for the understanding ofdense matter under extreme conditions. A typical neutron star has a mass of the order of1 − M ⊙ and a radius of the order of 11 Km, its temperature stands around 11 K rightafter its birth, followed by a rapid cooling process led by neutrino emission. Although theconventional models of neutron stars assume that dense matter is composed of hadrons andleptons, as the density increases inwards, the Fermi level of the nucleons increases to valuesabove the mass threshold of heavier particles, opening the possibility that another particleis created, reducing the total energy. Baryon number conservation, violation of strangenessand the Pauli exclusion principle guarantee this mechanism. The same phenomenon isresponsible for the reduction of the total pressure. On the other hand, the Bodmer-Wittenconjecture [1–3] states that quarks could be deconfined from the hadrons, forming a stablequark matter. This raises questions about the true constitution of ground state matterat high densities [4, 5] and arises the possibility that compact stars can be constituted ofpure deconfined quark matter or perhaps be hybrid stars, containing in their core a purequark phase or a non-homogeneous mixed quark-hadron phase, whose existence is a sourceof intense discussions in the literature [5–17]. Most neutron stars have masses of the order of1 . M ⊙ , but more recently, at least two pulsars, PSR J1614-2230 [18] and PSR J0348+0432[19] were confirmed to bear masses of the order of 2 M ⊙ . If one believes that a uniqueequation of state (EOS) has to be able to account for all possible observational data, a hardEOS at high densities is then mandatory.Neutron stars generally manifest themselves as pulsars, which are powered by their rota-tion energy or as accreting X-ray binaries, which are powered by the gravitational energy.Some compact objects, known as magnetars, do not fit into any of these categories. Theyare normally isolated neutron stars whose main power source is the magnetic field and twoclasses have been discovered: the soft gamma-repeaters that are x-ray transient sources andthe anomalous x-ray pulsars, a class of persistent x-ray sources with no sign of a binarycompanion. Although not very many magnetars have been confirmed so far (around twodozens), they are expected to constitute up to 10% of the total neutron star population.Hence, magnetars are extremely magnetized neutron stars, with magnetic fields reaching B = 10 G at the surface and central magnetic fields that could reach even higher values221, 22].At such high range of magnitudes magnetic fields can interfere on the thermodynamicand hydrodynamic properties [23–26], causing anisotropy. According to calculations withfree fermion systems at zero [27] and finite temperature [28], an upper limit for the valueof the magnetic field can be established if anisotropic effects are to be disregarded. Thisupper limit depends both on the temperature of the system and on the inclusion (or not) ofanomalous magnetic moments. Temperature washes out anisotropic effects on the pressureof the system and anomalous magnetic moments enhance them [28]. If the same problemis tackled in a system subject to stellar matter conditions, one has to take into account notonly anisotropic effects due to matter contribution [31], but also due to the pure magneticfield contribution that arises from the eletromagnetic tensor [25, 29, 30]. In this case,calculations done with different models indicate that the maximum magnetic field that canbe used in order to avoid anisotropic effect is of the order of B = 10 G, for a baryonicchemical potential equal to µ B = 1500 MeV in zero temperature systems. Nevertheless,when a magnetic field that varies with density (or analogously, with the baryonic chemicalpotential) is considered as in [32], a slightly higher magnetic field can be considered becausestability is maintained up to higher densities [29]. Hence, in the present work we considerthe maximum possible magnetic field that can still be used so that anisotropic effects canbe neglected and it is of the order of B = 10 . G ≃ . × G. One should bear inmind that these very high densities (corresponding to a baryon chemical potential of theorder of 1500 MeV) are not always reached in the core of hadronic and hybrid stars. Animportant consideration in the choice of the values of reasonable magnetic fields is the errorin the use of the Tolman-Oppenheimer-Volkoff equations, valid only for homogeneous andisotropic systems. According to a recent estimate [20], the error in the calculation of thestellar mass is very small, i.e., around 10 − − − M ⊙ for fields of the order of B = 10 Gand 10 − − − M ⊙ for fields of the order of B = 10 G. Hence, we believe that our choicefor the maximum magnetic field as 3 . × G is very reasonable.Another possible complication that the introduction of a magnetic field can bring is thatit is a source of gravitational energy. The virial theorem sets an upper limit so that magneticfields do not imply on a gravitational collapse of the magnetar. As said above, we applya density dependent magnetic field [32] on our equations, and this ensures no gravitationalsetbacks. 3he influence of strong magnetic fields on the quark-hadron phase transition was firstdiscussed in [32], using a Dirac-Hartree-Fock approach within a mean-field approximationto describe both the hadronic and the quark phases. For the hadronic matter a system ofprotons, neutrons and electrons was considered, and for the quark phase the MIT bag modelwith one-gluon exchange was used. A very hard quark equation of state (EOS) was obtainedso that the hybrid star did not have a quark core. The authors concluded that compactstars have a smaller maximum mass in the presence of strong magnetic fields, a result thatdoes not agree with other more recent works where hadronic stars [34–37] or quark stars[38, 39] in the presence of strong magnetic fields have been studied.If strong magnetic fields are considered, contributions from the anomalous magnetic mo-ments (AMM) of the nucleons and hyperons should also be taken into account. Experimentalmeasurements find that k p = µ N ( g p / −
1) for protons and k n = µ N g p / µ N is the nuclear magneton, g p = 5 .
58 and g n = − .
