The Density-Dependent Correlations among Observables in Nuclear Matter and Hyperon-Rich Neutron Stars

Abstract

The conserving 𝜎-𝜔-𝜌 mean-field 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 density-dependent 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, 𝐾, symmetry energy, 𝑎₄, hyperon-onset 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 density-dependent correlations.

1. Introduction

The linear and nonlinear hadronic mean-field 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 many-body systems. The high-density matter such as a neutron star has been actively investigated, observed masses of hadronic neutron stars are above 1.3𝑀_⊙, and the maximum masses of neutron stars are expected to be below 2.5𝑀_⊙ [4, 5]. The neutron stars are also speculated to be hyperon-mixed nuclear matter whose equation of state will provide us with important conditions to understand interactions of nuclear physics [6–9].

The current nonlinear mean-field approximation is constructed with 𝜎-𝜔-𝜌 self-interactions and mixing interactions of mesons. The nonlinear interactions of mesons are renormalized as the self-consistent effective masses and effective coupling constants by the requirement of thermodynamic consistency, or equivalently the theory of conserving approximations [10–21]. The self-consistent effective masses and effective coupling constants of mesons are essential to maintain self-consistency in nonlinear approximations and examine correlations among properties of nuclear and high-density, hyperonic matter.

The density-dependent 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 self-consistently to maintain conditions of conserving approximations [20, 21]. The nonlinear 𝜎-𝜔-𝜌 mean-field approximation suggests that density-dependent correlations induced by nonlinear interactions be significant and so, the analysis helps us understand nuclear many-body interactions.

The Lagrangian of nonlinear 𝜎-𝜔-𝜌 mean-field approximation which yields density-dependent effective masses and coupling constants is [20, 21] ℒ_NHA=_𝐵𝜓_𝐵[𝑖𝛾_𝜇𝜕^𝜇−𝑔^∗_𝜔𝐵𝛾₀𝑉₀−𝑔^∗_𝜌𝐵2𝛾₀𝜏₃𝑅₀−(𝑀_𝐵−𝑔^∗_𝜎𝐵𝜙₀)]𝜓_𝐵−12𝑚²_𝜎𝜙²₀−𝑔_𝜎3𝜙3!³⁰−𝑔_𝜎4𝜙4!⁴⁰+12𝑚²_𝜔𝑉²₀+𝑔_𝜔4𝑉4!⁴⁰+𝑔_𝜎𝜔4𝜙²₀𝑉²₀+12𝑚²_𝜌𝑅²₀+𝑔_𝜌4𝑅4!⁴⁰+𝑔_𝜎𝜌4𝜙²₀𝑅²₀+𝑔_𝜔𝜌4𝑉²₀𝑅²₀+_𝑙𝜓_𝑙(𝑖𝛾_𝜇𝜕^𝜇−𝑚_𝑙)𝜓_𝑙, (1.1) where 𝜓_𝐵 (𝐵=𝑛,𝑝,Λ,Σ,…) and 𝜓_𝑙 (𝑙=𝑒⁻,𝜇⁻) denote the field of baryons and leptons, respectively. The meson-fields operators are replaced by expectation values in the ground state: 𝜙₀ for the 𝜎-field, 𝑉₀ for the vector-isoscalar 𝜔-meson, 𝑉_𝜇𝑉^𝜇=𝑉²₀−𝐕², (𝜇=0,1,2,3); the neutral 𝜌-meson mean-field, 𝑅₀, is chosen for 𝜏₃-direction in isospin space. The masses in (1.1) are: 𝑀=939 MeV, 𝑚_𝜎=550 MeV, 𝑚_𝜔=783 MeV, and 𝑚_𝜌=770 MeV, in order to compare the effects of nonlinear interactions and hyperon-matter with those of the linear 𝜎-𝜔 approximation discussed by Serot and Walecka [1].

