Δ isobars in hyperon stars in a modified quark meson coupling model
aa r X i v : . [ nu c l - t h ] F e b ∆ isobars in hyperon stars in a modified quark meson coupling model H.S. Sahoo, R.N. Mishra, P.K. Panda, and N. Barik Department of Physics, Ravenshaw University, Cuttack-753 003, India Department of Physics, Utkal University, Bhubaneswar-751 004, India
The possibility of the appearance of ∆ isobars in neutron star matter and the so called ∆ puzzle isstudied in a modified quark meson coupling model where the confining interaction for quarks insidea baryon is represented by a phenomenological average potential in an equally mixed scalar-vectorharmonic form. The hadron-hadron interaction in nuclear matter is then realized by introducingadditional quark couplings to σ , ω , and ρ mesons through mean-field approximations. The couplingsof the ∆ to the meson fields are fixed from available constraints while the hyperon couplings arefixed from the optical potential values. It is observed that within the constraints of the mass of theprecisely measured massive pulsars, PSR J0348+0432 and PSR J1614-2230, neutron stars with acomposition of both ∆ isobars and hyperons is possible. It is also observed that with an increase inthe vector coupling strength of the ∆ isobars there is a decrease in the radius of the neutron stars. PACS numbers: 26.60.-c, 21.30.-x, 21.65.Mn, 95.30.Tg
I. INTRODUCTION
The investigations pertaining to the formation ofbaryons heavier than the nucleon at the core of neutronstars and the effects of such formation on the mass andradius of neutron stars is a subject of active researchin nuclear astrophysics. It is expected that high den-sity nuclear matter may consist not only of nucleons andleptons but also several exotic components such as hy-perons, mesons as well as quark matter in different formsand phases. While many studies have been conducted toaddress the appearance of hyperons and on the so called hyperon puzzle [1], little work has been done to studythe appearance of ∆ (1232) isobars in neutron stars. Anearlier work [2] indicated the appearance of ∆ at muchhigher densities than the typical densities of the core ofneutron stars and hence was considered of little signifi-cance to astrophysical studies. However, recent studies[3, 4] suggest the possibility of an early appearance of∆ isobars with consequent softening of the equation ofstate (EOS) of dense matter. This leads to a reductionin the maximum mass of neutron stars below the currentobservational limit of 2 . ± .
04 M ⊙ [5].In the present work, we would like to address the∆-puzzle in a modified quark-meson coupling model(MQMC) [6, 7] which has already been adopted success-fully studying various bulk properties of symmetric andasymmetric nuclear matter. The MQMC model is basedon a confining relativistic independent quark potentialmodel rather than a bag to describe the baryon struc-ture in vacuum. The baryon-baryon interactions are re-alized by making additional quark couplings to σ , ω , and ρ mesons through mean-field approximations. More re-cently [8] it was extended to study the EOS of nuclearmatter with the inclusion of hyperons as new degrees offreedom and the effect of a non-linear ω - ρ term keeping inview the constraints of the mass of the precisely measuredmassive pulsars, PSR J0348+0432 and PSR J1614-2230. Here, we include the delta isobars (∆ − , ∆ , ∆ + , ∆ ++ )together with hyperons as new degrees of freedom in dense hadronic matter relevant for neutron stars. The in-teractions between nucleons, ∆’s and hyperons in densematter is studied and the possibility of the existence ofthe ∆ baryon at densities relevant to neutron star coreas well as its effects on the mass of the neutron star isanalysed. The nucleon-nucleon interaction is well knownfrom nuclear properties. But the extrapolation of suchinteractions to densities beyond nuclear saturation den-sity is quite challenging. Most of the hyperon-nucleoninteraction are known experimentally, which we use toset the hyperon-nucleon interaction potential at satura-tion density. For the Λ, Σ and Ξ hyperons the respec-tive potentials are U Λ = −
28 MeV, U Σ = 30 MeV and U Ξ = −
18 MeV respectively. However, the coupling ofthe ∆ isobars with the mesons are poorly constrained.The ∆ isobars are commonly treated with the same cou-pling strengths as the nucleons. Studies [9, 10] based onthe quark counting argument suggest universal couplingsbetween nucleons, ∆ isobars and mesons, giving the valueof x ω ∆ = g ω ∆ /g ωN = 1. Theoretical studies of Gamow-Teller transitions and M1 giant resonance in nuclei byBohr and Mottelson [11] observed a 25 −
40% reductionin transition strength due to the couplings to ∆ isobars,indicating weaker coupling of the isoscalar mesons to the∆ isobars. Further, the difference between x σ and x ω was found to be x σ − x ω = 0 . x ω ∆ = 0 .
