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

Authors: Schun T. Uechi; Hiroshi Uechi
Journal: Advances in High Energy Physics
 (2009)

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## Abstract

The conservingσ-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.


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## Body

## 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 withσ-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 nonlinearσ-ω-ρ mean-field
approximation which yields density-dependent effective masses and coupling
constants is
[20, 21] (1.1)ℒNHA=∑Bψ¯B[iγμ∂μ−gωB∗γ0V0−gρB∗2γ0τ3R0−(MB−gσB∗ϕ0)]ψB−12mσ2ϕ02−gσ33!ϕ03−gσ44!ϕ04+12mω2V02+gω44!V04+gσω4ϕ02V02+12mρ2R02+gρ44!R04+gσρ4ϕ02R02+gωρ4V02R02+∑lψ¯l(iγμ∂μ−ml)ψl, where ψB (B=n,p,Λ,Σ,…) and ψl (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, VμVμ=V02−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, mσ=550 MeV, mω=783 MeV, and mρ=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 linearσ-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 kF=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 Mmax(n,p,e)=2.50M⊙  [20, 21]. In this way, the density-dependent correlations
among properties of nuclear matter and hyperonic neutron stars, Mstar(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 asnaturalness 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 byσ-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 by(2.1)gσN∗=gσN+(gσσN2)ϕ0,gωN∗=gωN+gσωNϕ0,gρN∗2=gρN2+gσρNϕ0.The effective masses compatible
with the effective coupling constants (2.1) are required to be(2.2)mσ∗2=mσ2(1+gσ32mσ2ϕ0+gσ43!mσ2ϕ02−gσω2mσ2V02−gσρ2mσ2R02−gσσN2mσ2ρsN),mω∗2=mω2(1+gω43!mω2V02+gσω2mω2ϕ02+gωρ2mω2R02),mρ∗2=mρ2(1+gρ43!mρ2R02+gσρ2mρ2ϕ02+gωρ2mρ2V02).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, ρsN. 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 nonlinearσσN-vertex interaction
is equivalent to define the effective mass of nucleon as(2.3)MN∗=MN−gσN∗ϕ0=MN−gσNϕ0−(gσσN2)ϕ02,and the effective mass of
hyperon H is(2.4)MH∗=MH−gσHϕ0.The effective masses of nucleons
and hyperons are obtained from (2.3) and (2.4):(2.5)MH−MH∗=gσHgσN∗(MN−MN∗).The scalar sources of nucleons (N) and hyperons (H) are respectively given by [17]
(2.6)ΣNs=igσN∗mσ∗2∫d4q(2π)4=Tr{(gσN*−gσωNV0γ0−gσρNR0γ0τ3)GD(q)}=−gσN*2mσ*2ρsN*,
where ρs* is the modified scalar density defined by gσN*ρs*=gσN*ρsN−gσωNV0ρB−gσρNR0ρ3.  The hyperon sources are(2.7)ΣHs=−gσN*mσ*2∑HgσH/gσN*π2∫0kFHdqq2MH*EH*(q). where kFB is the Fermi-momentum of the hyperon H, and EH∗(k)=(k2+MH∗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 gσH/gσN∗. 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
by(2.8)ρsB=∑BgσB/gσN∗π2∫0kFBdqq2MB∗EB∗(q),where gσB/gσN∗≡1 with B=(n,p). The ω-meson and ρ-meson
contributions to the self-energy are given by(2.9)Σωμ=−gωN∗2mω∗2ρωδμ,0,Σρ(np)μ=∓gρN∗24mρ∗2ρ3δμ,0,where the isoscalar density, ρω, is given by(2.10)ρω=ρp+ρn+∑HrHNωρH,and the density-dependent  ratios
of hyperon-nucleon  coupling constants on ω, rHNω, are defined self-consistently that will be explained
in the next section. The self-energies, Σρpμ and Σρnμ, are briefly denoted as Σρ(np)μ; the isovector density is denoted as ρ3=(kFp3−kFn3)/3π2 where the Fermi
momentum kFp is for proton
and kFn for neutron.
The baryon-isovector density, ρ3B, and the ratios of  hyperon-nucleon coupling constants
on ρ-meson are also
defined, for example, ρ3B=ρ3+(gρΣ/gρN∗)ρ3Σ and ρ3Σ=ρΣ+−ρΣ−.The energy density, pressure of isospin-asymmetric,
and charge-neutral nuclear matter are calculated by way of the energy-momentum
tensor as(2.11)ℰNHA=∑B1π2∫0kFBdkk2EB(k)+mσ22ϕ02+gσ33!ϕ03+gσ44!ϕ04−mω22V02−gω44!V04−gσω4ϕ02V02−(mρ22+gρ44!R02+gσρ4ϕ02+gωρ4V02)R02+∑l=e−,μ−1π2∫0kFldkk2El(k),pNHA=13π2∑B∫0kFBdkk4EB∗(k)−mσ22ϕ02−gσ33!ϕ03−gσ44!ϕ04+mω22V02+gω44!V04+gσω4ϕ02V02+(mρ22+gρ44!R02+gσρ4ϕ02+gωρ4V02)R02+∑l=e−,μ−13π2∫0kFldkk4El∗(k),where kFB is the Fermi
momentum for baryons. One can check that the thermodynamic relations, such as ℰNHA+pNHA=ρBEn(kFn) and the
chemical potential, μ=∂ℰNHA/∂ρB=En(kFn)=E∗(kFn)−Σ0(kFn), are exactly satisfied for a given baryon density, ρB=2kF3/3π2. 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 rΣ−Nσ=gσΣ−/gσN and rΛNσ=gσΛ/gσN. The phase transition begins at kFΣ−~1.6 fm−1  and kFΛ~1.7 fm−1. As one can notice from Figures 1 and 2, the onset
