Research Article  Open Access
The DensityDependent Correlations among Observables in Nuclear Matter and HyperonRich Neutron Stars
Abstract
The conserving meanfield approximation with nonlinear interactions of hadrons has been applied to examine properties of nuclear matter and hyperonic neutron stars. The nonlinear interactions that will produce densitydependent effective masses and coupling constants of hadrons are included in order to examine density correlations among properties of nuclear matter and neutron stars such as binding energy, incompressibility, , symmetry energy, , hyperononset density, and maximum masses of neutron stars. The conditions of conserving approximations in order to maintain thermodynamic consistency to an approximation are essential for the analysis of densitydependent correlations.
1. Introduction
The linear and nonlinear hadronic meanfield approximations have been extensively applied to finite nuclei, nuclear matter, and neutron stars [1–3]. The ground state of symmetric nuclear matter has always been a fundamental physical system in the understanding of complicated normal and high density, exotic nuclear manybody systems. The highdensity matter such as a neutron star has been actively investigated, observed masses of hadronic neutron stars are above , and the maximum masses of neutron stars are expected to be below [4, 5]. The neutron stars are also speculated to be hyperonmixed nuclear matter whose equation of state will provide us with important conditions to understand interactions of nuclear physics [6–9].
The current nonlinear meanfield approximation is constructed with  selfinteractions and mixing interactions of mesons. The nonlinear interactions of mesons are renormalized as the selfconsistent effective masses and effective coupling constants by the requirement of thermodynamic consistency, or equivalently the theory of conserving approximations [10–21]. The selfconsistent effective masses and effective coupling constants of mesons are essential to maintain selfconsistency in nonlinear approximations and examine correlations among properties of nuclear and highdensity, hyperonic matter.
The densitydependent correlations among properties of nuclear matter and neutron stars have been discussed in terms of effective masses and coupling constants of mesons and baryons, which are defined selfconsistently to maintain conditions of conserving approximations [20, 21]. The nonlinear  meanfield approximation suggests that densitydependent correlations induced by nonlinear interactions be significant and so, the analysis helps us understand nuclear manybody interactions.
The Lagrangian of nonlinear  meanfield approximation which yields densitydependent effective masses and coupling constants is [20, 21] where () and () denote the field of baryons and leptons, respectively. The mesonfields operators are replaced by expectation values in the ground state: for the field, for the vectorisoscalar meson, , ; the neutral meson meanfield, , is chosen for direction in isospin space. The masses in (1.1) are: MeV, MeV, MeV, and MeV, in order to compare the effects of nonlinear interactions and hyperonmatter with those of the linear  approximation discussed by Serot and Walecka [1].
The nonlinear model is motivated by preserving the structure of Serot and Walecka's linear  meanfield approximation [1], Lorentzinvariance and renormalizability, thermodynamic consistency: Landau's hypothesis of quasiparticles [22, 23], the Hugenholtzvan Hove theorem [24], and the virial theorem [25], and conditions of conserving approximations [11, 12, 20, 21]. The concept of effective masses and effective coupling constants is naturally generated by nonlinear interactions of mesons and baryons. The conditions of conserving approximations will require the functional form of single particle energy, effective masses, and coupling constants for selfconsistency, and then empirical values of lowdensity nuclear matter and highdensity neutron matter will be restricted with the effective masses and coupling constants [20, 21]. In other words, the admissible values of effective coupling constants and masses are confined in certain values due to strong densitydependent correlations among physical quantities of nuclear matter and neutron stars. The purpose of the analysis is to study densitydependent correlations among properties of nuclear matter and neutron stars with the minimum constraints at nuclear matter saturation and the maximum masses of hyperonic neutron stars.
The properties at saturation of symmetric nuclear matter and neutron stars are taken so as to fix nonlinear coupling constants. The binding energy at saturation is fixed as MeV at , and the symmetry energy, MeV. Then, the minimum value of incompressibility, , is determined by simultaneously maintaining the maximum masses of isospinasymmetric neutron stars to be [20, 21]. In this way, the densitydependent correlations among properties of nuclear matter and hyperonic neutron stars, , are investigated. The constraints will confine nonlinear parameters within certain values and suppress the effect of nonlinear interactions.