8. We also study the effect of moderatevariations in the value of x ω ∆ on the radius of neutronstars.The paper is organized as follows: In Sec. 2, a briefoutline of the model describing the baryon structure invacuum is discussed. The baryon mass is then realizedby appropriately taking into account the center-of-masscorrection, pionic correction, and gluonic correction. TheEOS with the inclusion of the ∆ isobars and the hyperonsis then developed in Sec. 3. The results and discussionsare made in Sec. 4. We summarize our findings in Sec.5. II. MODIFIED QUARK MESON COUPLINGMODEL
The modified quark-meson coupling model has beensuccessful in obtaining various bulk properties of sym-metric and asymmetric nuclear matter as well as hyper-onic matter within the accepted constraints [6–8]. Wenow extend this model to include the ∆ isobars (∆ − ,∆ , ∆ + , ∆ ++ ) along with nucleons and hyperons in neu-tron star matter under conditions of beta equilibrium andcharge neutrality. We begin by considering baryons ascomposed of three constituent quarks confined inside thehadron core by a phenomenological flavor-independentpotential, U ( r ). Such a potential may be expressed as anadmixture of equal scalar and vector parts in harmonicform [6], U ( r ) = 12 (1 + γ ) V ( r ) , with V ( r ) = ( ar + V ) , a > . (1)Here ( a, V ) are the potential parameters. The confininginteraction provides the zeroth-order quark dynamics ofthe hadron. In the medium, the quark field ψ q ( r ) satisfiesthe Dirac equation[ γ ( ǫ q − V ω − τ q V ρ ) − ~γ.~p − ( m q − V σ ) − U ( r )] ψ q ( ~r ) = 0(2)where V σ = g qσ σ , V ω = g qω ω and V ρ = g qρ b . Here σ , ω , and b are the classical meson fields, and g qσ , g qω ,and g qρ are the quark couplings to the σ , ω , and ρ mesons,respectively. m q is the quark mass and τ q is the thirdcomponent of the Pauli matrices. We can now define ǫ ′ q = ( ǫ ∗ q − V /
2) and m ′ q = ( m ∗ q + V / , (3)where the effective quark energy, ǫ ∗ q = ǫ q − V ω − τ q V ρ and effective quark mass, m ∗ q = m q − V σ . We now intro-duce λ q and r q as( ǫ ′ q + m ′ q ) = λ q and r q = ( aλ q ) − . (4)The ground-state quark energy can be obtained fromthe eigenvalue condition( ǫ ′ q − m ′ q ) r λ q a = 3 . (5)The solution of equation (5) for the quark energy ǫ ∗ q im-mediately leads to the mass of baryon in the medium inzeroth order as E ∗ B = X q ǫ ∗ q (6)We next consider the spurious center-of-mass correc-tion ǫ c.m. , the pionic correction δM πB for restoration of chiral symmetry, and the short-distance one-gluon ex-change contribution (∆ E B ) g to the zeroth-order baryonmass in the medium.We have used a fixed center potential to calculate thewavefunctions of a quark in a baryon. To study the prop-erties of the baryon constructed from these quarks, wemust extract the contribution of the center-of-mass mo-tion in order to obtain physically relevant results. Here,we extract the center of mass energy to first order in thedifference between the fixed center and relative quark co-ordinate, using the method described by Guichon et al. [13, 14]. The centre of mass correction is given by: e c.m. = e (1) c.m. + e (2) c.m. , (7)where, e (1) c.m. = X i =1 " m q i P k =1 m q k r q i (3 ǫ ′ q i + m ′ q i ) (8) e (2) c.m. = a (cid:20) P k m q k X i m i h r i i + 2 P k m q k X i m i h γ ( i ) r i i − P k m q k ) X i m i h r i i− P k m q k ) X i h γ (1) m i r i i − P k m q k ) X i h γ (2) m i r i i− P k m q k ) X i h γ (3) m i r i i (cid:21) (9)In the above, we have used for i = ( u, d, s ) and k =( u, d, s ) and the various quantities are defined as h r i i = (11 ǫ ′ qi + m ′ qi ) r qi ǫ ′ qi + m ′ qi ) (10) h γ ( i ) r i i = ( ǫ ′ qi + 11 m ′ qi ) r qi ǫ ′ qi + m ′ qi ) (11) h γ ( i ) r j i i = j = ( ǫ ′ qi + 3 m ′ qi ) h r j