83 are the Landau g-factors of protonsand neutrons, respectively. In [33] the proton and neutron AMM controbutions to hadronicEOS were computed for the first time and later, it was extended to include the contribu-tion from the eight lightes baryons [34]. Although the general conclusion was that a strongmagnetic field softens the EOS, which is not true if the pure electromagetic field contribu-tion is adequately included, the authors pointed out that the AMM stiffens the EOS. Thisproblem was then revisited and the EOS was obtained with a density dependent model andthe inclusion of the scalar-isovector δ mesons, which were shown to be important for lowmass stars [6]. In neither of these works the stellar maximum masses were computed andthe magnetic fields considered were always very high.In the present work we first study magnetars composed of hadronic matter only. We con-sider the inclusion of the anomalous magnetic moments of all the particles in the baryonicoctet and its effects on stellar properties. We describe the hadronic matter within the frame-work of the relativistic non-linear Walecka model (NLW) [40]. We also consider a magneticfield that increases, in a density dependent way, from the surface (10 G ) to the interior ofthe star. Comparing our work with [36] we show that significant differences on maximummasses and their respective radii for stronger magnetic fields can be obtained depending onthe choice of the parameters for slow and fast decays of the density dependent magneticfield. This means that different combinations of parameters can generate controllable valuesfor masses and radii, as expected from the results obtained in [30].4dditionally we study hybrid stars under the influence of magnetic fields. The structure ofhybrid stars is based on the Maxwell condition (without mixed phase), the hadronic matteris again described by the NLW [40] and the quark matter by the Nambu-Jona-Lasinio (NJL)model [41] composed of quarks up ( u ), down ( d ) and strange ( s ) in β -equilibrium. We alsoassume the density-dependent magnetic field and we choose the same two sets of valuesfor the parameters β and γ for slow and fast decays of the density dependent magneticfield as in the hadronic case. We show that hybrid stars have a larger maximum mass inthe presence of strong magnetic field as compared with the results presented in [42] andthat the slow decays produce smaller maximum masses, but larger radii. Our result for thevalues of maximum masses and radii for a weak magnetic field are in agreement with theresults obtained for B = 0 G [42]. The macroscopic properties of hybrid stars under effectsof magnetic fields have already been studied in the literature [43], where the quark phasewas described by the MIT bag model. A qualitative analysis shows that the properties ofthe stars obtained in both cases are very similar, but a quantitative analysis shows that weobtain higher maximum masses and radii for weaker magnetic fields with the NJL modeldescribing the quark core. This consideration is model and parameter dependent and, hence,has to be taken with care. We discuss these differences again when we present our results.The organization of this work follows: in Sec. II, we give a brief review of the formalismused to describe the hadronic and quark phases under a magnetic fieldand discuss the condi-tions for building of a hybrid star with the Maxwell construction. In Sec. III we present ourresults for the inclusion of a density dependent magnetic field on the total energy densityand total pressure, particle fractions and mass-radius relation for hadronic and hybrid stars.Finally, in Sec. IV we present our main conclusions. II. FORMALISM
In this section we present an overview of the equations used to describe the hadronic (sub-section A), quark (subsection B) and hybrid (subsection C) phases. We describe hadronicmatter within the framework of the relativistic non-linear Walecka model (NLW) [40]. Thequark matter is described by SU(3) version of the the Nambu-Jona-Lasinio (NJL) model[45]. Hybrid matter is built using the Maxwell conditions.5 . Hadronic phase under a magnetic field
For the description of the equation of state (EOS) of hadronic matter, we employ a field-theoretical approach in which the baryons interact via the exchange of σ − ω − ρ mesons inthe presence of a magnetic field B along the z − axis. The total lagrangian density reads: L H = X b L b + L m + X l L l + L B . (1)where L b , L m , L l and L B are the baryons, mesons, leptons and electromagnetic field La-grangians, respectively, and are given by L b = ψ b ( iγ µ ∂ µ − q b γ µ A µ − m b + g σb σ − g ωb γ µ ω µ − g ρb τ b γ µ ρ µ − k b σ µν F µν ) ψ b , (2) L m = 12 ( ∂ µ σ∂ µ σ − m σ σ ) − U ( σ ) + 12 m ω ω µ ω µ −
14 Ω µν Ω µν + 12 m ρ ~ρ µ · ~ρ µ − P µν P µν , (3) L l = ψ l ( iγ µ ∂ µ − q l γ µ A µ − m l ) ψ l , (4) L B = − F µν F µν . (5)where he b -sum runs over the baryonic octet b ≡ N ( p, n ) , Λ , Σ ± , , Ξ − , , ψ b is the corre-sponding baryon Dirac field, whose interactions are mediated by the σ scalar, ω µ isoscalar-vector and ρ µ isovector-vector meson fields. The baryon mass and isospin projection aredenoted by m b and τ b , respectively, and the masses of the mesons are m σ = 512 MeV, m ω = 783 MeV and m ρ = 770 MeV. The strong interaction couplings of the nucleons withthe meson fields are denoted by g σN = 8 . g ωN = 10 .
610 and g ρN = 8 . g iH = X iH g iN , where the values of X iH are chosen as X σH = 0 . X ωH = X ρH = 0 .
783 [5]. The term U ( σ ) = bm n ( g σN σ ) − c ( g σN σ ) denotesthe scalar self-interactions [47–49], with c = − . b = 0 . µν = ∂ µ ω ν − ∂ ν ω µ , P µν = ∂ µ ~ρ ν − ∂ ν ~ρ µ − g ρb ( ~ρ µ × ~ρ ν ) and F µν = ∂ µ A ν − ∂ ν A µ . The baryon anomalous magneticmoments (AMM) are introduced via the coupling of the baryons to the electromagnetic fieldtensor with σ µν = i [ γ µ , γ ν ] and the strength κ b = ( µ b /µ N ) − q b ( m p /m b ), where q p and m p µ b and m b are the magnetic moment and massesof the baryons, whose values can be seen in TABLE I. The l -sum runs over the two lightestleptons l ≡ e, µ and ψ l is the lepton Dirac field. The symmetric nuclear matter propertiesat saturation density adopted in this work are given by the GM1 parametrization [46], withcompressibility K = 300 (MeV), binding energy B/A = − . a sym = 32 . L = 94 (MeV), saturation density ρ = 0 .