It can be checked numerically that the baryons and an electron, , are sufficient to determine the masses of hyperonic neutron stars; other hyperons can be included, but because of charge neutrality and selfconsistency, the other hyperononset densities are pushed up to high densities where EOSs of the hyperons are not so important to determine the properties of neutron stars. Consequently, other hyperons produce small densitydependent correlations to properties of nuclear matter and neutron stars compared to matter. In other words, the EOS of matter dominates the densityregion decisive to properties of neutron stars. This is one of the important results obtained in the current conserving nonlinear meanfield approximation.
The selfconsistency required by thermodynamic consistency restricts values of nonlinear coefficients. The suppressions of nonlinear coefficients and nonlinear interactions are directly observed in selfconsistent effective masses and selfenergies of mesons and baryons, which are discussed as naturalness of nonlinear corrections [20, 21]. The more accurately we can determine the observables and constraints for nuclear and highdensity matter, the better we would be able to understand interactions and correlations, or limitations of hadronic models. The conserving meanfield approximation is applied in order to extract densitydependent correlations among properties of nuclear matter and highdensity, hyperonic matter.
2. SelfConsistent Effective Masses and Coupling Constants of Mesons
The densitydependent, effective coupling constants are assumed to be induced by field, preserving Lorentzinvariance as simple as possible. We have assumed that only nucleonmeson coupling constants are densitydependent in the current analysis since we are interested in the density correlations among properties of symmetric nuclear matter and highdensity matter. The densitydependent nucleonmeson coupling constants are given byThe effective masses compatible with the effective coupling constants (2.1) are required to beThe effective masses of mesons and coupling constants have to be determined selfconsistently. Note that the effective mass depends on the () scalar source of nucleons, . The nonlinear meanfield approximation is thermodynamically consistent only if effective masses of mesons and coupling constants are renormalized as (2.1) and (2.2).
The introduction of nonlinear vertex interaction is equivalent to define the effective mass of nucleon asand the effective mass of hyperon isThe effective masses of nucleons and hyperons are obtained from (2.3) and (2.4):
The scalar sources of nucleons () and hyperons () are respectively given by [17] where is the modified scalar density defined by The hyperon sources are where is the Fermimomentum of the hyperon , and . The sum is performed to baryons, and is used to denote proton and neutron: ; the hyperons are denoted as, Although the hyperon coupling constants are not densitydependent in the current investigation, the densitydependent interactions of nucleons and selfconsistency will effectively modify hyperon coupling constants as . The density dependence of nucleon coupling constants and correlations are mainly investigated, since it is important to distinguish the densitydependences of nucleon coupling constants from those of hyperons for quantitative analyses of nuclear matter. The densitydependent interactions of hyperon coupling constants will be examined quantitatively in the near future.
The scalar sources of baryons are, respectively, given bywhere with . The meson and meson contributions to the selfenergy are given bywhere the isoscalar density, , is given byand the densitydependent ratios of hyperonnucleon coupling constants on , , are defined selfconsistently that will be explained in the next section. The selfenergies, and , are briefly denoted as ; the isovector density is denoted as where the Fermi momentum is for proton and for neutron. The baryonisovector density, , and the ratios of hyperonnucleon coupling constants on meson are also defined, for example, and .
The energy density, pressure of isospinasymmetric, and chargeneutral nuclear matter are calculated by way of the energymomentum tensor aswhere is the Fermi momentum for baryons. One can check that the thermodynamic relations, such as and the chemical potential, , are exactly satisfied for a given baryon density, . Hence, the HugenholtzVan Hove theorem to the approximation is exactly maintained in all densities. In Figures 1 and 2, the binding energies of ()() and ()() matter are shown. By comparing binding energies of phase transitions from () to () matter, it is clearly examined that the equation of state (EOS) becomes softer when a hyperon, , is produced. Note that the bare hyperoncoupling ratios are defined by and . The phase transition begins at and . As one can notice from Figures 1 and 2, the onset densities do not change with the given ratios, , expected from effective quark models [26]. Although properties of nuclear matter and EOS of neutron stars are sensitive to nonlinear interactions, the hyperononset densities confined by conservation laws and phaseequilibrium conditions indicate that the hyperononset densities are fairly fixed with respect to the change of nonlinear interactions in the current conserving meanfield approximation (see Table 1). The hyperononset densities seem to be densityindependent, though properties of nuclear and neutron matter are strongly densitydependent.

