Influence of a scalar-isovector δ-meson field on the quark phase structure in neutron stars

Transcription

1 Research in Astron. Astrophys. 21 Vol. 1 No. 12, Research in Astronomy and Astrophysics Influence of a scalar-isovector δ-meson fid on the quark phase structure in neutron stars Grigor Bakhshi Alaverdyan Yerevan State University, Yerevan 25, Armenia; galaverdyan@ysu.am Received 21 Feruary 15; accepted 21 July 14 Astract The deconfinement phase transition from hadronic matter to quark matter in the interior of compact stars is investigated. The hadronic phase is descried in the framework of rativistic mean-fid theory, where the scalar-isovector δ-meson effective fid is also taken into account. The MIT ag mod for descriing a quark phase is used. The changes of the parameters of phase transition caused y the presence of a δ-meson fid are explored. Finally, alterations in the integral and structural parameters of hyrid stars due to oth a deconfinement phase transition and inclusion of a δ-meson fid are discussed. Key words: dense matter equation of state mean-fid stars: neutron quark 1 INTRODUCTION The structure of compact stars functionally depends on the equation of state (EOS) of matter in a sufficiently wide range of densities, from 7.9 gcm 3 (the endpoint of thermonuclear urning) to one order of magnitude higher than nuclear saturation density. Therefore, the study of properties and composition of the constituents of matter in the extremy high density region is of great interest in oth nuclear and neutron star physics. The rativistic mean-fid (RMF) theory (Walecka 1974; Serot & Walecka 1986, 1997) has een effectivy applied to descrie the structure of finite nuclei (Lalazissis et al. 1997; Typ & Wolter 1997), the features of heavy-ion collisions (Ko & Li 1996; Prassa et al. 27), and the equation of state (EOS) of nuclear matter (Müller & Serot 1995). Inclusion of the scalar-isovectorδ-meson in this theoretical scheme and investigation of its influence on low density asymmetric nuclear matter was performed in Refs. Kuis & Kutschera (1999); Liu et al. (22); Greco et al. (23). At sufficiently high density, different exotic degrees of freedom, such as pion and kaon condensates, in addition to deconfined quarks, may appear in the strongly interacting matter. The modern concept of a hadron-quark phase transition is ased on the feature of that transition, that is the presence of two conserved quantities in this transition: aryon numer and ectric charge (Glendenning 1992). It is known that, depending on the value of surface tension σ s, the phase transition of nuclear matter into quark matter can occur in two scenarios (Heiserg et al. 1993; Heiserg & Hjorth-Jensen 2): ordinary first order phase transition with a density jump (Maxwl construction), or formation of mixed hadron-quark matter with a continuous variation of pressure and density (Glendenning construction) (Glendenning 1992). Uncertainty of the surface tension values does not allow us to determine the phase transition scenario, which actually takes place. In our recent paper (Alaverdyan 29a), under the assumption that the transition to quark