153 ( f m − ) and nucleonmass m = 938 (MeV). Baryon p n Λ Σ + Σ Σ − Ξ Ξ − M b (MeV) 938 938 1116 1193 1193 1193 1318 1318 q b µ b /µ N k b meson, contrary to [34, 43]. The following equations present the scalar and vector densities for the charged and un-charged baryons [43], respectively: ρ sb = | q b | Bm ∗ b π ν max X ν X s ¯ m cb p m ∗ b + 2 ν | q b | B ln (cid:12)(cid:12)(cid:12)(cid:12) k bF,ν,s + E bF ¯ m cb (cid:12)(cid:12)(cid:12)(cid:12) , (6) ρ vb = | q b | B π ν max X ν X s k bF,ν,s , (7) ρ sb = m ∗ b π X s (cid:20) E bF k bF,s − ¯ m b ln (cid:12)(cid:12)(cid:12)(cid:12) k bF,s + E bF ¯ m b (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) , (8) ρ vb = 12 π X s (cid:20)
13 ( k bF,s ) − sµ N k b B (cid:18) ¯ m b k bF,s + ( E bF ) (cid:18) arcsin (cid:18) ¯ m b E bF (cid:19) − π (cid:19)(cid:19)(cid:21) . (9)where m ∗ b = m b − g σ σ , ¯ m cb = p m ∗ b + 2 ν | q b | B − sµ N k b B and ¯ m b = m ∗ b − sµ N k b B . ν = n + − sgn( q b ) s = 0 , , , ... are the Landau levels for the fermions with electric charge q b , s is the spin and assumes values +1 for spin up and − E bν,s = r ( k bz ) + ( q m ∗ b + 2 ν | q b | B − sµ N k b B ) + g ωb ω + τ b g ρb ρ (10) E bs = r ( k bz ) + ( q m ∗ b + k ⊥ − sµ N k b B ) + g ωb ω + τ b g ρb ρ , (11)where k ⊥ = k x + k y . The Fermi momenta k bF,ν,s of the charged baryons and k bF,s of theuncharged baryons and their relationship with the Fermi energies of the charged baryons E bF,ν,s and uncharged baryons E bF,s can be written as:( k bF,ν,s ) = ( E bF,ν,s ) − ( ¯ m cb ) (12)( k bF,s ) = ( E bF,s ) − ¯ m b . (13)For the leptons, the vector density is given by: ρ vl = | q l | B π ν max X ν X s k lF,ν,s , (14)where k lF,ν,s is the lepton Fermi momentum, which is related to the Fermi energy E lF,ν,s by:( k lF,ν,s ) = ( E lF,ν,s ) − ¯ m l , l = e, µ, (15)with ¯ m l = m l + 2 ν | q l | B . The summation over the Landau level runs until ν max , this is thelargest value of ν for which the square of Fermi momenta of the particle is still positive andcorresponds to the closest integer, from below to: ν max = (cid:20) ( E lF ) − m l | q l | B (cid:21) , leptons (16) ν max = (cid:20) ( E bF + sµ N k b B ) − m ∗ b | q b | B (cid:21) , charged baryons . (17)The chemical potentials of baryons and leptons are: µ b = E bF + g ωb ω + τ b g ρb ρ , (18) µ l = E lF = q ( k lF,ν,s ) + m l + 2 ν | q l | B . (19)From the Lagrangian density (1), and mean-field approximation, the energy density isgiven by ε m = X b ( ε cb + ε nb ) + 12 m σ σ + U ( σ ) + 12 m ω ω + 12 m ρ ρ , (20)8here the expressions for the energy densities of charged baryons ε cb and neutral baryons ε nb are, respectively, given by: ε cb = | q b | B π ν max X ν X s (cid:20) k bF,ν,s E bF + ( ¯ m cb ) ln (cid:12)(cid:12)(cid:12)(cid:12) k bF,ν,s + E bF ¯ m cb (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) , (21) ε nb = 14 π X s (cid:20) k bF,ν,s ( E bF ) − sµ N k b B ( E bF ) (cid:18) arcsin (cid:18) ¯ m b E bF (cid:19) − π (cid:19) − (cid:18) sµ N k b B + 14 ¯ m b (cid:19)(cid:18) ¯ m b k bF,ν,s E bF + ¯ m b ln (cid:12)(cid:12)(cid:12)(cid:12) E bF + k bF,ν,s ¯ m b (cid:12)(cid:12)(cid:12)(cid:12)(cid:19)(cid:21) . (22)The expression for the energy density of leptons ε l reads ε l = | q l | B π X l ν max X ν X s (cid:20) k lF,ν,s E lF + ¯ m l ln (cid:12)(cid:12)(cid:12)(cid:12) k lF,ν,s + E lF ¯ m l (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) . (23)The pressures of baryons and leptons are: P m = µ n X b ρ vb − ε m ,P l = X l µ l ρ vl − ε l , (24)where the expression of the vector densities ρ vb and ρ vl are given in (7) and (14), respectively.The total energy density and the total pressure of the system can be written by adding thecorresponding contributions of the magnetic field: ε H = ε m + ε l + (cid:18) B (cid:16) ρρ (cid:17) (cid:19) , P H = P m + P l + (cid:18) B (cid:16) ρρ (cid:17) (cid:19) B. Quark phase under a magnetic field