(a)  
 
(b)  

The equations of motion, selfenergies (2.6) and (2.9) enable one to obtain the effective coupling constants and masses, (2.1) and (2.2). In Figures 3 and 4, the effective masses of nucleons and hyperons (, ) after hyperononset densities are shown, respectively. The hyperon effective masses, and , behave almost the same as those of nucleons in high densities when the hyperon coupling ratios are . However, the other values of ratios, , indicate that density dependence of hyperons to effective masses are small in high densities and generate softer EOS, resulting in the lower maximum masses of neutron stars (see Table 1). As the softer EOS is examined in the twofold hyperon matter, (), it may be conjectured that manyhyperon matter () with the ratio, , would generate much softer EOS and be unable to support observed masses of neutron stars. In addition, many studies with hadronic field theory model independently indicate strong densitydependent interactions and correlations among properties of nuclear matter and neutron stars. Hence, the coupling ratios, , predicted by quarkbased effective models may not be compatible with those of hadronic models, which should be rigorously investigated to examine consistency and restriction of both hadronic and effective quark models.
3. The Phase Transition Conditions and HyperonOnset Densities
The hyperononset densities at phase transition are given by chemical potentials aswhere , , and , are the hyperon, neutron, and electron chemical potentials, and is the hyperon charge in the unit of . The phase transition conditions (3.1) are generally obtained by minimizing the energy density , and the baryons are restricted by the baryonnumber conservation and charge neutrality. The leptons are produced to maintain charge neutrality, and the lepton densities slowly increase for a low density region, but they decrease rapidly and vanish in high densities since the energies of leptons are absorbed and used to produce higher energy hyperons. The muon can be generated but restricted in a region narrower than that of an electron with the phaseequilibrium condition, , and so, the effect of the muon chemical potential is smaller than that of an electron.
The hyperononset densities are determined by chemical potentials which are equal to the single particle energy. The single particle energies of baryons, , are related to selfenergies which depend on effective masses and coupling constants induced by nonlinear interactions. The phaseequilibrium conditions (3.1) are complicated equations which interrelate the densitydependent interactions with hyperononset densities. The hyperononset densities are important to determine the maximum masses of neutron stars, since the generation of hyperons will soften the EOS of hyperonmixed nuclear matter. The EOS of hyperons depends also on binding energy and hyperon coupling constants given by densitydependent effective masses and coupling constants of nucleons. In this way, the correlations between properties of nuclear matter and hyperonic matter are intimately constructed to each other. The coupling constants of hyperons, and , play an essential role to determine onset densities.
The hyperon coupling constants, , can be calculated in terms of the effective masses, coupling constants, and binding energies of hyperons in the current conserving meanfield approximation. For example, suppose that () phase is generated after () phase. The hyperononset density is determined by the phase transition conditions (3.1), and the binding energy at the onsetdensity, , should be the lowest energy level of the hyperon (the hyperon single particle energy at saturation). The HugenholtzVan Hove theorem of a selfbound system at the onset density () leads toBy employing the effective masses of baryons (2.5) and the selfenergy of meson (2.9) with , one can obtainwhere , since ; is the lowest binding energy of isospin symmetric hyperon matter. The hyperonnucleon coupling ratio is determined by the densitydependent ratio, . Hence, the hyperoncoupling constants and the lowest binding energies of hyperons are constrained with effective coupling constants, masses of hadrons, nonlinear interactions, nuclear observables, and masses of neutron stars. The hyperononset density and hyperon EOS are intimately related to nonlinear interactions and properties of nuclear matter.