The nonlinear model is motivated by preserving the structure of Serot and Walecka's linear 𝜎-𝜔 mean-field approximation [1], Lorentz-invariance and renormalizability, thermodynamic consistency: Landau's hypothesis of quasiparticles [22, 23], the Hugenholtz-van 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 self-consistency, and then empirical values of low-density nuclear matter and high-density 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 density-dependent correlations among physical quantities of nuclear matter and neutron stars. The purpose of the analysis is to study density-dependent 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 −15.75 MeV at 𝑘_𝐹=1.30 fm⁻¹, and the symmetry energy, 𝑎₄=30.0 MeV. Then, the minimum value of incompressibility, 𝐾, is determined by simultaneously maintaining the maximum masses of isospin-asymmetric neutron stars to be 𝑀_max(𝑛,𝑝,𝑒)=2.50𝑀_⊙ [20, 21]. In this way, the density-dependent correlations among properties of nuclear matter and hyperonic neutron stars, 𝑀_star(𝑛,𝑝,Σ⁻,Λ,𝑒), 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, (𝑛,𝑝,Σ⁻,Λ,𝑒), are sufficient to determine the masses of hyperonic neutron stars; other hyperons can be included, but because of charge neutrality and self-consistency, the other hyperon-onset 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 density-dependent correlations to properties of nuclear matter and neutron stars compared to (𝑛,𝑝,Σ⁻,Λ,𝑒) matter. In other words, the EOS of (𝑛,𝑝,Σ⁻,Λ,𝑒) matter dominates the density-region decisive to properties of neutron stars. This is one of the important results obtained in the current conserving nonlinear mean-field approximation.

The self-consistency required by thermodynamic consistency restricts values of nonlinear coefficients. The suppressions of nonlinear coefficients and nonlinear interactions are directly observed in self-consistent effective masses and self-energies 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 high-density matter, the better we would be able to understand interactions and correlations, or limitations of hadronic models. The conserving mean-field approximation is applied in order to extract density-dependent correlations among properties of nuclear matter and high-density, hyperonic matter.

2. Self-Consistent Effective Masses and Coupling Constants of Mesons

The density-dependent, effective coupling constants are assumed to be induced by 𝜎-field, preserving Lorentz-invariance as simple as possible. We have assumed that only nucleon-meson coupling constants are density-dependent in the current analysis since we are interested in the density correlations among properties of symmetric nuclear matter and high-density matter. The density-dependent nucleon-meson coupling constants are given by𝑔^∗_𝜎𝑁=𝑔_𝜎𝑁𝑔+(^𝜎𝜎𝑁2)𝜙₀,𝑔^∗_𝜔𝑁=𝑔_𝜔𝑁+𝑔_𝜎𝜔𝑁𝜙₀,𝑔^∗_𝜌𝑁2=𝑔_𝜌𝑁2+𝑔_𝜎𝜌𝑁𝜙₀. (2.1)The effective masses compatible with the effective coupling constants (2.1) are required to be𝑚_𝜎^∗2=𝑚²_𝜎𝑔(1+^𝜎32𝑚²_𝜎𝜙₀+𝑔_𝜎43!𝑚²_𝜎𝜙²₀−𝑔_𝜎𝜔2𝑚²_𝜎𝑉²₀−𝑔_𝜎𝜌2𝑚²_𝜎𝑅²₀−𝑔_𝜎𝜎𝑁2𝑚²_𝜎𝜌_𝑠𝑁𝑚),_𝜔∗2=𝑚²_𝜔𝑔(1+^𝜔43!𝑚²_𝜔𝑉²₀+𝑔_𝜎𝜔2𝑚²_𝜔𝜙²₀+𝑔_𝜔𝜌2𝑚²_𝜔𝑅²₀𝑚),_𝜌∗2=𝑚²_𝜌𝑔(1+^𝜌43!𝑚²_𝜌𝑅²₀+𝑔_𝜎𝜌2𝑚²_𝜌𝜙²₀+𝑔_𝜔𝜌2𝑚²_𝜌𝑉²₀). (2.2)The effective masses of mesons and coupling constants have to be determined self-consistently. Note that the effective mass depends on the (𝑛,𝑝) scalar source of nucleons, 𝜌_𝑠𝑁. The nonlinear mean-field 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 𝜎𝜎𝑁-vertex interaction is equivalent to define the effective mass of nucleon as𝑀^∗_𝑁=𝑀_𝑁−𝑔^∗_𝜎𝑁𝜙₀=𝑀_𝑁−𝑔_𝜎𝑁𝜙₀𝑔−(^𝜎𝜎𝑁2)𝜙²₀, (2.3)and the effective mass of hyperon 𝐻 is𝑀^∗_𝐻=𝑀_𝐻−𝑔_𝜎𝐻𝜙₀. (2.4)The effective masses of nucleons and hyperons are obtained from (2.3) and (2.4):𝑀_𝐻−𝑀^∗_𝐻=𝑔_𝜎𝐻𝑔^∗_𝜎𝑁(𝑀_𝑁−𝑀^∗_𝑁). (2.5)