For the description of the equation of state (EOS) of quark matter, we consider a (threeflavor) quark matter in β equilibrium with magnetic fields. We introduce the lagrangiandensity L Q = L f + L l + L B , (26)where the quark sector is described by the SU(3) version of the Nambu-Jona-Lasinio model(NJL) [50], which includes a scalar-pseudoscalar interaction and the t`Hooft six-fermioninteraction. The lagrangian density L l and L B are given by (4) and (5), respectively. Thelagrangian density L f is defined by L f = ¯ ψ f [ γ µ ( i∂ µ − q f A µ ) − ˆ m c ] ψ f + L sym + L det , (27)9ith L sym = G X a =0 h(cid:0) ¯ ψ f λ a ψ f (cid:1) + (cid:0) ¯ ψ f iγ λ a ψ f (cid:1) i , (28) L det = − K ( d + + d − ) , (29)where G and K are coupling constants, d ± = det f (cid:2) ¯ ψ f (1 ± γ ) ψ f (cid:3) , ψ f = ( u, d, s ) T representsa quark field with three flavors, ˆ m c = diag f ( m u , m d , m s ) is the corresponding (current) massmatrix while q f represents the quark electric charge; λ = p / I , where I is the unit matrixin the three flavor space; and 0 < λ a ≤ m u = m d = m s . In the mean-field approximation the lagrangian density (27) can be writtenas [39] L MFA f = ¯ ψ f h γ µ ( i∂ µ − q f A µ ) − ˆ M i ψ f − G (cid:0) φ u + φ d + φ s (cid:1) + 4 Kφ u φ d φ s , (30)where ˆ M is a diagonal matrix with elements defined by the effective quark masses M i = m i − Gφ i + 2 Kφ j φ k (31)with ( i, j, k ) being some permutation of ( u, d, s ).Now, we need to evaluate the grand-canonical thermodynamical potential for the three-flavor quark sector, which can be written as Ω f = − P f = ε f − P f µ f ρ f − Ω , where P f represents the pressure, ε f the energy density, µ f the chemical potential and Ω ensures thatΩ f = 0 in the vacuum. In the mean-field approximation the pressure can be written as P f = θ u + θ d + θ s − G (cid:0) φ u + φ d + φ s (cid:1) + 4 Kφ u φ d φ s . (32)So, to determine the EOS for the SU(3) NJL model at finite density and in the presenceof a magnetic field we need to know the condensates φ f and the contribution from the gasof quasiparticles θ f . Both quantities have been evaluated with great detail in Refs. [38, 39].For this model we split the degeneracy of each quark into the spin degeneracy and colordegeneracy N c . The difference now is that both spin projections contribute for Landaulevels ν >
0, but only one of them contributes for ν = 0. The contribution from the gas ofquasiparticles for each flavor θ f = (cid:0) θ vac f + θ mag f + θ med f (cid:1) M f contains 3 different contributions:the vacuum, the magnetic and the medium one given by θ vac f = − N c π (cid:26) M f ln (cid:20) (Λ + ǫ Λ ) M f (cid:21) − ǫ Λ Λ (cid:0) Λ + ǫ (cid:1)(cid:27) , (33)10 mag f = N c ( | q f | B ) π (cid:20) ζ (1 , ( − , x f ) −
12 ( x f − x f ) ln x f + x f (cid:21) , (34) θ med f = X ν α ν N c | q f | B π h µ f q µ f − s f ( ν, B ) − s f ( ν, B ) ln µ f + q µ f − s f ( ν, B ) s f ( ν, B ) , (35)where s f ( ν, B ) = q M f + 2 | q f | Bν is the constituent mass of each quark modified by themagnetic field, ǫ Λ = q Λ + M f with Λ representing a non covariant ultra violet cut off [51], x f = M f / (2 | q f | B ) and ζ (1 , ( − , x f ) = dζ ( z, x f ) /dz | z = − with ζ ( z, x f ) being the Riemann-Hurwitz zeta function.Each of the quark condensates, φ f = h ¯ ψ f ψ f i = ( φ vac f + φ mag f + φ med f ) M f also contains 3different contributions: the vacuum, the magnetic and the medium one given by [38] φ vac f = − N c M f π (cid:20) Λ ǫ Λ − M f ln (cid:18) Λ + ǫ Λ M f (cid:19)(cid:21) , (36) φ mag f = − N c M f | q f | B π (cid:20) ln Γ( x f ) −
12 ln(2 π )+ x f −
12 (2 x f −
1) ln( x f ) (cid:21) , (37) φ med f = X ν α ν N c M f | q f | B π × ln µ f + q µ f − s f ( ν, B ) s f ( ν, B ) . (38)The quark contribution to the energy density is ε f = − P f + X f µ f ρ f + Ω , (39)where the density ρ f corresponds to each different flavor and is given by ρ f = X ν α ν N c | q f | B π k F,f , (40)with k F,f = q µ f − s f ( ν, B ) . 11he leptonic contribution for the pressure reads P l = X l X ν α ν | q l | B π (cid:20) µ l q µ l − s l ( ν, B ) − s l ( ν, B ) ln µ l + p µ l − s l ( ν, B ) s l ( ν, B ) ! , (41)and finally the vector density and energy density for leptons are given by the equations (14)and (23), respectively. Therefore, the total energy density and the total pressure of thesystem are given by adding the corresponding contribution of the magnetic field ε Q = ε f + ε l + (cid:16) B (cid:16) ρρ (cid:17)(cid:17) , P Q = P f + P l + (cid:16) B (cid:16) ρρ (cid:17)(cid:17) . (42)The parameter sets of the NJL model used in the present work are given in TABLE II. Λ G Λ K Λ m u,d m s Parameter set (MeV) (MeV) (MeV)SU(3) HK [45, 52] 631.4 1.835 9.29 5.5 135.7SU(3) RKH[45, 53] 602.3 1.835 12.36 5.5 140.7TABLE II. Parameter sets for the NJL SU(3) model.