With a given ratio , the hyperononset densities are calculated by the phaseequilibrium conditions (3.1) and the hyperonnucleon coupling ratio (3.3), which are complicated functions of single particle energy and selfenergies, effective masses, and coupling constants of hadrons. Figures 5 and 6 show the EOS of ()() and ()() matter with coupling ratios on , ( is omitted to be concise). The EOSs after hyperononset densities become softer in the density range important to determine the masses of neutron stars. In addition, it can be checked numerically that the onset density, in the twofold hyperon production such as (), for example, is different from that of (). The onset density in () is pushed up to a high density: . This holds in general for other hyperons, since the additional new hyperon production requires high energy and pressure for nucleons to absorb energies of leptons so that nucleons can transform to hyperons. The hyperon phase, (), exists in the density range relevant to determine properties of neutron stars, and furthermore, the EOS is again softened by production. Hence, the equilibrium matter, (), and the hyperon matter, (), (), would be more important than () in order to study the maximum masses of stable neutron stars and properties of nuclear matter.
4. Incompressibilities and Symmetry Energies for High Density
The equation of state (EOS) given by (, , ) and TolmanOppenheimerVolkoff (TOV) equation [27–29] will enable one to calculate properties of nuclear matter at saturation and neutron stars. The values of nonlinear coupling constants are adjusted so that the binding energy at saturation is MeV at , and the symmetry energy is MeV, searching simultaneously the lower bound of nuclear incompressibility, , which corresponds to the maximum mass of neutron stars. The results are listed in Table 1. The coupling constants and effective masses, nonlinear interactions are strictly confined with these imposed constraints, and consequently, physical quantities exhibit strong densitydependent correlations. The derivation of equation of state, incompressibility and symmetry energy, correlations among properties of nuclear and neutron matter in the conserving nonlinear meanfield approximation have been discussed in detail [20, 21]. We exhibit characteristic densitydependent correlations of properties of nuclear matter such as incompressibility and symmetry energy of ()() and ()() matter.
The incompressibility, , and nucleon symmetry energy, , are respectively, calculated in the conserving meanfield approximation as [30, 31] The computation of nucleon symmetry energy must be performed by maintaining phaseequilibrium conditions, which will fix meanfields, , , and and the ground state energy, ; then, the derivative of the energy density can be calculated by changing and with fixed and mean fields. The hyperon onset and softening of EOS are perceived as the discontinuity and abrupt reduction of incompressibility shown in Figure 7. This characteristic property can be understood from the decreasing slope of binding energy curves in Figures 1 and 2 and would significantly change incompressibility, symmetry energy, and Landau parameters in high densities, which should be examined, for example, in heavyion collision experiments as a signal for the hyperon production [30–33]. The symmetry energies are monotonically increasing around saturation density, while they saturate in high densities [20, 21], as shown in Figure 8; the saturation of symmetry energy in a high density is also numerically checked in hyperon matter. The theoretical calculations of and depend on interactions of baryons, manybody interactions, and constraints such as isospin asymmetry and charge neutrality. The current results are different in high densities from those discussed in articles [30–35].
The lowest binding energies of and are fixed in the current calculation as MeV and MeV, respectively, [26, 36–38]. The lowest binding energies are related to the hyperoncoupling strength as shown in (3.3). We have checked the hyperononset densities by changing the values of binding energies to examine whether onset densities can be noticeably changed or not. In the numerical analysis, the hyperononset densities are fairly fixed with changes of , if is confined smaller than ; experimental values of are typically smaller than . It suggests that effective masses and coupling constants be more important to determine hyperononset densities than binding energies are. However, the binding energies of hyperons , effective masses, and coupling constants are important factors to determine the EOS and properties for neutron stars. Therefore, hyperononset densities, binding energies, and nonlinear interactions of hadrons will intimately interrelate properties of nuclear matter with those of neutron stars.
5. Remarks
The current conserving meanfield approximation and renormalized nonlinear interactions have exhibited interesting densitydependent correlations among observables of nuclear and highdensity hyperonic matter.