2 1256 G. B. Alaverdyan matter is a usual first-order phase transition, which can e descried y the Maxwl construction, we have shown that the presence of the δ-meson fid leads to the decrease of transition pressure P and coexistence of aryon numer densities n N and n Q. In this article, we investigate the hadron-quark phase transition of neutron star matter, when the transition proceeds through a mixed phase. The calculation results of the mixed phase structure (Glendenning construction) are compared with the results of a usual first-order phase transition (Maxwl construction). Also, the influence of a δ-meson fid on phase transition characteristics is discussed. Finally, using the EOS otained, we calculate the integral and structural characteristics of neutron stars with quark degrees of freedom. 2 EQUATION OF STATE OF NEUTRON STAR MATTER 2.1 Nuclear Matter In this section, we consider the EOS of matter in the region of nuclear and supranuclear density (n.1 fm 3 ). For the lower density region, corresponding to the outer and inner crust of the star, we have used the EOS of Baym-Bethe-Pethick (BBP) (Baym et al. 1971). To descrie the hadronic phase we use the rativistic nonlinear Lagrangian density of a many-particle system consisting of nucleons (p, n), ectrons and isoscalar-scalar (σ), isoscalar-vector (ω), isovector-scalar (δ), and isovector-vector (ρ) - exchanged mesons 1 L = ψ N [γ μ (i μ g ω ω μ (x) 1 2 g ρτ N ρ μ (x)) (m N g σ σ(x) g δ τ N δ(x))]ψ N ( μσ(x) μ σ(x) m σ σ(x) 2 ) U(σ(x)) m2 ωω μ (x)ω μ (x) 1 4 Ω μν(x)ω μν (x) ( μδ(x) μ δ(x) m 2 δ δ(x)2 )+ 1 2 m2 ρ ρμ (x)ρ μ (x) 1 4 R μν(x)r μν (x) + ψ e (iγ μ μ m e ) ψ e, (1) where x = x μ =(t, x, y, z), σ(x), ω μ (x), δ(x), andρ μ (x) are the fids of the σ, ω, δ, andρ exchange mesons, respectivy, U(σ) is the nonlinear part of the potential of the σ-fid, given y Boguta & Bodmer (1977) U(σ) = 3 m N(g σ σ) 3 + c 4 (g σσ) 4, (2) ( ) ψp m N, m e, m σ, m ω, m δ and m ρ are the masses of the free particles, ψ N = is the isospin ψ n doulet for nucleonic ispinors, and τ are the isospin 2 2 Pauli matrices. The old face type denotes vectors in isotopic spin space. This Lagrangian also includes antisymmetric tensors of the vector fids ω μ (x) and ρ μ (x) given y Ω μν (x) = μ ω ν (x) ν ω μ (x), R μν (x) = μ ρ ν (x) ν ρ μ (x). (3) In the RMF theory, the meson fids σ (x), ω μ (x), δ (x) and ρ μ (x) are replaced y the effective mean-fids σ, ω μ, δ and ρ μ. This Lagrangian density (1) contains the meson-nucleon coupling constants, g σ, g ω, g ρ and g δ, as wl as parameters of the σ-fid sf-interacting terms and c. In our calculations, we take a δ =(g δ /m δ ) 2 =2.5 fm 2 for the δ coupling constant, as given in Refs. Liu et al. (22); Greco et al. (23); Alaverdyan (29a); for the are nucleon mass m N = MeV, for the nucleon effective mass m N =.78 m N, for the aryon numer density at saturation n =.153 fm 3, for the inding 1 We use the natural system of units with h = c =1.

3 Influence of a Scalar-Isovector δ-meson Fid on the Quark Phase Structure 1257 energy per aryon f = 16.3 MeV, for the incompressiility modulus K = 3 MeV, and for the asymmetry energy E sym () =32.5MeV. Five other constants, a σ =(g σ /m σ ) 2, a ω =(g ω /m ω ) 2, a ρ =(g ρ /m ρ ) 2, and c, can then e numerically determined (Alaverdyan 29). In Tale 1 we list the values of the mod parameters with and without the isovector-scalar δ meson interaction chann (The mods RMFσωρδ and RMFσωρ, respectivy). Tale 1 Mod Parameters with and without a δ -Meson Fid a σ (fm 2 ) a ω (fm 2 ) a δ (fm 2 ) a ρ (fm 2 ) (1 2 fm 1 ) c (1 2 ) RMFσωρδ RMFσωρ The knowledge of the mod parameters makes it possile to solve the set of four equations in a sf-consistent way and to determine the re-denoted mean-fids, σ g σ σ, ω g ω ω, δ g δ δ(3), and ρ g ρ ρ (3), depending on aryon numer density n and asymmetry parameter α =(n n n p )/n. Here δ (3) and ρ (3) are the third isospin components of corresponding mean-fids. The standard QHD procedure allows us to otain expressions for energy density ε(n, α) and pressure P (n, α) of nuclear npe plasma ε NM (n, α, μ e )= 1 π 2 k (n,α) + 3 m N σ 3 + c 4 σ k 2 + [ m p (σ, δ)] 2 k 2 dk + 1 π 2 ( σ 2 a σ + ω2 a ω + δ 2 a δ + ρ 2 a ρ ) + 1 π 2 k +(n,α) μ 2 e m 2 e k 2 +[m n (σ, δ)]2 k 2 dk k2 + m e2 k 2 dk, (4) P NM (n, α, μ e ) = 1 π π 2 k (n,α) k +(n,α) ([k (n, α)] 2 + [ m p (σ, δ)] 2 k 2 + [ ) m p (σ, δ)] 2 ( [k + (n, α)] 2 +[m n(σ, δ)] 2 3 m N σ 3 c 4 σ4 + 1 ) ( σ2 + ω2 δ2 + ρ2 2 a σ a ω a δ + 1 3π 2 μ ( e μ 2 e m 2 3/2 1 e) π 2 where μ e is the chemical potential of ectrons, μ 2 e m 2 e k 2 dk k 2 +[m n(σ, δ)] 2 ) k 2 dk a ρ k2 + m e2 k 2 dk, (5) m p (σ, δ) =m N σ δ, m n (σ, δ) =m N σ + δ (6) are the effective masses of the proton and neutron, respectivy, and [ 3π 2 ] 1/3 n k ± (n, α) = (1 ± α). (7) 2