C. Hybrid star
There are two ways of constructing a hybrid star, one with a mixed phase and anotherwithout a mixed phase (hadron and quark phases are in direct contact). In the first case,neutron and electron chemical potentials are continuous throughout the stellar matter, basedon the standard thermodynamical rules for phase coexistence known as Gibbs condition[5, 42, 54, 55]. In the second case, the electron chemical potential suffers a discontinuitybecause only the neutron chemical potential is imposed to be continuous. The conditionunderlying the fact that only a single chemical potential is common to both phases is knownas Maxwell condition. Recently, some authors calculated macroscopic quantities as radiiand masses for hybrid stars with and without mixed phase and they concluded that thedifferences were not relevant [42, 54, 55]. Inspired by these results, in the present work wechoose the simpler construction for a hybrid star which is based on the Maxwell condition.12or the construction of a hybrid star with the Maxwell condition, we just need to findthe point where µ H n = µ Q n and P H = P Q . (43)To construct a hybrid star we consider a system constituted by 8 baryons in the hadron phaseand 3 quarks in the quark phase. For the EOS of the hadronic phase we use equation (25)with κ b = 0 in the equations (17), (21) and (22) (i.e. without anomalous magnetic moment)and for the EOS of the quark phase we use equation (42). III. RESULTS
In the sequel we consider two different systems under a strong magnetic field: (A) bary-onic, and (B) hybrid matters. In both cases the effects of strong magnetic fields on themacroscopic properties of compact stars were obtained from the integration of the Tolman-Oppenheimer-Volkoff (TOV) equations [56], using as input the EOS obtained from subsec-tions II A and II C for baryonic and hybrid matters, respectively.We assume that the density-dependent magnetic field B in the EOS is given by [32, 36,39, 43, 57]: B (cid:18) ρρ (cid:19) = B surf + B (cid:26) − exp (cid:20) − β (cid:18) ρρ (cid:19) γ (cid:21)(cid:27) , (44)where ρ = P b ρ vb is the baryon density, ρ is the saturation density, B surf is the magneticfield on the surface taken equal to 10 G in agreement with observational values and B is the magnetic field for larger densities. The remaining parameters β and γ are chosen toreproduce two behaviors of the magnetic field: a fast decay with γ = 3 .
00 and β = 0 . γ = 2 .
00 and β = 0 .
05 whose curves can be seen in Fig. 1.In [30], different profiles for the density dependence of the magnetic field were studiedin the context of hadronic stars and the authors concluded that the equation of state isinsensitive to magnetic fields lower than 10 G, a behaviour already observed in [36, 38, 39]for different models. Moreover, they found that for magnetic fields higher than 10 G,matter becomes unstable due to the increase of anisotropic effects on the pressure. Takinginto account that 10 G seems to establish two different boundaries and the anisotropiceffects around 3 . × G is small [29], resulting in a small error in the stellar masses if theTOV equations are used [20], we next use two values for the magnetic field, namely 10 Gand 3 . × G and include the effects of the AMM.13
0 0.2 0.4 0.6 0.8 1 1.2 B ( G ) ρ (fm -3 ) B FAST B SLOW
FIG. 1. Variable density dependent magnetic fields for B = 10 G (lower curves) and B =3 . × G (upper curves), for FAST (green line) and SLOW (red line) decays.
A. Baryonic matter
In Fig. 2 we show the equation of state for hadronic matter under the influence of B =10 G (left panels) and B = 3 . × G (right panels) magnetic fields, and with threepossible conditions for the inclusion of the anomalous magnetic moment, ( k b = 0) for nocorrections, ( k n ) and ( k p ) for the inclusion of the neutron and proton anomalous magneticmoments and ( k n,p,hyp ) for the inclusion of the corrections for all the baryons, both for slow(upper panels) and fast (lower panels) decays. We see no great difference in any of the casesstudied for B = 10 G . As expected, they practically coincide with the non-magnetizedcurve (in red), as can be seen in the zoomed boxes.At B = 3 . × G we notice the stiffening caused by the larger magnetic field applied,on both fast and slow cases. On the zoomed boxes it is possible to notice the stiffeningeffects of the inclusion of the corrections due to the magnetic moments, even when only κ n and κ p are included. This effect is stronger for higher energy densities, which coincidesqualitatively with the the effects caused on nucleonic matter found in [6]. The effect of theinclusion of the anomalous magnetic moment of all the hyperons only becomes evident athigher values of the energy density.In Fig. 3 we present the particle fractions for hadronic matter with the inclusion ofthe anomalous magnetic moment of all particles for B = 10 G (left panel) and B =3 . × G (right panel). Comparing the two graphs, we see different behaviors of someabundances caused by the increase in the intensity of the magnetic field, like the kinks14 P ( f m - ) ε (fm -4 )B = 10 GB= 0 Gk b =0k n ,k p k n,p,hyp P ( f m - ) ε (fm -4 )B = 3.1x 10 G 1.2 1.5 3.9 4.2 0 0.5 1 1.5 2 2.5 0 1 2 3 4 5 6 P ( f m - ) ε (fm -4 )B = 10 G 1.5 1.6 1.7 1.8 5.4 5.7 6 0 0.5 1 1.5 2 2.5 0 1 2 3 4 5 6 P ( f m - ) ε (fm -4 )B = 3.1x 10 G 1.5 1.8 4.5 4.8
