Hyperon star in a modified quark meson coupling model

# Hyperon star in a modified quark meson coupling model

R.N. Mishra    H.S. Sahoo Department of Physics, Ravenshaw University, Cuttack-753 003, India    P.K. Panda    N. Barik Department of Physics, Utkal University, Bhubaneswar-751 004, India    T. Frederico Instituto Tecnólogico de Aeronática, DCTA, 12228-900 São José dos Campos, SP, Brazil
###### Abstract

We determine the equation of state (EOS) of nuclear matter with the inclusion of hyperons in a self-consistent manner by using a Modified Quark Meson Coupling Model (MQMC) where the confining interaction for quarks inside a baryon is represented by a phenomenological average potential in an equally mixed scalar-vector harmonic form. The hadron-hadron interaction in nuclear matter is then realized by introducing additional quark couplings to , , and mesons through mean-field approximations. The effect of a nonlinear - term on the equation of state is studied. The hyperon couplings are fixed from the optical potential values and the mass-radius curve is determined satisfying the maximum mass constraint of  M for neutron stars, as determined in recent measurements of the pulsar PSR J0348+0432. We also observe that there is no significant advantage of introducing the nonlinear - term in the context of obtaining the star mass constraint in the present set of parametrizations.

###### pacs:
26.60.+c, 21.30.-x, 21.65.Qr, 95.30.Tg

## I Introduction

Over the last few decades intensive theoretical investigations are being pursued to understand the microscopic composition and properties of dense nuclear matter. It has been realized by now from such studies saakyan (); pring (); aakmal (); jrstone (); jrstone1 (); fweber (); weise (); bdlackey (); debarati () that high density nuclear matter may consist not only of nucleons and leptons but also several exotic components such as hyperons, mesons as well as quark matter in different forms and phases. Hyperons in particular are expected to appear in the inner core of neutron stars at densities times the normal saturation density fm. This is because at such high densities the nucleon chemical potential becomes large enough to facilitate the formation of hyperons to be energetically favorable by the inverse beta decay process of nucleons in the -stable nuclear matter. As a consequence the Fermi pressure exerted by the baryons is reduced and the Equation of State (EOS) describing such dense matter in neutron stars with hyperon core becomes softer leading to the reduction of the maximum mass of the star glend (); glenmos (); knorren (); balberg (); prakash (); taurines (). However relativistic Hartree-Fock models miyatsu (); miyatsu1 (), relativistic mean field models bednarek (); weissenborn () or quantum hadrodynamic model jiang () show relatively weaker effects on the EOS due to the presence of strange baryons in neutron star core.

Until recently the reliability requirement for any model EOS was only to predict a maximum neutron star mass compatible with the canonical value of M, since most of the precisely measured neutron star mass were clustered around these values only. This constraint was probably not stringent enough for which without any discrmination, most relativistic models even with the inclusion of hyperons glend (); glenmos (); knorren (); balberg (); prakash () have succeeded to this extent. But recent discovery of the unusually high mass of the millisecond pulsars PSR J1903+0327 ( M) djchamp (); freire (); freire1 (), PSR J1614-2230 ( M) demorest () and PSR J0348+0432 ( M) antoniadis () show that the neutron star mass distribution is much wider extending firmly up to  M. Also there has been considerable progress in the measurement of the neutron star radii by reducing their uncertainties with a better understanding of the sources of systematic errors to estimate them in km range for a  M neutron star ozel (). Another study by Fortin et al fortin () has shown that the observational constraint on the maximum mass implies that the hyperonic stars with masses in the range  M must be larger than km due to a pre-hyperonic stiffening of EOS. It has been found by Providência and Rabhi cprovindencia () that the radius of a hyperonic star of a given mass decreases linearly with the increase of the total hyperon content. These observations may serve to further constrain the EOS in achieving greater reliability.

Various studies have established that the presence of hyperons in the neutron star core leads to softening of the EOS and consequent reduction in the maximum mass of the star. This has provided a challenge to develop an equation of state (EOS) stiff enough to give such high mass with the inclusion of hyperons. In fact most relativistic models obtain maximum star masses in the range with the inclusion of hyperons glenmos (). However there are some exceptional cases rikovska () where maximum mass of the hyperonic star have been realized in the range  M.