4 1258 G. B. Alaverdyan Fig. 1 Asymmetry parameter as a function of the aryon numer density n for a β-equilirium charge-neutral npe -plasma. The solid and dotted lines correspond to the RMFσωρδ and RMFσωρ mods, respectivy. Fig. 2 Re-denoted meson mean-fids as a function of the aryon numer density n in case of a β-equilirium charge-neutral npe-plasma with and without the δ-meson fid. The solid and dashed lines correspond to the RMFσωρδ and RMFσωρ mods, respectivy. The chemical potentials of the proton and neutron are given y μ p (n, α) = [k (n, α)] 2 + [ m p(σ, δ) ] ω + ρ, (8) 2 μ n (n, α) = [k + (n, α)] 2 +[m n (σ, δ)] 2 + ω 1 ρ. (9) 2 In Figure 1, we show the asymmetry parameter α for the β-equilirium charge-neutral npeplasma as a function of the aryon numer density, n (Alaverdyan 29a). The solid and dotted lines correspond to the RMFσωρδ and RMFσωρ mods, respectivy. One can see that the asymmetry parameter falls off monotonically with the increase of aryon numer density n. Forafixed aryon numer density n, the inclusion of the δ-meson effective fid reduces the asymmetry parameter α. The presence of a δ-fid reduces the neutron density n n and increases the proton density n p. In Figure 2, we plot the effective mean-fids of exchanged mesons, σ, ω, ρ and δ, as a function of the aryon numer density n for the charge-neutral β-equilirium npe-plasma. The solid and dashed lines correspond to the RMFσωρδ and RMFσωρ mods, respectivy. From Figures 1 and 2, one can see that the inclusion of the scalar-isovector virtual δ(a (98)) meson results in significant changes of species aryon numer densities n p and n n,aswlasthe ρ and δ meson effective fids. This can result in changes of the deconfinement phase transition parameters and, thus, alter the structural characteristics of neutron stars. The results of our analysis show that the scalar - isovector δ-meson fid inclusion leads to the increase of the EOS stiffness of nuclear matter due to the splitting of proton and neutron effective masses, and also to the increase of asymmetry energy (for details see Ref. Alaverdyan 29a).

5 Influence of a Scalar-Isovector δ-meson Fid on the Quark Phase Structure Quark Matter To descrie the quark phase, an improved version of the MIT ag mod (Chodos et al. 1974) is used, in which the interactions etween u, d and s quarks inside the ag are taken into account in the one-gluon exchange approximation (Farhi & Jaffe 1984). The quark phase consists of three quark flavors, u, d, ands, and ectrons, which are in equilirium with respect to weak interactions. We choose m u =5MeV, m d =7MeV and m s = 15 MeV for quark masses, and α s =.5 for the strong interaction constant. 2.3 Deconfinement Phase Transition Parameters There are two independent conserved charges in the hadron-quark phase transition: aryonic charge and ectric charge. The constituent chemical potentials of the npe-plasma in β-equilirium are expressed through two potentials, μ (NM) and μ (NM), according to conserved charges as follows μ n = μ (NM), μ p = μ (NM) μ (NM), μ e = μ (NM). (1) In this case, the pressure P NM, energy density ε NM and aryon numer density n NM are functions of potentials, μ (NM) and μ (NM). The particle species chemical potentials for udse-plasma in β-equilirium are expressed through the chemical potentials μ (QM) and μ (QM) as follows ( μ (QM) ) 2 μ (QM), μ u = 1 3 μ d = μ s = 1 ( ) μ (QM) + μ (QM), (11) 3 μ e = μ d μ u = μ (QM). In this case, the thermodynamic characteristics, pressure P QM, energy density ε QM and aryon numer density n QM are functions of chemical potentials μ (QM) and μ (QM). The mechanical and chemical equilirium conditions (Gis conditions) for the mixed phase are μ (QM) The volume fraction of the quark phase is = μ (NM) = μ, μ (QM) = μ (NM) = μ, (12) P QM (μ,μ )=P NM (μ,μ ). (13) χ = V QM / (V QM + V NM ), (14) where V QM and V NM are volumes occupied y quark matter and nucleonic matter, respectivy. We applied the gloal ectrical neutrality condition for mixed quark-nucleonic matter, which according to Glendenning is (Glendenning 1992, 2), (1 χ)[n p (μ,μ ) n e (μ )] [ 2 +χ 3 n u(μ,μ ) 1 3 n d(μ,μ ) 1 ] 3 n s(μ,μ ) n e (μ ) =. (15) The aryon numer density in the mixed phase is determined as n =(1 χ)[n p (μ,μ )+n n (μ,μ )] χ [n u(μ,μ )+n d (μ,μ )+n s (μ,μ )], (16)