(1) The hyperononset densities are fairly fixed, respectively, although densitydependent interactions prominently affect the EOS and properties of nuclear and neutron matter. Therefore, the hyperononset density could be one of the important constraints on theoretical and experimental models of high density, exotic nuclear matter. The signals of hyperon production and onset density should be investigated further in heavyion collision experiments, and the results obtained in the current investigation should be examined carefully for nonlinear interactions including all other hyperons.
(2) The onset density of in the twofold hyperon phase, (), shifts to a higher density than that of (), and the EOS becomes softer. The twofold hyperon production requires high energy and pressure restricted by phaseequilibrium conditions, and Fermi energies of baryons will be redistributed among baryons to maintain the phaseequilibrium conditions and constraints, resulting in the reduction of Fermi energies (chemical potentials). The chemical potentials of leptons tend to be converted to those of baryons in high densities, and leptons vanish so that nuclear matter become baryonsonly phase (e.g., ()() phase transition in Figure 8). The conversion of chemical potentials among baryons in order to satisfy phasetransition conditions and constraints can be observed numerically with newly generated hyperons. The characteristic feature increases the hyperononset density higher and makes the EOS softer. Hence, it suggests that properties of neutron stars be mainly determined by (), (), and () matter rather than () matter; manyhyperon matter could be possible in a high density, such as in the core of neutron stars.
(3) The softening of EOS and discontinuity of incompressibility are interrelated to the strength of the hyperon coupling constants and effective masses of mesons and hyperons; hence, theoretical and experimental analyses of incompressibility and EOS in high densities are essential to determine physical quantities. The discontinuous change is also obtained for the symmetry energy for ()()() matter. The symmetry energy is monotonically increasing in the density range, , but it saturates in a high density (see Figure 8); the saturation of symmetry energy is the effect of both nonlinear interactions and isospin asymmetry [20, 21]. The theoretical predictions for the symmetry energy are very different in high densities. The value should be investigated actively in heavyion collision and other experiments to discriminate these predictions [34, 35].
(4) The binding energies, effective masses, and coupling constants of hyperons generate strong density correlations among properties of nuclear matter and neutron stars. Therefore, the binding energies and coupling ratios of hyperons, the hyperononset densities and signals of phase transition of (), (), and (), will, respectively, exhibit important information on saturation properties (the binding energy and density, incompressibility, and symmetry energy) not only for isospinsymmetric but also for isospinasymmetric nuclear matter and neutron stars [39].
(5) The values of hyperon coupling ratios, (), yield consistent results with the central energy density and the maximum mass configuration [5]. However, the hyperon coupling ratios, , suggested by effective quark models indicate that density interactions of baryons are weak in high densities. It seems to be inconsistent with predictions suggested by theoretical models of hadrons that densitydependent interactions be significant for nuclear matter and neutron stars. This aspect should be investigated further for both hadronic and effective quark models.
The densities of hyperon onset and phase transitions, () (), are sensitive to coupling ratios given by densitydependent effective masses and coupling constants of nucleons. The hyperononset densities and binding energies of hyperons are important to determine properties of EOS and neutron stars. Hence, the consistency of coupling strengths and binding energies of hyperons could be evaluated from certain astronomical data. The results suggest that the analyses of nuclear matter and neutron stars may provide important information on the models of nuclear and astronomical physics. The signals of the abrupt change of EOS, discontinuous change of incompressibility, and the saturation property of symmetry energy are essential to understand high density, exotic nuclear matter. The nonlinear meanfield approximation has exhibited interesting correlations among effective coupling constants and masses of hadrons, incompressibility, symmetry energy, and masses of hyperonic neutron stars. The properties of nuclear matter, neutron stars, and nuclear astrophysics are abundant in interesting physics to one another; the interdisciplinary progresses of these fields would be anticipated in the near future.
Acknowledgments
The authors would like to acknowledge Professor T. Muto of Chiba Institute of Technology for his valuable comments on binding energies of hyperons. The work is supported by Osaka Gakuin Junior College research grant for the 2008 Academic Year.
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Copyright © 2009 Schun T. Uechi and Hiroshi Uechi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.