i ǫ ′ qi + m ′ qi (12)The pionic corrections in the model for the nucleonsbecome δM πN = − I π f NNπ , (13)where, f NNπ is the pseudo-vector nucleon-pion couplingconstant. Taking w k = ( k + m π ) / , I π becomes I π = 1 πm π Z ∞ dk. k u ( k ) w k , (14)with the axial vector nucleon form factor given as u ( k ) = h − k λ q (5 ǫ ′ q + 7 m ′ q ) i e − k r / . (15) TABLE I. The coefficients a ij and b ij used in the calculationof the color-electric and and color-magnetic energy contribu-tions due to one-gluon exchange.Baryon a uu a us a ss b uu b us b ss N -3 0 0 0 0 0∆ 3 0 0 0 0 0Λ -3 0 0 1 -2 1Σ 1 -4 0 1 -2 1Ξ 0 -4 1 1 -2 1 The pionic correction for Σ and Λ become δM π Σ = − f NNπ I π , (16) δM π Λ = − f NNπ I π . (17)Similarly the pionic correction for Σ − and Σ + is δM π Σ + , Σ − = − f NNπ I π . (18)The pionic correction for Ξ and Ξ − is δM π Ξ − , Ξ = − f NNπ I π . (19)For ∆ baryon, the pionic correction is given by δM π ∆ = − f NNπ I π . (20)The one-gluon exchange interaction is provided by theinteraction Lagrangian density L gI = X J µai ( x ) A aµ ( x ) , (21)where A aµ ( x ) are the octet gluon vector-fields and J µai ( x )is the i -th quark color current. The gluonic correction canbe separated in two pieces, namely, one from the colorelectric field ( E ai ) and another from the magnetic field( B ai ) generated by the i -th quark color current density J µai ( x ) = g c ¯ ψ q ( x ) γ µ λ ai ψ q ( x ) , (22)with λ ai being the usual Gell-Mann SU (3) matrices and α c = g c / π . The contribution to the mass can be writtenas a sum of color electric and color magnetic part as(∆ E B ) g = (∆ E B ) Eg + (∆ E B ) Mg . (23)Finally, taking into account the specific quark fla-vor and spin configurations in the ground state baryonsand using the relations h P a ( λ ai ) i = 16 / h P a ( λ ai λ aj ) i i = j = − / E B ) Eg = α c ( b uu I Euu + b us I Eus + b ss I Ess ) , (24) and due to color magnetic contributions, as(∆ E B ) Mg = α c ( a uu I Muu + a us I Mus + a ss I Mss ) , (25)where a ij and b ij are the numerical coefficients dependingon each baryon and are given in Table I. In the above,we have I Eij = 163 √ π R ij h − α i + α j R ij + 3 α i α j R ij i I Mij = 2569 √ π R ij ǫ ′ i + m ′ i ) 1(3 ǫ ′ j + m ′ j ) , (26)where R ij = 3 h ǫ ′ i − m ′ i ) + 1( ǫ ′ j − m ′ j ) i α i = 1( ǫ ′ i + m ′ i )(3 ǫ ′ i + m ′ i ) . (27)The color electric contributions to the bare mass for nu-cleon and the ∆ baryon are (∆ E N ) Eg = 0 and (∆ E ∆ ) Eg =0. Therefore the one-gluon contribution for ∆ becomes(∆ E ∆ ) Mg = 256 α c √ π h ǫ ′ u + m ′ u ) R uu i (28)The details of the gluonic correction for the nucleons andhyperons is given in [8].Treating all energy corrections independently, the massof the baryon in the medium becomes M ∗ B = E ∗ B − ǫ c.m. + δM πB + (∆ E B ) Eg + (∆ E B ) Mg . (29) III. THE EQUATION OF STATE
The total energy density and pressure at a particu-lar baryon density, encompassing all the members of thebaryon octet, for the nuclear matter in β -equilibrium canbe found as E = 12 m σ σ + 12 m ω ω + 12 m ρ b + + γ π X B Z k f,B [ k + M ∗ B ] / k dk + X l π Z k l [ k + m l ] / k dk, (30a) P = − m σ σ + 12 m ω ω + 12 m ρ b + + γ π X B Z k f,B k dk [ k + M ∗ B ] / + 13 X l π Z k l k dk [ k + m l ] / , (30b)where γ is the spin degeneracy factor for nuclear mat-ter. For the nucleons and hyperons γ = 2 and for the ∆baryons γ = 4. Here B = N, ∆ , Λ , Σ ± , Σ , Ξ − , Ξ and l = e, µ .The chemical potentials, necessary to define the β − equilibrium conditions, are given by µ B = q k B + M ∗ B + g ω ω + g ρ τ B b (31)where τ B is the isospsin projection of the baryon B.The lepton Fermi momenta are the positive real solu-tions of ( k e + m e ) / = µ e and ( k µ + m µ ) / = µ µ . Theequilibrium composition of