6 126 G. B. Alaverdyan and the energy density is ε =(1 χ)[ε p (μ,μ )+ε n (μ,μ )] + χ [ε u (μ,μ )+ε d (μ,μ )+ε s (μ,μ )] + ε e (μ ). (17) In the case of χ =, the chemical potentials μ N and μn, corresponding to the lower threshold of a mixed phase, are determined y solving Equations (13) and (15). This allows us to find the lower oundary parameters P N, ε N and n N. Similarly, we calculate the upper oundary values of mixed phase parameters, P Q, ε Q and n Q,forχ =1. The system of Equations (13), (15), (16) and (17) makes it possile to determine the EOS of the mixed phase etween these critical states. Note that in the case of an ordinary first-order phase transition, oth nuclear and quark matter are assumed to e separaty ectrically neutral, and at some pressure P, corresponding to the coexistence of the two phases, their aryon chemical potentials are equal, i.e., μ NM (P )=μ QM (P ). (18) Such a phase transition scenario is known as a phase transition with constant pressure (Maxwl construction). Tale 2 represents the parameter sets of the mixed phase with and without a δ-meson fid. It is shown that the presence of the δ-fid alters the threshold characteristics of the mixed phase. For B =6MeV fm 3, the lower threshold parameters, n N, ε N and P N, are increased; meanwhile the upper ones, n Q, ε Q and P Q, are slowly decreased. For B = 1 MeV fm 3 this ehavior changes to the opposite. Tale 2 Mixed Phase Threshold Parameters with and without the δ -Meson Fid for Bag Parameter Values B =6MeV fm 3 and B = 1 MeV fm 3 Mod n N n Q P N P Q ε N ε Q (fm 3 ) (fm 3 ) (MeV fm 3 ) (MeV fm 3 ) (MeV fm 3 ) (MeV fm 3 ) B6σωρδ B6σωρ B1σωρδ B1σωρ In Figures 3 and 4, we plot the EOS of compact star matter with the deconfinement phase transition for two values of the ag constant, B =6MeV fm 3 and B = 1 MeV fm 3, respectivy. The dotted lines correspond to pure nucleonic and quark matter without any phase transition, while the solid lines correspond to two alternative phase transition scenarios. Open circles show the oundary points of the mixed phase. In Figure 5, we plot the particle species numer densities as a function of aryon density n for the Glendenning construction. Quarks appear at the critical density n N =.241 fm 3. The hadronic matter complety disappears at n Q =1.448 fm 3, where the pure quark phase occurs. The solid lines correspond to the case when the δ- meson effective fid is also taken into account in addition to the σ, ω, andρ meson fids (mod B1 σωρδ). The dashed lines represent the results in the case where we neglect the δ-meson fid (mod B1 σωρ). One can see that inclusion of the δ- meson fid leads to the increase of numer densities of quarks and protons, and simultaneously to the reduction of numer densities of neutrons and ectrons. In Tale 2 we have already shown the mixed phase oundaries changes, which are caused y the inclusion of the δ - meson effective fid. Figure 6 shows the constituents numer density as a function of aryon numer density n for B = 1 MeV fm 3, when phase transition is descried according to the Maxwl construction. The Maxwl constructionleads to the appearance of a discontinuity. In this case, the charge neutral

7 Influence of a Scalar-Isovector δ-meson Fid on the Quark Phase Structure 1261 Fig. 3 EOS of neutron star matter with the deconfinement phase transition for a ag constant B = 6 MeV fm 3. For comparison we plot oth the Glendenning and Maxwl constructions. Open circles represent the mixed phase oundaries. Fig. 4 Same as in Fig. 3, ut for B=1 MeV fm 3. Fig. 5 Constituents numer density versus aryon numer density n for B=1 MeV fm 3 in the case of the Glendenning construction. Vertical dotted lines represent the mixed phase oundaries. The dashed lines show appropriate results of the mod without the δ-meson fid. Fig. 6 Same as in Fig. 5, ut for the Maxwl construction. Vertical dotted lines represent the density jump oundaries.