The scalar sources of nucleons (𝑁) and hyperons (𝐻) are respectively given by [17] Σ^𝑠_𝑁𝑔=𝑖^∗𝜎𝑁𝑚_𝜎^∗2𝑑⁴𝑞(2𝜋)⁴=Tr{(𝑔^∗_𝜎𝑁−𝑔_𝜎𝜔𝑁𝑉₀𝛾⁰−𝑔_𝜎𝜌𝑁𝑅₀𝛾⁰𝜏₃)𝐺_𝐷𝑔(𝑞)}=−^∗2𝜎𝑁𝑚_𝜎^∗2𝜌^∗_𝑠𝑁, (2.6) where 𝜌^∗_𝑠 is the modified scalar density defined by 𝑔^∗_𝜎𝑁𝜌^∗_𝑠=𝑔^∗_𝜎𝑁𝜌_𝑠𝑁−𝑔_𝜎𝜔𝑁𝑉₀𝜌_𝐵−𝑔_𝜎𝜌𝑁𝑅₀𝜌₃. The hyperon sources areΣ^𝑠_𝐻𝑔=−^∗𝜎𝑁𝑚_𝜎^∗2_𝐻𝑔_𝜎𝐻/𝑔^∗_𝜎𝑁𝜋^{2𝑘𝐹𝐻}₀𝑑𝑞𝑞²𝑀^∗_𝐻𝐸^∗_𝐻(𝑞). (2.7) where 𝑘_𝐹𝐵 is the Fermi-momentum of the hyperon 𝐻, and 𝐸^∗_𝐻(𝑘)=(𝑘²+𝑀_𝐻^∗2)^1/2. The sum is performed to baryons, and 𝑁 is used to denote proton and neutron: 𝑁=(𝑝,𝑛); the hyperons are denoted as, 𝐻=Λ,Σ⁻,Σ⁰,Σ⁺,…. Although the hyperon coupling constants are not density-dependent in the current investigation, the density-dependent interactions of nucleons and self-consistency will effectively modify hyperon coupling constants as 𝑔_𝜎𝐻/𝑔^∗_𝜎𝑁. The density dependence of nucleon coupling constants and correlations are mainly investigated, since it is important to distinguish the density-dependences of nucleon coupling constants from those of hyperons for quantitative analyses of nuclear matter. The density-dependent interactions of hyperon coupling constants will be examined quantitatively in the near future.

The scalar sources of baryons are, respectively, given by𝜌_𝑠𝐵=_𝐵𝑔_𝜎𝐵/𝑔^∗_𝜎𝑁𝜋^{2𝑘𝐹𝐵}₀𝑑𝑞𝑞²𝑀^∗_𝐵𝐸^∗_𝐵(𝑞), (2.8)where 𝑔_𝜎𝐵/𝑔^∗_𝜎𝑁≡1 with 𝐵=(𝑛,𝑝). The 𝜔-meson and 𝜌-meson contributions to the self-energy are given byΣ^𝜇_𝜔𝑔=−^∗2𝜔𝑁𝑚_𝜔^∗2𝜌_𝜔𝛿_𝜇,0,Σ^𝜇_{𝜌(𝑝𝑛)}𝑔=∓^∗2𝜌𝑁4𝑚_𝜌^∗2𝜌₃𝛿_𝜇,0, (2.9)where the isoscalar density, 𝜌_𝜔, is given by𝜌_𝜔=𝜌_𝑝+𝜌_𝑛+_𝐻𝑟^𝜔_𝐻𝑁𝜌_𝐻, (2.10)and the density-dependent ratios of hyperon-nucleon coupling constants on 𝜔, 𝑟^𝜔_𝐻𝑁, are defined self-consistently that will be explained in the next section. The self-energies, Σ^𝜇_𝜌𝑝 and Σ^𝜇_𝜌𝑛, are briefly denoted as Σ^𝜇_{𝜌(𝑝𝑛)}; the isovector density is denoted as 𝜌₃=(𝑘³_𝐹𝑝−𝑘³_𝐹𝑛)/3𝜋² where the Fermi momentum 𝑘_𝐹𝑝 is for proton and 𝑘_𝐹𝑛 for neutron. The baryon-isovector density, 𝜌_3𝐵, and the ratios of hyperon-nucleon coupling constants on 𝜌-meson are also defined, for example, 𝜌_3𝐵=𝜌₃+(𝑔_𝜌Σ/𝑔^∗_𝜌𝑁)𝜌_3Σ and 𝜌_3Σ=𝜌_Σ+−𝜌_Σ−.