the star is obtained by solv-ing the equations of motion of meson fields in conjunctionwith the charge neutrality condition, given in Eq. (32),at a given total baryonic density ρ = P B γk B / (6 π ).The effective masses of the baryons are obtained self-consistently in this model.Since the neutron star time scale is quite long we needto consider the occurence of weak processes in its mat-ter. Moreover, for stars in which the strongly interactingparticles are baryons, the composition is determined bythe requirements of charge neutrality and β -equilibriumconditions under the weak processes B → B + l + ν l and B + l → B + ν l . After de-leptonization, the chargeneutrality condition yields q tot = X B q B γk B π + X l = e,µ q l k l π = 0 , (32)where q B corresponds to the electric charge of baryonspecies B and q l corresponds to the electric charge oflepton species l . Since the time scale of a star is ef-fectively infinite compared to the weak interaction timescale, weak interaction violates strangeness conservation.The strangeness quantum number is therefore not con-served in a star and the net strangeness is determined bythe condition of β -equilibrium which for baryon B is thengiven by µ B = b B µ n − q B µ e , where µ B is the chemicalpotential of baryon B and b B its baryon number. Thusthe chemical potential of any baryon can be obtainedfrom the two independent chemical potentials µ n and µ e of neutron and electron respectively.In the present work, the baryon couplings are given by, g σB = x σB g σN , g ωB = x ωB g ωN , g ρB = x ρB g ρN , where x σB , x ωB and x ρB are equal to 1 for the nucleonsand acquire different values in different parameterisationsfor the other baryons. Information about the hyperoncouplings can be obtained from the levels in Λ hyper-nuclei [15]. We note that the s -quark is unaffected bythe σ - and ω - mesons i.e. g sσ = g sω = 0. The couplingof the ∆ resonances are constrained poorly due to theirunstable nature. Earlier works [9, 10] based on the quarkcounting argument considered simple universal choice ofcouplings of the ∆ with the mesons. Wehrberger et al. [12] carried out studies of ∆ − baryon excitation in finitenuclei in linear Walecka model and reproduced propertiesof some finite nucleus. They constrained the scaling to0 . x σ ∆ − x ω ∆ . . ω and b are determinedthrough ω = g ω m ω X B x ωB ρ B b = g ρ m ρ X B x ρB τ B ρ B , (33) where g ω = 3 g qω and g ρ = g qρ . Finally, the scalar mean-field σ is fixed by ∂ E ∂σ = 0 . (34)The iso-scalar scalar and iso-scalar vector couplings g qσ and g ω are fitted to the saturation density and bindingenergy for nuclear matter. The iso-vector vector coupling g ρ is set by fixing the symmetry energy at J = 32 . ω , b , and σ are calculatedfrom Eq. (33) and (34), respectively.The relation between the mass and radius of a star withits central energy density can be obtained by integratingthe Tolman-Oppenheimer-Volkoff (TOV) equations [16,17] given by, dPdr = − Gr [ E + P ] (cid:2) M + 4 πr P (cid:3) ( r − GM ) , (35) dMdr = 4 πr E , (36)with G as the gravitational constant and M ( r ) as theenclosed gravitational mass. We have used c = 1. Givenan EOS, these equations can be integrated from the ori-gin as an initial value problem for a given choice of thecentral energy density, ( ε ). Of particular importance isthe maximum mass obtained from and the solution of theTOV equations. The value of r (= R ), where the pres-sure vanishes defines the surface of the star. The surfacegravitational redshift Z s is defined as, Z s = (cid:18) − GMR (cid:19) − / − IV. RESULTS AND DISCUSSION
The MQMC model has two potential parameters, a and V which are obtained by fitting the nucleon mass M N = 939 MeV and charge radius of the proton h r N i =0 .