FIG. 2. Equations of State for hadronic matter, with the inclusion of the baryonic octet. Threecases for the inclusion of the anomalous magnetic moments are considered, for slow (upper panels)and fast (lower panels) decays, and for B = 10 G (left panels) and B = 3 . × G (rightpanels). produced on the populations of charged particles, due to the filling of Landau levels.In Fig. 4 we plot the mass-radius relation of hadronic matter under the influence of B = 10 G (left panel) and B = 3 . × G (right panel) magnetic fields, and withthe three possible conditions for the inclusion of the magnetic moment corrections presentedbefore, both for slow (upper panels) and fast (lower panels) decays. The tails of the hadronicmatter were obtained with the insertion of the BPS EOS [58]. On the left as expected, thecurves for B = 10 G present maximum masses and radii that do not differ greatly fromthose found for the non-magnetized EOS (in red), used for comparison, as already expectedfrom previous results in the literature [33]. On the right, for the upper and lower panels,15 -3 -2 -1
0 1 2 3 4 5 6 7 8 Y i ρ / ρ e - µ np Λ Σ - Σ Σ + Ξ - Ξ B = 10 G 10 -3 -2 -1
0 1 2 3 4 5 6 7 8 Y i ρ / ρ e - µ np Λ Σ - Σ Σ + Ξ - Ξ B = 3.1x 10 G FIG. 3. Particle fractions for hadronic matter, with the inclusion of the anomalous magneticmoment for all the baryonic octet, for B = 10 G (left panel) and B = 3 . × G (rightpanel). due to the stiffening of the curves for B = 3 . × G , caused by the stronger magneticfield, the effects caused by the different behaviors on the decay of the equation (44), becomemore evident, and the extensions of these effects on equation (25), from slow to fast decay,generate higher maximum masses and lower radii, for all of the anomalous magnetic momentconditions considered. This can be seen in TABLE III.16 M / M R (Km) B = 10 GB=0 Gk b =0k n ,k p k n,p,hyp M / M R (Km) B = 3.1x 10 G 0 0.5 1 1.5 2 2.5 10 12 14 16 18 20 M / M R (Km) B = 10 G 0 0.5 1 1.5 2 2.5 10 12 14 16 18 20 M / M R (Km) B = 3.1x 10 G FIG. 4. Mass-radius curves for hadronic matter, with the inclusion of the baryonic octet. Threecases of anomalous magnetic moments are considered, k b = 0 without magnetic moment corrections, k n and k p with corrections for neutrons and protons and k n,p,hyp with the anomalous magneticmoment of all baryons. For B = 10 G (left panel) and B = 3 . × G (right panel), for slow(upper panels) and fast (lower panels) decays. It is well known that the inclusion of the hyperons softens the EOS, reducing the max-imum stellar mass. However, for high values of the magnetic fields, we see from III thatthe progressive inclusion of the anomalous magnetic moment stiffens the EOS, first with thecurves with only the neutron ( k n ) and proton ( k p ) corrections and then with the AMM of allbaryons ( k n,p,hyp ), causing the increase of the maximum mass. This happens both with theFAST and SLOW cases. For the lower value of the magnetic field, the macroscopic resultsdepend very little on the inclusion of the AMM, as they also depend only slightly on B .For the higher value of the magnetic field, when we compare FAST and SLOW cases, we17 agnetic Field AMM FAST SLOW M max R ε c µ n ( ε c ) µ e ( ε c ) R( M = 1 . M ) M max R ε c µ n ( ε c ) µ e ( ε c ) R( M = 1 . M )( M ) (Km) (fm − ) (MeV) (MeV) (Km) ( M ) (Km) (fm − ) (MeV) (MeV) (Km) B = 0 G κ b = 0 2.00 11.87 5.93 1577.5 122.1 13.90 2.00 11.87 5.93 1577.5 122.1 13.90 B = 10 G κ n,p κ n,p,hyp κ b = 0 2.36 12.37 5.27 1427.3 150.2 14.02 2.29 12.58 5.11 1446.1 150.2 14.29 B = 3 . × G κ n,p κ n,p,hyp µ n ( ε c ) and µ e ( ε c ) are the chemicalpotentials for neutron and electron at the central energy density ε c , R( M = 1 . M ) is the radius for a M = 1 . M for this configurations. otice that independently of the AMM condition, the maximum masses of the FAST casesare always larger than those of the respective SLOW case. We attribute this to the greaterstiffness of the EOS caused by the faster decay in equation (44). Notice, however, that thisbehaviour depends on the strenght of the magnetic field in the core, as discussed in [30, 43].Comparing our results with [36] we confirm that the inclusion of low magnetic fields, ofthe order of B = 10 G do not produce any significant effect neither on the EOS nor onthe particle fractions. No nozzles are noticed, because due to the low magnetic field thereare several Landau levels to be filled, even at low densities. Still comparing our results for B = 10 G , we found for our parametrization without the anomalous magnetic momentcorrections ( k b = 0) a maximum mass M max = 2 . M and a radius R = 11 .
87 Kmcompatible with the M max = 2 . M and R = 11 .
86 Km found in ref. [36].When we compare our results for B = 3 . × G with those in [36] we confirm thenozzles on the particle fractions at lower densities, related to the van Alphen oscillationsrelated to the creation of a new Landau levels, and a behavior close to the continuous athigher densities, due to the higher number of filled Landau levels. On the other hand,we also find some different results, mainly because of the choice of the decay parametersof equation (44). For instance, our results for the maximum mass of the EOS with nomagnetic moment corrections ( k b = 0) are M max = 2 . M and R = 12 .
37 Km. In ref. [36],for β = 6 . × − and γ = 3 .
5, the authors obtained M max = 2 . M and R = 11 .