densities do not change with the given ratios, rHNσ=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 Mmax(n,p,e)=2.50M⊙ are listed. The
coupling constants are chosen from the data NHA2.50  in the paper
[20, 21].
gσgωgρgσ3 (MeV)gσ4gω4gρ4gσωgσρgωρ9.32610.4214.76510.020.020.04.0018.0−18.0−18.0gσσNgσωNgσρNgσ∗gω∗gρ∗−0.0180.0130.0489.06310.8007.567MN∗/Mmσ∗/mσmω∗/mωK (MeV)a4 (MeV)0.701.021.0132930.0Table 2
The maximum
masses,Mmax, and central energy densities, ℰC (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 rHNσ=gσH/gσN=1.0,2/3,1/3  [26].
(a)(n,p,e)-(n,p,Σ−,e)rΣ−NσMmaxℰc1.002.222.142/32.082.241/31.931.21(b)(n,p,e)-(n,p,Λ,e)rΛNσMmaxℰc1.002.222.152/31.671.001/31.561.02Figure 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)
rΣ−Nσ=gσΣ−/gσN=1.0, and (dashed line)
rΣ−Nσ=2/3, (dotted-dashed line)
rΣ−Nσ=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)
rΛNσ=gσΛ/gσN=1.0, and (dashed line)
rΛNσ=2/3, (dotted-dashed  line)
rΛNσ=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, MΣ−∗ and MΛ∗, behave almost the same as those of nucleons in high
densities when the hyperon coupling ratios are rHNσ=1. However, the other values of ratios, rHNσ=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, rHNσ<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, rHNσ<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 ofN
and Σ−. Note that the effective mass of
hyperon shows MΣ−∗ / MΣ−~1 as rΣ−Nσ=1/3. The smaller coupling ratios mean
less density-dependent interactions for the hyperon.Figure 4
The effective masses ofN
and Λ. Note that the effective mass of
hyperon shows MΛ∗/MΛ~1 as rΛNσ=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 as(3.1)μH=μn−qHμe,where μH, μn, and μe, are the hyperon, neutron, and electron chemical
potentials, and qH 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, μμ−=μe−, 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,EB(k)=(k2+MB∗2)1/2−Σω,ρ0(kB), 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, rHNσ and rHNω, play an essential role to determine onset densities.The hyperon coupling constants,rHNω, 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, αH, 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 (ρH=0) leads to(3.2)αH=((ℰρB)H−MH)ρH=0=EH(0)−MH=EH∗(0)−ΣωH0−MH=gωHV0+MH∗−MH.By employing the   effective
masses of baryons (2.5) and the self-energy of ω-meson (2.9) with Σω0=−gωN∗V0, one can obtain(3.3)rHNω=mω∗2gωNgωN∗ρω(gσHgσN∗(MN−MN∗)+αH)=mω∗2gωNgωN∗ρω(MH−MH∗+αH),where ρω=ρp+ρn, since ρH=0; αH is the lowest
binding energy of isospin symmetric hyperon matter. The hyperon-nucleon
coupling ratio is determined by the density-dependent ratio, gσH/gσN∗. 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 ratiorHNσ, the hyperon-onset densities are calculated by the
phase-equilibrium conditions (3.1) and the hyperon-nucleon coupling ratio rHNω (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, rHNσ=1.0,(1/3) (rHNσ=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, kFΛ 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: kFΛ~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 rΣ−Nσ=1/3 becomes softer in (n,p,Σ−,e) phase.Figure 6
The equation of state forΛ from (n,p,e) to (n,p,Λ,e). The EOS for rΛNσ=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 (ℰNHA, pNHA, ρB) 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 kF=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] (4.1)K=9ρB∂2ℰ∂ρB2,a4=12ρN[[∂2ℰ∂ρ32]ρN]ρ3=0.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(ρp,ρn); then, the derivative of the energy density ℰ(ρp,ρn) can be
calculated by changing ρp and ρn with fixed ρN=ρp+ρn 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 rΣ−Nσ=1.0 and rΛNσ=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 ofΛ 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 αH, if |αH| is confined
smaller than gσH/gσN∗(MN−MN∗); experimental values of |αH| are typically
smaller than gσH/gσN∗(MN−MN∗). 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 αH, 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 ofΛ 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, ρB/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, (rΛNσ~1,1/3,2/3), yield consistent results with the central energy
density and the maximum mass configuration
[5]. However, the hyperon coupling ratios, rHNσ≲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.



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*Source: 640919-2009-05-18.xml*