87 fm in free space. Keeping the value of the potentialparameter a same as that for nucleons, we obtain V forthe Λ, ∆, Σ and Ξ baryons by fitting their respectivemasses to M Λ = 1115 . M ∆ = 1232 MeV, M Σ =1193 . M Ξ = 1321 . g qσ , g ω = 3 g qω , and g ρ = g qρ are fitted self-consistently for the nucleons to obtain thecorrect saturation properties of nuclear matter bindingenergy, E B.E. ≡ B = E /ρ B − M N = − . P = 0, and symmetry energy J = 32 . ρ B = ρ = 0 .
15 fm − .We have taken the standard values for the mesonmasses; namely, m σ = 550 MeV, m ω = 783 MeV and m ρ = 763 MeV. The values of the quark meson cou-plings, g qσ , g ω , and g ρ at quark mass 150 MeV is given inTable III. M B * / M B ρ B (fm -3 )m q =150 MeV N ∆ FIG. 1. Effective baryon mass as a function of baryon density. P ( f m - ) ε (fm -4 ) m q = 150 MeV NPNP + ∆ NP + ∆ + Hyp FIG. 2. Total pressure as a function of the energy densityfor various composition of the stellar matter at quark mass m q = 150 MeV. The shaded region shows the empirical EOSobtained by Steiner et al from a heterogeneous set of sevenneutron stars. Y i ρ Β (fm -3 ) npe µ Ξ - Ξ ∆ - ∆ + ∆ ∆ ++ FIG. 3. Particle fraction as a function of the baryon densityindicating the onset of the ∆ isobars at quark mass m q = 150MeV and x ω ∆ = 0 .
8. TABLE II. The potential parameter V obtained for thequark mass m u = m d = 150 MeV, m s = 300 MeV with a = 0 . − .Baryon M B (MeV) V (MeV) N
939 44.05∆ 1232 102.40Λ 1115.6 50.06Σ 1193.1 66.44Ξ 1321.3 66.82 M ( M ⊙ ) R (km)x ω∆ =0.7x ω∆ =0.8x ω∆ =0.9x ω∆ =1.0 FIG. 4. Gravitational mass as a function of radius for variouscouplings of the ∆ isobars at quark mass m q = 150 MeV. The couplings of the hyperons to the σ -meson neednot be fixed since we determine the effective masses ofthe hyperons self-consistently. The hyperon couplings tothe ω -meson are fixed by determining x ωB . The value of x ωB is obtained from the hyperon potentials in nuclearmatter, U B = − ( M ∗ B − M B ) + x ωB g ω ω for B = Λ , Σand Ξ as −
28 MeV, 30 MeV and −
18 MeV respectively.For the quark masses 150 MeV the corresponding valuesfor x ωB are given in Table IV. The ∆-coupling to the ω -meson is fixed at x ω ∆ = 0 .