The energy density, pressure of isospin-asymmetric, and charge-neutral nuclear matter are calculated by way of the energy-momentum tensor asℰ_NHA=_𝐵1𝜋^{2𝑘𝐹𝐵}₀𝑑𝑘𝑘²𝐸_𝐵𝑚(𝑘)+^2𝜎2𝜙²₀+𝑔_𝜎3𝜙3!³⁰+𝑔_𝜎4𝜙4!⁴⁰−𝑚²_𝜔2𝑉²₀−𝑔_𝜔4𝑉4!⁴⁰−𝑔_𝜎𝜔4𝜙²₀𝑉²₀𝑚−(^2𝜌2+𝑔_𝜌4𝑅4!²⁰+𝑔_𝜎𝜌4𝜙²₀+𝑔_𝜔𝜌4𝑉²₀)𝑅²₀+_{𝑙=𝑒−,𝜇−}1𝜋^{2𝑘𝐹𝑙}₀𝑑𝑘𝑘²𝐸_𝑙𝑝(𝑘),_NHA=13𝜋²_𝐵^𝑘𝐹𝐵₀𝑘𝑑𝑘⁴𝐸^∗_𝐵−𝑚(𝑘)2^2𝜎𝜙²₀−𝑔_𝜎3𝜙3!³⁰−𝑔_𝜎4𝜙4!⁴⁰+𝑚²_𝜔2𝑉²₀+𝑔_𝜔4𝑉4!⁴⁰+𝑔_𝜎𝜔4𝜙²₀𝑉²₀𝑚+(^2𝜌2+𝑔_𝜌4𝑅4!²⁰+𝑔_𝜎𝜌4𝜙²₀+𝑔_𝜔𝜌4𝑉²₀)𝑅²₀+∑_{𝑙=𝑒−,𝜇−}13𝜋²∫^𝑘𝐹𝑙₀𝑘𝑑𝑘⁴𝐸^∗_𝑙,(𝑘) (2.11)where 𝑘_𝐹𝐵 is the Fermi momentum for baryons. One can check that the thermodynamic relations, such as ℰ_NHA+𝑝_NHA=𝜌_𝐵𝐸_𝑛(𝑘_𝐹𝑛) and the chemical potential, 𝜇=𝜕ℰ_NHA/𝜕𝜌_𝐵=𝐸_𝑛(𝑘_𝐹𝑛)=𝐸^∗(𝑘_𝐹𝑛)−Σ⁰(𝑘_𝐹𝑛), are exactly satisfied for a given baryon density, 𝜌_𝐵=2𝑘³_𝐹/3𝜋². Hence, the Hugenholtz-Van Hove theorem to the approximation is exactly maintained in all densities. In Figures 1 and 2, the binding energies of (𝑛,𝑝,𝑒)-(𝑛,𝑝,Σ⁻,𝑒) and (𝑛,𝑝,𝑒)-(𝑛,𝑝,Λ,𝑒) matter are shown. By comparing binding energies of phase transitions from (𝑛,𝑝,𝑒) to (𝑛,𝑝,𝐻,𝑒) matter, it is clearly examined that the equation of state (EOS) becomes softer when a hyperon, 𝐻, is produced. Note that the bare hyperon-coupling ratios are defined by 𝑟^𝜎_Σ⁻_𝑁=𝑔_𝜎Σ−/𝑔_𝜎𝑁 and 𝑟^𝜎_Λ𝑁=𝑔_𝜎Λ/𝑔_𝜎𝑁. The phase transition begins at 𝑘_𝐹Σ−∼1.6 fm⁻¹ and 𝑘_𝐹Λ∼1.7 fm⁻¹. As one can notice from Figures 1 and 2, the onset densities do not change with the given ratios, 𝑟^𝜎_𝐻𝑁=1,2/3,1/3, expected from effective quark models [26]. Although properties of nuclear matter and EOS of neutron stars are sensitive to nonlinear interactions, the hyperon-onset densities confined by conservation laws and phase-equilibrium conditions indicate that the hyperon-onset densities are fairly fixed with respect to the change of nonlinear interactions in the current conserving mean-field approximation (see Table 1). The hyperon-onset densities seem to be density-independent, though properties of nuclear and neutron matter are strongly density-dependent.