8 1262 G. B. Alaverdyan nucleonic matter at aryon density n 1 =.475 fm 3 coexists with the charge neutral quark matter at aryon density n 2 =.65 fm 3. Thus, the density range n 1 <n<n 2 is foridden. In case of the Maxwl construction, the chemical potential of ectrons, μ e, has a jump at the coexistence pressure P. Notice that such discontinuity ehavior takes place only in a usual first-order phase transition, i.e., in the Maxwl construction case. 3 PROPERTIES OF HYBRID STARS Using the EOS otained in the previous section, we calculate the integral and structural characteristics of neutron stars with quark degrees of freedom. The hydrostatic equilirium properties of spherically symmetric and isotropic compact stars in terms of general rativity are descried y the Tolman-Oppenheimer-Volkoff (TOV) equations (Tolman 1939; Oppenheimer & Volkoff 1939) dp dr = G (P + ε)(m +4πr 3 P ) r 2, (19) 1 2Gm/r dm dr =4πr2 ε, (2) where G is the gravitational constant, r is the distance from the center of the star, m(r) is the mass inside a sphere of radius r, andp (r) and ε(r) are the pressure and energy density at the radius r, respectivy. To integrate the TOV equations, it is necessary to know the EOS of neutron star matter in a form ε(p ). Using the neutron star matter EOS otained in the previous section, we have integrated the TOV equations and otained the gravitational mass M and the radius R of compact stars (with and without quark degrees of freedom) for the different values of central pressure P c. Figures 7 and 8 illustrate the M(R) dependence of neutron stars for the two values of ag constant B = 6MeV fm 3 and B = 1 MeV fm 3, respectivy. We can see that the ehavior of the mass-radius dependence significantly differs for the two types of phase transitions. Figure 7 shows that for B =6MeV fm 3 there are unstale regions where dm/dp c < etween the two stale ranches of compact stars, corresponding to configurations with and without quark matter. In this case, there is a nonzero minimum value of the quark phase core radius. Accretion of matter on a critical neutron star configuration will then result in a catastrophic rearrangement of the star, forming a star with a quark matter core. The range of mass values for stars, containing the mixed phase, is [.85 M ;1.853 M ] for B =6MeV fm 3,andis[.997 M ;1.78 M ] for B = 1 MeV fm 3. In the case of a Maxwlian type phase transition, the analogous range is [.216 M ;1.828 M ] for B =6MeV fm 3. From Figure 8, one can oserve that in the case of B = 1 MeV fm 3 the star configurations with deconfined quark matter are unstale. Thus, the stale neutron star maximum mass is M. Our analysis shows that for B = 1 MeV fm 3, the pressure upper threshold value for the mixed phase is larger than the pressure, corresponding to the maximum mass configuration. Hence in this case, the mixed phase can exist in the center of compact stars, ut no pure quark matter can exist. The dash-dotted curve in Figure 8 represents the results in the case when we neglect the δ-meson fid (mod B1 σωρ). One can see that for a fixed gravitational mass, the star with the δ-meson fid has a larger radius than the corresponding star without the δ-meson fid. The influence of the δ-meson fid on the hyrid star properties is demonstrated in Tale 3, where we display the hyrid star properties with and without the δ-meson fid for minimum and maximum mass configurations. The results show that the minimum mass of hyrid stars and the corresponding radius are increased with the inclusion of the δ-meson fid. Notice that the influence of the δ-meson fid on the maximum mass configuration properties is insignificant.