8. The value of x ρB = 1 isfixed for all baryons.The Λ hyperon potential has been chosen from themeasured single particle levels of Λ hypernuclei frommass numbers A = 3 to 209 [18, 19] of the binding ofΛ to symmetric nuclear matter. Studies of Σ nuclear in-teraction [20, 21] from the analysis of Σ − atomic dataindicate a repulsive isoscalar potential in the interior ofnuclei. Measurements of the final state interaction of Ξhyperons produced in ( K − , K + ) reaction on C in E224experiment at KEK [22] and E885 experiment at AGS[23] indicate a shallow attractive potential U Ξ ∼ − U Ξ ∼ −
14 or less respectively. In view of thiswe consider the Ξ hyperon potential at U Ξ = −
18 MeV.Fig. 1 shows the effective mass of the nucleons and ∆.With increasing density the effective mass decreases dueto the attractive σ field for the baryons.The EOS for different compositions of neutron starmatter is shown in Fig. 2. It is observed that with the TABLE III. Parameters for nuclear matter. They are determined from the binding energy per nucleon, E B.E = B ≡ E /ρ B − M N = − . P = 0 at saturation density ρ B = ρ = 0 .
15 fm − . Also shown are the values of the nuclearmatter incompressibility K and the slope of the symmetry energy L for the quark mass m q = 150 MeV. m q g qσ g ω g ρ M ∗ N /M N K L(MeV) (MeV) (MeV)150 4.39952 6.74299 8.79976 0.87 292 86.39TABLE IV. x ωB determined by fixing the potentials for the hyperons. m q (MeV) x ω Λ x ω Σ x ω Ξ U Λ = −
28 MeV U Σ = 30 MeV U Ξ = −
18 MeV150 0.81309 1.58607 0.24769 inclusion of ∆, the EOS becomes softer than for mat-ter containing only the nucleons. For matter containingthe nucleons, delta and the hyperons, we observe signifi-cant decrease of stiffness. Infact, when both the hyperonsand the ∆ baryons are present, the softness appears ata density of ρ B = 0 .
41 fm − , which is lower than thedensity of ρ B = 0 .
54 fm − , when the softness increasesfor matter containing nucleons and ∆’s. The shaded re-gion shows the empirical EOS obtained by Steiner et al. from a heterogeneous set of seven neutron stars with welldetermined distances [24].The composition of the matter is shown in Fig. 3.which shows the particle fractions for β -equilibriatedmatter. At densities below the saturation value the β -decay of neutrons to muons are allowed and thus muonsstart to populate. At higher densities the lepton fractionbegins to fall since charge neutrality can now be main-tained more economically with the appearance of nega-tive baryon species. Since the ∆ − can replace the neutronand electron at the top of the Fermi sea, it appears firstat a density of ρ B = 0 .
41 fm − . This is followed by theappearance of Ξ − . The sequence of appearance of the ∆resonances is consistent with the notion of charge-favoredor unfavored species [2]. As such, the first ∆ species toappear is ∆ − , followed by the ∆ , ∆ + and ∆ ++ . Theslope of the symmetry energy L also plays a key role inthe appearance of ∆ resonances. Drago et al. [25] con-straining L in the range 40 < L <
62 MeV have observedthe appearance of ∆ close to twice the saturation density.At high densities all baryons tend to saturate. Moreover,the Σ hyperon is not present in the matter distributionfor the given set of potentials since we have chosen arepulsive potential for it.Since the vector coupling of the ∆ are not constrainedby the properties of saturated nuclear matter, we studythe effect of moderate variations in the strength of thevector coupling of the ∆ on the mass-radius of the neu-tron star. Considering only the nucleon and ∆ compo-sition of the matter, we plot in Fig. 4 the gravitationalmass as a function of radius by changing the couplingstrength x ω ∆ of the ∆ isobars. By decreasing the cou-pling strength from x ω ∆ = 1 . x ω ∆ = 0 .