In the present work, we have developed an EOS using a modified quark-meson coupling model (MQMC). The MQMC model is based on confining relativistic independent quark potential model rather than a bag to describe the baryon structure in vacuum. In such a picture the quarks inside the baryon are considered to be independently confined by a phenomenologically average potential with an equally mixed scalar-vector harmonic form. Such a potential has characteristically simplifying features in converting the independent quark Dirac equation into a Schrödinger like equation for the upper component of Dirac spinor which can be solved easily. The implications of such potential forms in the Dirac framework has been studied earlier barik (); prd (). The baryon-baryon interactions are realized by making additional quark couplings to , , and mesons through mean-field approximations, in an extension of previous works based on the MIT bag model guichon (); qmc (); frederico89 (). The MQMC model has already been well tested in determining various bulk properties of symmetric and asymmetric nuclear matter rnm (); hss (). The relevant parameters of the interaction are obtained self-consistently by realizing the saturation properties such as binding energy and pressure. Here, we study the role of hyperons on the properties of neutron stars. In the present work we have also introduced an additional non-linear coupling to study its effect on the stiffening of EOS necessary for the purpose.

We include hyperons as a new degree of freedom in dense hadronic matter relevant for neutron stars. The interactions between nucleons and the baryons of the baryon octet in dense matter is studied and its effects on the mass of the neutron star is analysed. The nucleon-nucleon interaction is well known from nuclear properties. But the extrapolation of such interactions to densities beyond nuclear saturation density is a great problem. Most of the hyperon-nucleon interaction are known experimentally. This has inspired us to set the hyperon-nucleon interaction potential at saturation density for the , and hyperons to MeV, MeV and MeV respectively. Because of the uncertainties in the measurement of the hyperon potentials, we make a variation in the and study the effects on the mass of the star. However, we do not include the hyperon-hyperon interactions which are experimentally least well known.

In this model we observe that the compressibility of the neutron star matter depends on the mass of the quark. The quark mass has been fixed at MeV giving us a compressibility of MeV which lies within the range predicted from experimental GMR studies stonemos () and also from theoretical predictions of infinite nuclear matter model lsatpathy (). We also compare our results at two different quark masses of MeV and MeV.

The paper is organized as follows: In Sec. II, a brief outline of the model describing the baryon structure in vacuum is discussed. The baryon mass is then realized by appropriately taking into account the center-of-mass correction, pionic correction, and gluonic correction in Sec. III. The EOS is then developed in Sec. IV. The results and discussions are made in Sec. V. We summarize our findings in Sec. IV.

## Ii Modified quark meson coupling model

The modified quark-meson coupling model has been extensively applied for the study of the bulk properties of both symmetric as well as asymmetric nuclear matter. Under such a model the nucleon-nucleon () interaction was realized in a mean-field approach through the exchange of effective mesonic fields coupling to the quarks inside the nucleon for the symmetric case rnm () and the additional iso-vector vector meson field () coupling to the quarks for the asymmetric case hss (). In our earlier work hss () this model was used to investigate the nature of the thermodynamic instabilities and the correlation of the symmetry energy with its slope. We now extend this model to investigate the role of nucleons and hyperons in neutron star matter under conditions of beta equilibrium and charge neutrality.

We begin by considering baryons as composed of three constituent quarks in a phenomenological flavor-independent confining potential, in an equally mixed scalar and vector harmonic form inside the baryon rnm (), where

 U(r)=12(1+γ0)V(r),

with

 V(r)=(ar2+V0),        a>0. (1)

Here are the potential parameters. The confining interaction provides the zeroth-order quark dynamics of the hadron. In the medium, the quark field satisfies the Dirac equation

 [γ0 (ϵq−Vω−12τ3qVρ)−→γ.→p−(mq−Vσ)−U(r)]ψq(→r)=0 (2)

where , and . Here , , and are the classical meson fields, and , , and are the quark couplings to the , , and mesons, respectively. is the quark mass and is the third component of the Pauli matrices. We can now define

 ϵ′q=(ϵ∗q−V0/2)   and   m′q=(m∗q+V0/2), (3)

where the effective quark energy, and effective quark mass, . We now introduce and as

 (ϵ′q+m′q)=λq    and    r0q=(aλq)−14. (4)

The ground-state quark energy can be obtained from the eigenvalue condition

 (ϵ′q−m′q)√λqa=3. (5)

The solution of equation (5) for the quark energy immediately leads to the mass of baryon in the medium in zeroth order as

 E∗0B=∑q ϵ∗q (6)

## Iii Effective mass of baryon

We next consider the spurious center-of-mass correction , the pionic correction for restoration of chiral symmetry, and the short-distance one-gluon exchange contribution to the zeroth-order baryon mass in the medium.