9 Influence of a Scalar-Isovector δ-meson Fid on the Quark Phase Structure 1263 Fig. 7 Mass-radius ration of a neutron star with different deconfinement phase transitionscenarios for the ag constant B = 6MeV fm 3.Open circles and squares denote the critical configurations for the Glendenning and Maxwlian type transitions, respectivy. Solid circles and squares denote hyrid stars with minimal and maximal masses, respectivy. Fig. 8 Same as in Fig. 7, ut for B=1 MeV fm 3. The mass-radius ration in the case of the Glendennig construction without the δ-meson effective fid is also displayed for comparison (dashdotted curve). Tale 3 Hyrid Star Critical Configuration Properties for B = 1 MeV fm 3 with and without the δ-meson Fid Minimum Mass Configuration Maximum Mass Configuration Mod ε c M min R ε c M max R (MeV fm 3 ) (M ) (km) (MeV fm 3 ) (M ) (km) B1σωρδ B1σωρ CONCLUSIONS In this paper, we have studied the deconfinement phase transition of neutron star matter, when the nuclear matter is descried in the RMF theory with the δ-meson effective fid. We show that the inclusion of the scalar isovector δ-meson fid terms leads to the stiff nuclear matter EOS. In a nucleonic star, oth the gravitational mass and corresponding radius of the maximum mass stale configuration increases with the inclusion of the δ fid. The presence of the scalar isovector δ- meson fid alters the threshold characteristics of the mixed phase. For B =6MeV fm 3,the lower threshold parameters, n N, ε N,andP N, are increased, meanwhile the upper ones, n Q, ε Q,and P Q, are slowly decreased. For B = 1 MeV fm 3, this ehavior changes to the opposite. In case of the ag constant value B = 1 MeV fm 3, the pressure upper threshold value for the mixed phase is larger than the pressure, corresponding to the maximum mass configuration. This means that in this case, the stale compact star can possess a mixed phase core, ut the density range does not allow it to possess a pure strange quark matter core.

10 1264 G. B. Alaverdyan Stars with a δ-meson fid have a larger radius than stars of the same gravitational mass without the δ-meson fid. Alterations of the maximum mass configuration parameters caused y the inclusion of the δ-meson fid are insignificant. For the ag constant value B = 6MeV fm 3, the maximum mass configuration has a gravitational mass M max = M with radius R = 1.71 km and central density ρ c = gcm 3. This star has a pure strange quark matter core with radius r Q.83 km; next it has a nucleon-quark mixed phase layer with a thickness of r MP 9.43 km, followed y a normal nuclear matter layer with a thickness of r N.45 km. Acknowledgements The author would like to thank Profs. Yu. L. Vartanyan and G. S. Hajyan for fruitful discussions on issues rated to the suject of this research. This work was partially supported y the Ministry of Education and Sciences of the Repulic of Armenia under grant References Alaverdyan, G. B. 29a, Astrophysics, 52, 132 Alaverdyan, G. B. 29, Gravitation & Cosmology, 15, 5 Baym, G., Bethe, H. A., & Pethick, C. J. 1971, Nucl. Phys. A, 175, 225 Boguta J., & Bodmer, A. R. 1977, Nucl. Phys. A, 292, 413 Chodos, A., Jaffe, R. L., Johnson, K., Thorn, C. B., & Weisskopf, V. F. 1974, Phys. Rev. D, 9, 3471 Farhi, E., & Jaffe, R. L. 1984, Phys. Rev. D, 3, 2379 Glendenning, N. K. 1992, Phys. Rev. D, 46, 1274 Glendenning, N. K. 2, Compact Stars: nuclear physics, particle physics, and general rativity / Norman K. Glendenning (New York: Springer), 2, (Astronomy and astrophysics lirary) Greco, V., Colonna, M., Di Toro, M., & Matera, F. 23, Phys. Rev. C, 67, 1523 Heiserg, H., Pethick, C. J., & Stauo, E. F. 1993, Phys. Rev. Lett., 7, 1355 Heiserg, H., & Hjorth-Jensen, M. 2, Phys. Rep., 328, 237 Ko, C. M., & Li, G. Q. 1996, Journal of Physics G Nuclear Physics, 22, 1673 Kuis, S., & Kutschera, M. 1997, Phys. Lett. B, 399, 191 Lalazissis, G. A., Konig, J., & Ring, P. 1997, Phys. Rev. C, 55, 54 Liu, B., Greco, V., Baran, V., Colonna, M., & Di Toro, M. 22, Phys. Rev. C, 65, 4521 Müller, H., & Serot, B. D. 1995, Phys. Rev. C, 52, 272 Oppenheimer, J., & Volkoff, G. 1939, Phys. Rev. 55, 374 Prassa, V., Ferini, G., Gaitanos, T., Wolter, H. H., Lalazissis, G. A., & Di Toro, M. 27, Nuclear Physics A, 789, 311 Serot, B. D., & Walecka, J. D. 1986, in Adv. in Nucl. Phys., eds. J. W. Nege, & E. Vogt, 16 Serot, B. D., & Walecka, J. D. 1997, Int. J. Mod. Phys. E, 6, 515 Tolman, R. 1939, Phys. Rev. 55, 364 Typ, S., & Wolter, H. H. 1999, Nucl. Phys. A 656, 331 Walecka, J. D. 1974, Ann. Phys., 83, 491