Table 1 

Properties of nuclear matter and (𝑛,𝑝,Σ⁻,𝑒), (𝑛,𝑝,Λ,𝑒) neutron stars. The properties of symmetric nuclear matter connected with isospin-asymmetric, beta-equilibrium matter (𝑛,𝑝,𝑒) whose EOS produces 𝑀_max(𝑛,𝑝,𝑒)=2.50𝑀_⊙ are listed. The coupling constants are chosen from the data NHA^2.50 in the paper [20, 21].

Table 2 

The maximum masses, 𝑀_max, and central energy densities, ℰ_𝐶 (10¹⁵g/cm³), of neutron stars produced by way of (a) (𝑛,𝑝,𝑒)-(𝑛,𝑝,Σ⁻,𝑒) and (b) (𝑛,𝑝,𝑒)-(𝑛,𝑝,Λ,𝑒) are listed, respectively, with the same coupling constants. The EOS of the hyperon phase (𝑛,𝑝,Σ⁻,𝑒) and (𝑛,𝑝,Λ,𝑒) is calculated with the ratio 𝑟^𝜎_𝐻𝑁=𝑔_𝜎𝐻/𝑔_𝜎𝑁=1.0,2/3,1/3 [26].

Figure 1 

The binding energies of (𝑛,𝑝,𝑒) and (𝑛,𝑝,Σ−,𝑒). The onset density of Σ− is about 𝑘𝐹=1.6 fm−1 as shown in the figure. The ratios of Σ−-coupling constant on 𝜎 are (dotted line) 𝑟𝜎Σ−𝑁=𝑔𝜎Σ−/𝑔𝜎𝑁=1.0, and (dashed line) 𝑟𝜎Σ−𝑁=2/3, (dotted-dashed line) 𝑟𝜎Σ−𝑁=1/3, respectively. The other coupling constants are fixed as in Table 1.

Figure 2 

The binding energies of (𝑛,𝑝,𝑒) and (𝑛,𝑝,Λ,𝑒). The onset density of Λ is about 𝑘𝐹=1.7 fm−1 as shown in the figure. The ratios of Λ-coupling constant on 𝜎 are (dotted line) 𝑟𝜎Λ𝑁=𝑔𝜎Λ/𝑔𝜎𝑁=1.0, and (dashed line) 𝑟𝜎Λ𝑁=2/3, (dotted-dashed line) 𝑟𝜎Λ𝑁=1/3, respectively. The other coupling constants are fixed as in Table 1.

The equations of motion, self-energies (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 (Σ⁻, Λ) after hyperon-onset densities are shown, respectively. The hyperon effective masses, 𝑀^∗_Σ⁻ and 𝑀^∗_Λ, behave almost the same as those of nucleons in high densities when the hyperon coupling ratios are 𝑟^𝜎_𝐻𝑁=1. However, the other values of ratios, 𝑟^𝜎_𝐻𝑁=1/3,2/3, 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 two-fold hyperon matter, (𝑛,𝑝,Σ⁻,Λ,𝑒), it may be conjectured that many-hyperon matter (𝑛,𝑝,Σ⁻,Λ,𝐻₁,𝐻₂,…,𝑒) with the ratio, 𝑟^𝜎_𝐻𝑁<1, 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 density-dependent interactions and correlations among properties of nuclear matter and neutron stars. Hence, the coupling ratios, 𝑟^𝜎_𝐻𝑁<1, predicted by quark-based 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.

Figure 3 

The effective masses of 𝑁 and Σ−. Note that the effective mass of hyperon shows 𝑀∗Σ− / 𝑀Σ−∼1 as 𝑟𝜎Σ−𝑁=1/3. The smaller coupling ratios mean less density-dependent interactions for the hyperon.

Figure 4 

The effective masses of 𝑁 and Λ. Note that the effective mass of hyperon shows 𝑀∗Λ/𝑀Λ∼1 as 𝑟𝜎Λ𝑁=1/3. The smaller coupling ratios indicate less density-dependent interactions for the hyperon.