Here, we extract the center of mass energy to first order in the difference between the fixed center and relative quark co-ordinate, using the method described by Guichon et al. guichon (). The centre of mass correction is given by:

 ec.m.=e(1)c.m.+e(2)c.m., (7)

where,

 e(1)c.m.=3∑i=1[mqi∑3k=1mqk6r20qi(3ϵ′qi+m′qi)] (8)
 e(2)c.m. = a2[2∑kmqk∑imi⟨r2i⟩+2∑kmqk∑imi⟨γ0(i)r2i⟩−3(∑kmqk)2∑im2i⟨r2i⟩ (9) − 1(∑kmqk)2∑i⟨γ0(1)m2ir2i⟩−1(∑kmqk)2∑i⟨γ0(2)m2ir2i⟩−1(∑kmqk)2∑i⟨γ0(3)m2ir2i⟩]

In the above, we have used for and and the various quantities are defined as

 ⟨r2i⟩=(11ϵ′qi+m′qi)r20qi2(3ϵ′qi+m′qi) (10)
 ⟨γ0(i)r2i⟩=(ϵ′qi+11m′qi)r20qi2(3ϵ′qi+m′qi) (11)
 ⟨γ0(i)r2j⟩i≠j=(ϵ′qi+3m′qi)⟨r2j⟩3ϵ′qi+m′qi (12)

The pseudo-vector nucleon pion coupling constant, can be obtained from Goldberg-Treiman relations by using the axial-vector coupling constant value in the model as

 √4πfNNπmπ=gA(N)2fπ, (13)

where

 gA(n→p)=59(5ϵ′u+7m′u)(3ϵ′u+m′u). (14)

The pionic corrections in the model for the nucleons become

 δMπN=−17125Iπf2NNπ. (15)

Taking becomes

 Iπ=1πmπ2∫∞0dk.k4u2(k)w2k, (16)

with the axial vector nucleon form factor given as

 u(k)=[1−32k2λq(5ϵ′q+7m′q)]e−k2r20/4 . (17)

The pionic correction for and become

 δMπΣ0=−125f2NNπIπ, (18)
 δMπΛ0=−10825f2NNπIπ. (19)

Similarly the pionic correction for and is

 δMπΣ+,Σ−=−125f2NNπIπ. (20)

The pionic correction for and is

 δMπΞ−,Ξ0=−2725f2NNπIπ. (21)

The one-gluon exchange interaction is provided by the interaction Lagrangian density

 LgI=∑Jμai(x)Aaμ(x) , (22)

where are the octet gluon vector-fields and is the -th quark color current. The gluonic correction can be separated in two pieces, namely, one from the color electric field () and another from the magnetic field () generated by the -th quark color current density

 Jμai(x)=gc¯ψq(x)γμλaiψq(x) , (23)

with being the usual Gell-Mann matrices and . The contribution to the mass can be written as a sum of color electric and color magnetic part as

 (ΔEB)g=(ΔEB)Eg+(ΔEB)Mg , (24)

where

 (ΔEB)Eg = 18π∑i,j8∑a=1∫d3rid3rj|ri−rj| (25) × ⟨B|J0ai(ri)J0aj(rj)|B⟩ ,

and

 (ΔEB)Mg = −18π∑i,j8∑a=1∫d3rid3rj|ri−rj| × ⟨B|→Jai(ri)→Jaj(rj)|B⟩ .

Finally, taking into account the specific quark flavor and spin configurations in the ground state baryons and using the relations and for baryons, one can write the energy correction due to color electric contribution, as

 (ΔEB)Eg=αc(buuIEuu+busIEus+bssIEss) , (27)

and due to color magnetic contributions, as

 (ΔEB)Mg=αc(auuIMuu+ausIMus+assIMss) , (28)

where and are the numerical coefficients depending on each baryon and are given in Table 1. In the above, we have

 IEij=163√π1Rij[1−αi+αjR2ij+3αiαjR4ij] IMij=2569√π1R3ij1(3ϵ′i+m′i)1(3ϵ′j+m′j) , (29)

where

 R2ij = 3[1(ϵ′i2−m′i2)+1(ϵ′j2−m′j2)] αi = 1(ϵ′i+m′i)(3ϵ′i+m′i) . (30)

The color electric contributions to the bare mass for nucleon . Therefore the one-gluon contribution for nucleon becomes

 (ΔEN)Mg=−256αc3√π[1(3ϵ′u+m′u)2R3uu] (31)