7, we observe
TABLE V. Mass-radius relationaship of neutron stars for dif-ferent vector coupling strength of the ∆ isobars at m q = 150MeV. x ω ∆ M max R( M ⊙ ) (km)0.70 2.14 14.880.80 2.19 14.400.90 2.22 14.281.00 2.24 14.15TABLE VI. Stellar properties obtained at different composi-tions of the star matter for quark mass m q = 150 MeV. m q Composition M max
R(MeV) ( M ⊙ ) (km)150 NP 2.25 14.0NP+∆ 2.19 14.4NP+∆+HYP 2.15 15.4 a gradual decrease in the maximum mass of the star, seeTable V. This follows from the fact that by decreasing theinteraction strength of the ∆ with respect to the nucle-ons, the EOS becomes softer with a consequent decreasein the maximum mass of the star. We also observe adecrease in the radius with increasing coupling strength.In Fig. 5 we plot the mass-radius relations for thethree possible compositions of neutron star matter at m q = 150 MeV. A stiffer EOS corresponding to mat-ter with nucleons only gives the maximum star mass of M star = 2 . M ⊙ . With the appearance of the ∆ isobars,mass decreases by 0 . M ⊙ to M star = 2 . M ⊙ . Theinclusion of the hyperons further softens the EOS result-ing in a corresponding decrease in the maximum massto M star = 2 . M ⊙ . The results are shown in Table VI.The recently observed pulsar PSR J0348+0432 providea mass constraint of 2 . ± . M ⊙ [5] while an earlieraccurately measured pulsar PSR J1614-2230 gives a massof 1 . ± . M ⊙ [26]. From our calculations we obtaina range of masses varying from 2 . M ⊙ to 2 . M ⊙ de-pending on the composition of the matter. Though in thepresent model we are able to meet the mass constraint, M ( M ⊙ ) R (km) PSR J0348+0432m q =150 MeV NPNP + ∆ NP + ∆ + Hyp FIG. 5. Gravitational mass as a function of radius for varyingcomposition of star matter at quark mass m q = 150 MeV and x ω ∆ = 0 . Z s M(M ⊙ ) FIG. 6. Surface gravitational redshift as a function of starmass at quark mass m q = 150 MeV and x ω ∆ = 0 . we do not get a lower radius. Such problems are alsofound in many other models. One of the way out in suchmodel based calculations is to consider a compact starwith mixed phases of hadrons and quarks. At presentwe consider only a hadronic phase. The work in this re-gard to meet the compactness in the present model is inprogress.Fig. 6 shows the gravitational redshift versus the gravi-tational mass of the neutron star at quark mass m q = 150MeV and x ω ∆ = 0 .
8. It also shows the maximum redshift (redshift corresponding to the maximum mass) which, forthe present work comes out to be Z maxs = 0 .
17. This iswell below the upper bound on the surface redshift forsubluminal equation of states, i.e. z CLs = 0 . V. CONCLUSION
In the present work we have studied the possibility ofthe appearance the ∆ isobars in dense matter relevant toneutron stars. We have developed the EOS using a mod-ified quark-meson coupling model which considers thebaryons to be composed of three independent relativisticquarks confined by an equal admixture of a scalar-vectorharmonic potential in a background of scalar and vectormean fields. Corrections to the centre of mass motion,pionic and gluonic exchanges within the nucleon are cal-culated to obtain the effective mass of the baryon. Thebaryon-baryon interactions are realised by the quark cou-pling to the σ , ω and ρ mesons through a mean field ap-proximation. The nuclear matter incompressibility K isdetermined to agree with experimental studies. Further,the slope of the nuclear symmetry is calculated whichalso agrees well with experimental observations.By varying the composition of the matter we observethe variation in the degree of stiffness of the EOS and thecorresponding effect on the maximum mass of the star.As predicted theoretically, we observe that the inclusionof the ∆ and hyperon degrees of freedom softens the EOSand hence lowers the maximum mass of the neutron star.The so called ∆ and hyperon puzzles state that the pres-ence of the ∆ isobars and hyperons would decrease themaximum star mass below the recently observed massesof the pulsars PSR J0348+0432 and PSR J1614-2230. Inthe present work, we are able to achieve the observedmass constraint and at the same time satisfy the theo-retical predictions of the possibility of existence of highermass baryons in highly dense matter. Moreover, we studythe effect of moderate variations in the strength of thevector coupling of the ∆ resonances and observe a de-crease in the radius of neutron stars with an increase inthe coupling strength. ACKNOWLEDGMENTS
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