3. The Phase Transition Conditions and Hyperon-Onset Densities

The hyperon-onset densities at phase transition are given by chemical potentials as𝜇_𝐻=𝜇_𝑛−𝑞_𝐻𝜇_𝑒, (3.1)where 𝜇_𝐻, 𝜇_𝑛, and 𝜇_𝑒, are the hyperon, neutron, and electron chemical potentials, and 𝑞_𝐻 is the hyperon charge in the unit of 𝑒. The phase transition conditions (3.1) are generally obtained by minimizing the energy density ℰ(𝑛,𝑝,𝐻,𝑒), and the baryons are restricted by the baryon-number 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 phase-equilibrium condition, 𝜇_𝜇−=𝜇_𝑒−, and so, the effect of the muon chemical potential is smaller than that of an electron.

The hyperon-onset densities are determined by chemical potentials which are equal to the single particle energy. The single particle energies of baryons, 𝐸_𝐵(𝑘)=(𝑘²+𝑀_𝐵^∗2)^1/2−Σ⁰_𝜔,𝜌(𝑘_𝐵), are related to self-energies which depend on effective masses and coupling constants induced by nonlinear interactions. The phase-equilibrium conditions (3.1) are complicated equations which interrelate the density-dependent interactions with hyperon-onset densities. The hyperon-onset densities are important to determine the maximum masses of neutron stars, since the generation of hyperons will soften the EOS of hyperon-mixed nuclear matter. The EOS of hyperons depends also on binding energy and hyperon coupling constants given by density-dependent 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, 𝑟^𝜎_𝐻𝑁 and 𝑟^𝜔_𝐻𝑁, play an essential role to determine onset densities.

The hyperon coupling constants, 𝑟^𝜔_𝐻𝑁, can be calculated in terms of the effective masses, coupling constants, and binding energies of hyperons in the current conserving mean-field approximation. For example, suppose that (𝑛,𝑝,𝐻,𝑒) phase is generated after (𝑛,𝑝,𝑒) phase. The hyperon-onset density is determined by the phase transition conditions (3.1), and the binding energy at the onset-density, 𝛼_𝐻, should be the lowest energy level of the hyperon 𝐻 (the hyperon single particle energy at saturation). The Hugenholtz-Van Hove theorem of a self-bound system at the onset density (𝜌_𝐻=0) leads to𝛼_𝐻ℰ=((𝜌_𝐵)_𝐻−𝑀_𝐻)_𝜌𝐻=0=𝐸_𝐻(0)−𝑀_𝐻=𝐸^∗_𝐻(0)−Σ⁰_𝜔𝐻−𝑀_𝐻=𝑔_𝜔𝐻𝑉₀+𝑀^∗_𝐻−𝑀_𝐻. (3.2)By employing the effective masses of baryons (2.5) and the self-energy of 𝜔-meson (2.9) with Σ⁰_𝜔=−𝑔^∗_𝜔𝑁𝑉₀, one can obtain𝑟^𝜔_𝐻𝑁=𝑚_𝜔^∗2𝑔_𝜔𝑁𝑔^∗_𝜔𝑁𝜌_𝜔(𝑔_𝜎𝐻𝑔^∗_𝜎𝑁(𝑀_𝑁−𝑀^∗_𝑁)+𝛼_𝐻𝑚)=^𝜔∗2𝑔_𝜔𝑁𝑔^∗_𝜔𝑁𝜌_𝜔(𝑀_𝐻−𝑀^∗_𝐻+𝛼_𝐻), (3.3)where 𝜌_𝜔=𝜌_𝑝+𝜌_𝑛, since 𝜌_𝐻=0; 𝛼_𝐻 is the lowest binding energy of isospin symmetric hyperon matter. The hyperon-nucleon coupling ratio is determined by the density-dependent ratio, 𝑔_𝜎𝐻/𝑔^∗_𝜎𝑁. Hence, the hyperon-coupling 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 hyperon-onset density and hyperon EOS are intimately related to nonlinear interactions and properties of nuclear matter.