The one-gluon contribution for becomes

 (ΔEΣ+,Σ−)Eg = αc163√π[1Ruu(1−2αuR2uu−3α2uR4uu) (32) − 2Rus(1−αu+αsR2us+3αuαsR4us) + 1Rss(1−2αsR2ss+3α2sR4ss)]
 (ΔEΣ+,Σ−)Mg = 256αc9√π[1(3ϵ′u+m′u)2R3uu (33) − 4R3us(3ϵ′u+m′u)(3ϵ′s+m′s)]
 (ΔEΣ+,Σ−)g=(ΔEΣ+,Σ−)Eg+(ΔEΣ+,Σ−)Mg (34)

The gluonic correction for is

 (ΔEΣ0)Eg = αc163√π[1Ruu(1−2αuR2uu−3α2uR4uu) (35) − 2Rus(1−αu+αsR2us+3αuαsR4us) + 1Rss(1−2αsR2ss+3α2sR4ss)]
 (ΔEΣ0)Mg = 256αc9√π[1(3ϵ′u+m′u)2R3uu (36) − 4R3us(3ϵ′u+m′u)(3ϵ′s+m′s)]
 (ΔEΣ0)g=(ΔEΣ0)Eg+(ΔEΣ0)Mg (37)

The gluonic correction for is

 (ΔEΣ0)Eg=(ΔEΛ)Eg (38)

The color magnetic contribution is different

 (ΔEΛ)Mg=−256αc3√π[1(3ϵ′u+m′u)2R3uu] (39)
 (ΔEΛ)g=(ΔEΛ)Eg+(ΔEΛ)Mg (40)

The color electric contributions for and are same as that of or but the color magnetic contributions to the correction of masses of baryon are different:

 (ΔEΞ−,Ξ0)Mg = 256αc9√π[1(3ϵ′s+m′s)2R3ss (41) − 4R3us(3ϵ′u+m′u)(3ϵ′s+m′s)]

Finally, the gluonic correction for and is given by:

 (ΔEΞ−,Ξ0)g=(ΔEΞ−,Ξ0)Eg+(ΔEΞ−,Ξ0)Mg (42)

Treating all energy corrections independently, the mass of the baryon in the medium becomes

 M∗B=E∗0B−ϵc.m.+δMπB+(ΔEB)Eg+(ΔEB)Mg. (43)

## Iv The Equation of state

The total energy density and pressure at a particular baryon density, encompassing all the members of the baryon octet, for the nuclear matter in -equilibrium can be found as

 E = 12m2σσ20+12m2ωω20+12m2ρb203+3g2ωg2ρΛνb203ω20 (44a) + γ2π2∑B∫kf,B[k2+M∗B2]1/2k2dk + ∑l1π2∫kl0[k2+m2l]1/2k2dk, P = − 12m2σσ20+12m2ωω20+12m2ρb203+g2ωg2ρΛνb203ω20 (44b) + γ6π2∑B∫kf,Bk4 dk[k2+M∗B2]1/2 + 13∑l1π2∫kl0k4dk[k2+m2l]1/2,

where is the spin degeneracy factor for nuclear matter, and . In the above expression for the energy density and pressure, a nonlinear coupling term is introduced with coupling coefficient, horowitz01 ().

Another important quantity for the study of nuclear matter is the symmetry energy, which is defined as

 Esym(ρB)=k26E∗2N+g2ρ8m2ρρB (45)

where , the index for neutrons and protons. The slope of the symmetry energy is then obtained as,

 L=3ρ0∂Esym(ρB)∂ρB∣∣∣ρB=ρ0 (46)

For obtaining a constraint on the quark mass we use the value of compressibility given by,

 K=9[dPdρB]ρB=ρ0 (47)

The chemical potentials, necessary to define the equilibrium conditions, are given by

 μB=√k2B+M∗B2+gωω0+gρτ3Bb03 (48)

where is the isopsin projection of the baryon B.

The lepton Fermi momenta are the positive real solutions of and . The equilibrium composition of the star is obtained by solving the equations of motion of meson fields in conjunction with the charge neutrality condition, given in equation (50), at a given total baryonic density . The effective masses of the baryons are obtained self-consistently in this model.

Since we consider the octet baryons, the presence of strange baryons in the matter plays a significant role. We define the strangeness fraction as

 fs=13∑i|si|ρiρ. (49)

Here refers to the strangeness number of baryon and is defined as .