80 Km,which corroborates the conclusions that the choice of parameters in the density dependentmagnetic field influences the macroscopic properties of the stars. We can observe that thecentral energy densities attained do not show a well established pattern: for low magneticfields, they decrease when all AMM are included and for high magnetic fields they oscillatewhen the AMM are added. As already mentioned in the Introduction, we can see thatbaryonic chemical potentials higher than 1500 MeV are not reached for strong magneticfields, independently of the chosen decay rate. For the sake of completeness, we have alsoadded the radii results for canonical 1.4 M ⊙ stars, where we can see that they are quite largefor our choice of magnetic field decay rates.19 P ( f m - ) ε (fm -4 )B =0 GB =10 GB =3.1x10 G 0 0.5 1 1.5 2 2.5 0 1 2 3 4 5 6 7 8 9 P ( f m - ) ε (fm -4 )B =0 GB =10 GB =3.1x10 G FIG. 5. EOS for the hybrid star without mixed phase built with the GM1 and SU(3) HK NJLparametrization, for slow (left panel) and fast (right panel) decays. P ( f m - ) ε (fm -4 )B =0 GB =10 GB =3.1x10 G 0 0.5 1 1.5 2 2.5 0 1 2 3 4 5 6 7 8 9 P ( f m - ) ε (fm -4 )B =0 GB =10 GB =3.1x10 G FIG. 6. EOS for the hybrid star without mixed phase built with the GM1 and SU(3) RKH NJLparametrizations, slow (left panel) and fast (right panel) decays.
B. Hybrid stars
Now we turn our attention to hybrid stars. To study the effects of strong magnetic fieldson the macroscopic properties of hybrid stars without mixed phase (Maxwell condition) wealso assume the density-dependent magnetic field given in Eq.(44). In this case, we choosethe same two sets of values for the parameters β and γ , a fast varying field defined by β = 0 . γ = 3 .
00 and a slowly varying field with β = 0 .
05 and γ = 2 .
00 as in the last section. InFigs. 5 and 6, we plot the EOS of hybrid stars under the influence of weak ( B = 10 G ) and20trong ( B = 3 . × G ) magnetic fields. We note that as in the hadronic case the EOSfor B = 10 G presents no great difference between FAST and SLOW cases. For the SU(3)HK (SU(3) RKH) parametrization, the onset of the quark phase occurs at P ≈ . f m − , µ n ≈ P ≈ . f m − , µ n ≈ B = 0 G . For B = 10 G , theonset of the quark phase (SU(3) HK) occurs at P ≈ . f m − , µ n ≈ P ≈ . f m − , µ n ≈ B = 10 G also forboth FAST and SLOW cases. For B = 3 . × G the contribution of the magnetic fieldmakes the EOS harder as compared with the EOS for B = 10 G and this effect is reflectedin the higher values of the maximum masses. The presence of a strong magnetic field alsoaffects the onset of the quark phase. In this case, the onset of the quark phase (SU(3) HK)occurs at P ≈ . f m − , µ n ≈ P ≈ . f m − , µ n ≈ P ≈ . f m − , µ n ≈ P ≈ . f m − , µ n ≈ B = 10 G ) andstrong ( B = 3 . × G ) magnetic fields. As in the case of hadronic matter the tails ofthe hybrid stars were obtained with the insertion of the BPS EOS [58]. The values of themaximum masses and radii for a hybrid star are shown in TABLES IV and V. As expected,for B = 10 G the maximum masses and radii do not differ significantly from those foundfor B = 0 in ref. [42]. It is worthwhile to mention that the small difference is consequenceof considering a different central energy equal to 0 . f m − instead of 0 . × − f m − as input to the TOV equations as done in [42]. From TABLES IV and V we can see thatfor B = 10 G the values of the maximum masses and radii for the two parametrizationsof the magnetic field (FAST and SLOW cases) are the same. This result is also expectedbecause the effects caused by the two parametrizations are not evident for a weak magnetic21eld, exactly as in the case previously discussed for hadronic stars.Comparing the chemical neutron chemical potential µ n of the onset of the quark phasewith the one at the central energy density µ n ( ε c ), we can check whether the star in hybridor if it remains a hadronic star. We can see that for the SU(3) RKH parametrizationonly the case of B = 3 . × G (SLOW case) is a hybrid star, all other case with thisparametrization are hadronic stars (including the case of B = 0 G ), since µ n > µ n ( ε c ). Thisdoes not happen when we use the SU(3) HK parametrization. In this case all of the casesconsidered resulted in hybrid stars because µ n < µ n ( ε c ).The macroscopic properties of hybrids stars under effects of magnetic fields have alreadybeen studied in the literature [43]. In ref. [43] the quark phase of the hybrid star is describedby MIT bag model with m u = m d = 5 . m s = 150 MeV and two values for theBag constant (165 MeV) and (180 MeV) . Comparing our results with [43], we obtainhigher maximum masses and radii for weaker magnetic fields. For instance, in ref. [43] for B ≈ . × G and Bag / = 180 MeV the authors obtain M = 1 . M , R = 11 .