With a given ratio 𝑟^𝜎_𝐻𝑁, the hyperon-onset densities are calculated by the phase-equilibrium conditions (3.1) and the hyperon-nucleon coupling ratio 𝑟^𝜔_𝐻𝑁 (3.3), which are complicated functions of single particle energy and self-energies, effective masses, and coupling constants of hadrons. Figures 5 and 6 show the EOS of (𝑛,𝑝,𝑒)-(𝑛,𝑝,Σ⁻,𝑒) and (𝑛,𝑝,𝑒)-(𝑛,𝑝,Λ,𝑒) matter with coupling ratios on 𝜎, 𝑟^𝜎_𝐻𝑁=1.0,(1/3) (𝑟^𝜎_𝐻𝑁=2/3 is omitted to be concise). The EOSs after hyperon-onset densities become softer in the density range important to determine the masses of neutron stars. In addition, it can be checked numerically that the Λ-onset density, 𝑘_𝐹Λ in the two-fold hyperon production such as (𝑛,𝑝,Σ⁻,Λ,𝑒), for example, is different from that of (𝑛,𝑝,Λ,𝑒). The Λ-onset density in (𝑛,𝑝,Σ⁻,Λ,𝑒) is pushed up to a high density: 𝑘_𝐹Λ∼2.2 fm⁻¹. 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, (𝑛,𝑝,Σ⁻,Λ,𝑒), exists in the density range relevant to determine properties of neutron stars, and furthermore, the EOS is again softened by Λ production. Hence, the 𝛽-equilibrium matter, (𝑛,𝑝,𝑒), and the hyperon matter, (𝑛,𝑝,Σ⁻,𝑒), (𝑛,𝑝,Σ⁻,Λ,𝑒), would be more important than (𝑛,𝑝,Σ⁻,Λ,𝐻₁,𝐻₂,…,𝑒) in order to study the maximum masses of stable neutron stars and properties of nuclear matter.

Figure 5 

The equation of state for (𝑛,𝑝,𝑒)-(𝑛,𝑝,Σ−,𝑒). Note that the equation of state for 𝑟𝜎Σ−𝑁=1/3 becomes softer in (𝑛,𝑝,Σ−,𝑒) phase.

Figure 6 

The equation of state for Λ from (𝑛,𝑝,𝑒) to (𝑛,𝑝,Λ,𝑒). The EOS for 𝑟𝜎Λ𝑁=1/3 becomes softer in (𝑛,𝑝,Λ,𝑒) phase.

4. Incompressibilities and Symmetry Energies for High Density

The equation of state (EOS) given by (ℰ_NHA, 𝑝_NHA, 𝜌_𝐵) and Tolman-Oppenheimer-Volkoff (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 −15.75 MeV at 𝑘_𝐹=1.30 fm⁻¹, and the symmetry energy is 𝑎₄=30.0 MeV, searching simultaneously the lower bound of nuclear incompressibility, 𝐾, 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 density-dependent correlations. The derivation of equation of state, incompressibility and symmetry energy, correlations among properties of nuclear and neutron matter in the conserving nonlinear mean-field approximation have been discussed in detail [20, 21]. We exhibit characteristic density-dependent correlations of properties of nuclear matter such as incompressibility and symmetry energy of (𝑛,𝑝,𝑒)-(𝑛,𝑝,Σ⁻,𝑒) and (𝑛,𝑝,𝑒)-(𝑛,𝑝,Λ,𝑒) matter.

The incompressibility, 𝐾, and nucleon symmetry energy, 𝑎₄, are respectively, calculated in the conserving mean-field approximation as [30, 31] 𝐾=9𝜌_𝐵𝜕²ℰ𝜕𝜌²_𝐵,𝑎₄=12𝜌_𝑁𝜕[[²ℰ𝜕𝜌²₃]_𝜌𝑁]_𝜌3=0. (4.1)The computation of nucleon symmetry energy must be performed by maintaining phase-equilibrium conditions, which will fix mean-fields, 𝜙₀, 𝑉₀, and 𝑅₀ and the ground state energy, ℰ(𝜌_𝑝,𝜌_𝑛); then, the derivative of the energy density ℰ(𝜌_𝑝,𝜌_𝑛) can be calculated by changing 𝜌_𝑝 and 𝜌_𝑛 with fixed 𝜌_𝑁=𝜌_𝑝+𝜌_𝑛 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 heavy-ion 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 𝐾 and 𝑎₄ depend on interactions of baryons, many-body 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].

Figure 7 

Incompressibilities of (𝑛,𝑝,𝑒) and (𝑛,𝑝,Λ,𝑒). The discontinuity of incompressibility occurs at 𝑘𝐹=1.7 fm−1, because of the phase transition from (𝑛,𝑝,𝑒) to (𝑛,𝑝,Λ,𝑒). 𝜌0 is the saturation density of symmetric nuclear matter.