For stars in which the strongly interacting particles are baryons, the composition is determined by the requirements of charge neutrality and -equilibrium conditions under the weak processes and . After deleptonization, the charge neutrality condition yields

 qtot=∑BqBγk3B6π2+∑l=e,μqlk3l3π2=0 , (50)

where corresponds to the electric charge of baryon species and corresponds to the electric charge of lepton species . Since the time scale of a star is effectively infinite compared to the weak interaction time scale, weak interaction violates strangeness conservation. The strangeness quantum number is therefore not conserved in a star and the net strangeness is determined by the condition of -equilibrium which for baryon is then given by , where is the chemical potential of baryon and its baryon number. Thus the chemical potential of any baryon can be obtained from the two independent chemical potentials and of neutron and electron respectively.

The hyperon couplings are not relevant to the ground state properties of nuclear matter, but information about them can be available from the levels in hypernuclei chrien ().

 gσB=xσB gσN,  gωB=xωB gωN,  gρB=xρB gρN

and , and are equal to for the nucleons and acquire different values in different parameterisations for the other baryons. We note that the -quark is unaffected by the sigma and omega mesons i.e. .

The vector mean-fields and are determined through

 ω0=gωm∗ω2∑BxωBρB     b03=gρ2m∗ρ2∑BxρBτ3BρB, (51)

where , , and . Finally, the scalar mean-field is fixed by

 ∂E∂σ0=0. (52)

The iso-scalar scalar and iso-scalar vector couplings and are fitted to the saturation density and binding energy for nuclear matter. The iso-vector vector coupling is set by fixing the symmetry energy at MeV. For a given baryon density, , , and are calculated from equations (51) and (52), respectively.

Following the determination of the EOS the relation between the mass and radius of a star with its central density can be obtained by integrating the Tolman-Oppenheimer-Volkoff (TOV) equations tov () given by,

 dPdr=−Gr[E+P][M+4πr3P](r−2GM), (53)
 dMdr=4πr2E, (54)

with as the gravitational constant and as the enclosed gravitational mass. We have used . Given an EOS, these equations can be integrated from the origin as an initial value problem for a given choice of the central energy density, . Of particular importance is the maximum mass obtained from and the solution of the TOV equations. The value of , where the pressure vanishes defines the surface of the star.

## V Results and Discussion

Our MQMC model has two potential parameters, and and we obtain them by fitting the nucleon mass MeV and charge radius of the proton fm in free space. Keeping the value of the potential parameter same as that for nucleons, we obtain for the , and baryons by fitting their respective masses to MeV, MeV and MeV. The set of potential parameters for the baryons along with their respective energy corrections at zero density are given in Table 2.

The quark meson couplings , , and are fitted self-consistently for the nucleons to obtain the correct saturation properties of nuclear matter binding energy, MeV, pressure, , and symmetry energy MeV at fm.

Table 3 shows the contribution to the spurious center-of-mass correction, the pionic correction and the gluonic correction to obtain the effective mass of the baryon. It is interesting to note that as the mass of the quark increases from MeV to MeV, the magnitude of the pionic correction increases whereas that due gluonic correction decreases for all baryon species.

We have taken the standard values for the meson masses; namely, MeV, MeV and MeV. The values of the quark meson couplings, , , and at quark masses MeV and MeV are given in Table 4.

By changing the value of the - coupling term there is a change in the value of . For and we obtain the value of to be and respectively.

Incompressibility of symmetric nuclear matter as well as the slope of the symmetry energy provide important constraints to the properties of nuclear matter. In the present work, we determine the value of the compression modulus at quark masses MeV and MeV which comes out to be MeV and MeV respectively. From various experimental giant monopole resonance (GMR) studies stonemos () and microscopic calculations of the GMR energies khan () the value of is predicted to lie in the range MeV and MeV respectively. The slope of the nuclear symmetry energy in the present work is calculated to be MeV and MeV for quark masses MeV and MeV, which agrees well with the value extracted from isospin sensitive observables in heavy-ion reactions xuli (). By increasing the value of the value of increases to for and for .

The couplings of the hyperons to the -meson need not be fixed since we determine the effective masses of the hyperons self-consistently. The hyperon couplings to the -meson are fixed by determining . The value of is obtained from the hyperon potentials in nuclear matter, for and as MeV, MeV and MeV respectively. For the quark masses MeV and MeV the corresponding values for are given in Table 5.

The value of is fixed for all baryons.

The hyperon potential has been chosen from the measured single particle levels of hypernuclei from mass numbers to millener () of the binding of