56 Km(SLOW) and M = 1 . M , R = 11 .
49 Km (FAST) and for B ≈ . × G andBag / = 165 MeV they obtain M = 1 . M , R = 9 .
80 Km (SLOW) and M = 1 . M , R = 10 .
07 Km (FAST). We obtain higher maximum masses and radii even for a weakmagnetic field B = 10 G . As it is well known, the mass and radius of compact starsobtained with the MIT bag model can be calibrated by increasing the value of the Bagconstant. Nonetheless, it is important to emphasize that it would be desirable that thevalues of the Bag constant were obtained through the study of stability windows [37, 44] forwhich quark matter is absolutely stable. Therefore, even with strong magnetic fields, hybridstars that contain a quark core described by the MIT model with Bag values within the rangeof the stability windows, cannot support large maximum masses. For instance, the stabilitywindows for the MIT bag model are 147 . < Bag / < . . < Bag / < . B = 0 G and B = 7 . × G , respectively [37]. Note, however, that if different correctionswere included in the MIT model, as in [59], hybrid stars with higher maximum masses couldbe attained. Certainly,the stability conditions for quark matter should also be verified whenthe quark phase is described by the NJL model. For zero temperature and zero magneticfield, the NJL model is not absolutely stable either, but for higher values of the magneticfields, it is consistent with the requirements for the existence of stable quark matter [37, 44].22 agnetic Field FAST SLOW M max R ε c µ n ( ε c ) R( M = 1 . M ) M max R ε c µ n ( ε c ) R( M = 1 . M )( M ) (Km) (fm − ) (MeV) (Km) ( M ) (Km) (fm − ) (MeV) (Km) B = 0 G 1.90 12.80 4.57 1360.8 13.86 1.90 12.80 4.57 1360.8 13.86 B = 10 G B = 3 . × G µ n ( ε c ) is the chemical potential for neutron at the central energy density ε c and R( M = 1 . M ) is the radius for a M = 1 . M . agnetic Field FAST SLOW M max R ε c µ n ( ε c ) R( M = 1 . M ) M max R ε c µ n ( ε c ) R( M = 1 . M )( M ) (Km) (fm − ) (MeV) (Km) ( M ) (Km) (fm − ) (MeV) (Km) B = 0 G 1.96 12.52 6.00 1402.7 13.86 1.96 12.52 6.00 1402.7 13.86 B = 10 G B = 3 . × G µ n ( ε c ) is the chemical potential for neutron at the central energy density ε c and R( M = 1 . M ) is the radius for a M = 1 . M . M / M R (Km) B =0 GB =10 GB =3.1x10 G 0 0.5 1 1.5 2 2.5 10 12 14 16 18 20 M / M R (Km) B =0 GB =10 GB =3.1x10 G FIG. 7. Mass-radius curves for hybrid stars without mixed phase built with the GM1 and SU(3)HK NJL parametrizations, SLOW (left panel) and FAST (right panel) cases. M / M R (Km) B =0 GB =10 GB =3.1x10 G 0 0.5 1 1.5 2 2.5 10 12 14 16 18 20 M / M R (Km) B =0 GB =10 GB =3.1x10 G FIG. 8. Mass-radius curves for hybrid stars without mixed phase built with the GM1 and SU(3)RKH NJL parametrizations, SLOW (left panel) and FAST (right panel) cases.
IV. CONCLUSIONS
In the present work we have revisited the calculations of magnetars composed of hadronicmatter only as in [6, 33, 34] and also composed of a quark core as in [43]. In the first case, themain targets were to compute the differences caused by the individual anomalous magneticmoments in the EOS with the inclusion of hyperons, their particle abundances and theresulting stellar properties. In the second case, our aim was to build a hybrid star with aquark core described by the NJL model, instead of the MIT bag model used in [43]. We25ave chosen the Maxwell conditions to construct the hybrid star because our main concernwas the evaluation of the macroscopic stellar properties obtained from different models andit was already shown in [42] that the Gibbs and Maxwell constructions result in practicallythe same results when the NJL model is used for the quark matter.All calculations were performed with two different values for the magnetic field, B = 10 G and B = 3 . × G, the last one being the stronger possible value, for which theanisotropic effects in the pressure can be circumvented, since we have opted to used anisotropic EOS. The magnetic fields were chosen to be density dependent and vary from asurface value of B = 10 G to the two values mentioned above.For the low value of the magnetic field, B = 10 G , the results do not differ from theones obtained for a non-magnetized star. These results are displayed in TABLES III, IVand V and were already expected. When a strong magnetic field, B = 3 . × G, isassumed, some conclusions can be drawn. For hadronic stars, the maximum masses increasewith the inclusion of the anomalous magnetic moments, as expected, since they stiffen theEOS. A fast decay mode for the density dependent magnetic field yields larger maximummasses, what is also seen in hybrid stars. Generally, hybrid stars present lower maximummasses due to softer equations of state, a well know result for non-magnetized stars andlarger radii. The central energy densities do not present a common pattern. Moreover, withthe models and constants chosen in the present work, we can describe the recently detectedneutron stars with masses of the order of 2 M ⊙ [18, 19], contrary to what was found, forinstance in [43] with a quark core described by the MIT bag model and a not too largevalue of the magnetic field. We have also seen that a hybrid star cannot be always obtainedwith both parametrizations of the NJL model used in the present work for magnetized andnon-magnetized matter. Although the EOS was built in such a way that the star could behybrid, the TOV results have shown that the onset of the quark phases sometimes takesplace at energy densities higher than the ones found in the core of the star.Finally, let’s make some comments on the possible values of neutron stars radii. Basedon chiral effective theory, the authors of ref. [60] estimate the radii of the canonical 1 . M ⊙ neutron star to lie in the range 9.7-13.9 Km. More recently, two different analysis of fivequiescent low-mass X-ray binaries in globular clusters resulted in different ranges for neutronstar radii. The first one, in which it was assumed that all neutron stars have the same radii,predicted that they should lie in the range R = 9 . +1 , − . [61]. The second calculation, based on26 Bayesian analysis, foresees radii of all neutron stars to lie in between 10 and 13.1 Km [62].If one believes those are definite constraints, all hadronic and hybrid stars with both zeroand large magnetic fields obtained with the choice of EOS studied in the present work wouldbe ruled out, as can be seen from Figs. 2 and 7. Nevertheless, as already explained, theradii depend on the choice of the magnetic field decay rate. Moreover, as pointed out in[62], better X-ray data is needed to determine the compositions of accreting neutron stars,as this can make 30% or greater changes in inferred neutron star radii.To conclude, let’s say that as we have obtained our results for two limits of the magneticfield, namely, the lowest possible one that could contribute at least slightly to the EOS andthe maximum value that allows us to avoid anisotropic pressures, all possible analysis forintermediate values are contemplated and the stellar maximum masses will always lie inbetween the values we have calculated. Based on previous experiences with the NJL model,we believe that our results will not change qualitatively if another parametrization wereused. However, had we chosen to introduce one of the possible vector interactions availablein the literature [63] and [64], the quark matter EOS would certainly be harder and theconsequences of including magnetic fields are presently under investigation. ACKNOWLEDGMENTS
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