Figure 8 

Nucleon symmetry energies of (𝑛,𝑝,𝑒), (𝑛,𝑝,Σ−,𝑒), and (𝑛,𝑝,Λ,𝑒) are shown with the solid, doted, and dot-dashed lines, respectively. The ratios of hyperon coupling constants are 𝑟𝜎Σ−𝑁=1.0 and 𝑟𝜎Λ𝑁=1.0. The discontinuity of symmetry energy, (𝑛,𝑝,Σ−,𝑒), occurs at 𝜌𝐵/𝜌0∼3.641, because of the phase transition from (𝑛,𝑝,Σ−,𝑒) to (𝑛,𝑝,Σ−).

The lowest binding energies of Λ and Σ⁻ are fixed in the current calculation as −28 MeV and 20 MeV, respectively, [26, 36–38]. The lowest binding energies are related to the hyperon-coupling strength as shown in (3.3). We have checked the hyperon-onset densities by changing the values of binding energies to examine whether onset densities can be noticeably changed or not. In the numerical analysis, the hyperon-onset densities are fairly fixed with changes of 𝛼_𝐻, if |𝛼_𝐻| is confined smaller than 𝑔_𝜎𝐻/𝑔^∗_𝜎𝑁(𝑀_𝑁−𝑀^∗_𝑁); experimental values of |𝛼_𝐻| are typically smaller than 𝑔_𝜎𝐻/𝑔^∗_𝜎𝑁(𝑀_𝑁−𝑀^∗_𝑁). It suggests that effective masses and coupling constants be more important to determine hyperon-onset densities than binding energies are. However, the binding energies of hyperons 𝛼_𝐻, effective masses, and coupling constants are important factors to determine the EOS and properties for neutron stars. Therefore, hyperon-onset 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 mean-field approximation and renormalized nonlinear interactions have exhibited interesting density-dependent correlations among observables of nuclear and high-density hyperonic matter.

(1) The hyperon-onset densities are fairly fixed, respectively, although density-dependent interactions prominently affect the EOS and properties of nuclear and neutron matter. Therefore, the hyperon-onset 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 heavy-ion 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 Λ in the two-fold hyperon phase, (𝑛,𝑝,Σ⁻,Λ,𝑒), shifts to a higher density than that of (𝑛,𝑝,Λ,𝑒), and the EOS becomes softer. The two-fold hyperon production requires high energy and pressure restricted by phase-equilibrium conditions, and Fermi energies of baryons will be redistributed among baryons to maintain the phase-equilibrium 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 baryons-only phase (e.g., (𝑛,𝑝,Σ⁻,𝑒)-(𝑛,𝑝,Σ⁻) phase transition in Figure 8). The conversion of chemical potentials among baryons in order to satisfy phase-transition conditions and constraints can be observed numerically with newly generated hyperons. The characteristic feature increases the hyperon-onset density higher and makes the EOS softer. Hence, it suggests that properties of neutron stars be mainly determined by (𝑛,𝑝,𝑒), (𝑛,𝑝,Σ⁻,𝑒), and (𝑛,𝑝,Σ⁻,Λ,𝑒) matter rather than (𝑛,𝑝,Σ⁻,Λ,𝐻₁,𝐻₂,…,𝑒) matter; many-hyperon 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 (𝑛,𝑝,𝑒)-(𝑛,𝑝,Σ⁻,𝑒)-(𝑛,𝑝,Σ⁻) matter. The symmetry energy is monotonically increasing in the density range, 𝜌_𝐵/𝜌₀≲3.0, 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 heavy-ion 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 hyperon-onset densities and signals of phase transition of (𝑛,𝑝,𝑒), (𝑛,𝑝,Σ⁻,𝑒), and (𝑛,𝑝,Σ⁻,Λ,𝑒), will, respectively, exhibit important information on saturation properties (the binding energy and density, incompressibility, and symmetry energy) not only for isospin-symmetric but also for isospin-asymmetric nuclear matter and neutron stars [39].

(5) The values of hyperon coupling ratios, (𝑟^𝜎_Λ𝑁∼1,1/3,2/3), yield consistent results with the central energy density and the maximum mass configuration [5]. However, the hyperon coupling ratios, 𝑟^𝜎_𝐻𝑁≲1, 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 density-dependent 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, (𝑛,𝑝,𝑒) → (𝑛,𝑝,Σ⁻,Λ,𝑒,…), are sensitive to coupling ratios given by density-dependent effective masses and coupling constants of nucleons. The hyperon-onset 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 mean-field 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.