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We review theoretical relations between macroscopic properties of neutron stars and microscopic quantities of nuclear matter, such as consistency of hadronic nuclear models and observed masses of neutron stars. The relativistic hadronic field theory, quantum hadrodynamics (QHD), and mean-field approximations of the theory are applied to saturation properties of symmetric nuclear and neutron matter. The equivalence between mean-field approximations and Hartree approximation is emphasized in terms of renormalized effective masses and effective coupling constants of hadrons. This is important to prove that the direct application of mean-field (Hartree) approximation to nuclear and neutron matter is inadequate to examine physical observables. The equations of state (EOS), binding energies of nuclear matter, self-consistency of nuclear matter, are reviewed, and the result of chiral Hartree-Fock approximation is shown. Neutron stars and history of nuclear astrophysics, nuclear model and nuclear matter, possibility of hadron and hadron-quark neutron stars are briefly reviewed. The hadronic models are very useful and practical for understanding astrophysical phenomena, nuclear matter and radiation phenomena of nuclei.

A neutron star is a high density object that is anticipated from the gravitational collapse of a massive star during a supernova explosion. Such stars are hypothesized as high density objects mainly composed of neutrons and supported against further collapse because of Pauli exclusion principle exerted by nuclear particles. Hence, the balance between gravitational force and quantum mechanical force produced by nucleons is considered as the reason for stable existence of neutron stars.

The Fermi energy and pressure are produced by nuclear strong interactions, and therefore, properties of astrophysical phenomena and nuclear strong interactions can be directly interconnected.

This is an active research field of nuclear astrophysics. Several relations between many-body system of strongly interacting particles (nuclear matter) and high density matter, such as neutron, hyperon and quark matter are theoretically expected and explained by using relativistic models of nuclear physics [

In seeking an explanation for the origin of supernova and high density stars, neutron stars were proposed to be formed in a supernova explosion process (see,

It is proposed that the release of the gravitational binding energy of neutron stars powers the supernova and if the central part of a massive star before its collapse contains about

A magnetar is a type of neutron star with an extremely powerful magnetic field, and the emission of copious amounts of high energy electromagnetic radiation is considered as X-rays and gamma rays. The radiation can only be observed when the beam of emission is pointing towards the Earth, which is called the lighthouse effect. The strong magnetic field may have significant effects on both the equation of state (EOS) and the structure of neutron stars.

The magnitude of the magnetic field strength,

where

The polytropic equation of state is a useful approximation to consider many cosmological phenomena, because any material will be expected to behave like an ideal gas at a high enough temperature. When the temperature is far enough above the chemical potential or single particle energy of material, its macroscopic behavior is considered well described by a polytropic EOS, for example,

and

where

The radiation pressure of blackbody and systems in which radiation is important or dominant is expressed as

where

The radius

is implied from thermal observations of neutron star’s surface. Redshift is defined as

Moments of inertia, I

The physics of neutron stars offers an interesting interplay between nuclear physics and astrophysical observables. The relevant densities of neutron stars are far from the normal nuclear matter density,

The relevant degrees of freedom may not be the same in the crust region where the density is much smaller than the saturation density of nuclear matter, and also, hadronic models may be questionable in the center of the star where density is so high. It is essential to study these questions quantitatively by comparing hadronic and QCD equations of state, and discrepancies of predictions or model-independent predictions of both hadronic and quark theories are important in order to understand nuclear physics and sub-nuclear microscopic dynamics.

Within non-relativistic lowest-order Brueckner theory, it suggests that all the new phase-shift equivalent nucleon-nucleon potentials yield essentially similar equations of state up to densities of

However, the relativistic approaches, three-body interactions, onset of hyperons, pion and kaon condensations, superfluidity, and hadron-quark phase transitions are known to yield significant corrections to EOS above saturation density. In order to obtain consistent results from different approaches, it is essential to examine conditions and constraints to nuclear models and calculations.

B. D. Serot and J. D. Walecka started applications and analyses of relativistic field theory of nucleons known as quantum hadrodynamics (QHD) [

As one of the extended QHD models, the chiral effective

where

If coulomb forces were negligible and the number of nucleus was so large that the nucleus was the size of neutron stars

The mean-field approximation of (7) and other analogous nonlinear hadronic models are defined by replacing meson quantum fields with classical fields as,

and pion does not appear directly in the level of mean-field (Hartree) approximation, but the pion contributions appear from HF approximation. The definition of mean-field approximation is equivalent to Hartree approximation [

When J. D. Walecka introduced the mean-field approximation by replacing meson fields with classical fields, it was shown by B. D. Serot [

Although the QHD calculations have been successful during the last quarter of 20^{th} century, the original QHD model and approximations to QHD have been extended by including nonlinear interactions of mesons and chiral symmetry (reviews in Chapters 2, 3 and 5, 6 in [

Nonlinear hadronic models and chiral effective models based on QHD generate nonlinear interaction terms in the form of

much less than 1). The requirement of renormalizability in the level of Lagrangian is abandoned in the effective model of hadrons, but the nonlinear terms are considered as effective interactions in the level of approximations to QHD.

It is shown that nonlinear interactions are interpreted as manifestations of many-body interactions of hadrons and renormalized as effective masses

The self-consistency of approximations is essential to prove that all nonlinear approximations with meson interactions are equivalent to the mean-field approximation (Hartree approximation). The requirement of self- consistency is rigorously examined by the functional derivative of energy density,

where

The equivalence of the mean-field approximation with the Hartree approximation indicates that the mean-field approximation is not sufficient as the first approximation to nuclear matter. The exchange contributions, Fock terms, should be included, which are as important as the Hartree approximation. Hence, the self-consistent Dirac-Hartree-Fock approximation is sufficient for the first approximation to nuclear matter.

It is important to emphasize that the fundamental requirement (9) is necessary to construct the self-consistent DHF approximation, which is termed as the conserving DHF approximation (CDHF) [

One should be careful that the construction of HF approximation as well as other sophisticated approximations by way of Feynman diagram method or the functional derivative method are not necessarily equivalent [

The CDHF approximation improves experimental values derived from Hartree (mean-field) approximation, and physical reasons for the improvement can be explained by the binding energy curve at nuclear matter saturation density (

The exchange interactions (Fock contributions) to binding energy are important at low densities compared to Hartree contributions and should not be neglected for nuclear matter calculations.

Therefore, mean-field (Hartree) approximations with nonlinear interactions proposed by other researchers are not sufficient to examine nuclear matter calculations. One can conclude that Hartree-Fock approximation is the correct ground state for nuclear matter calculations.

The equation of state (EOS) derived from the conserving effective CDHF

where

The exchange interactions produce attractive interactions at low densities similar to those of Hartree contributions, which render the EOS of Hartree approximation softer around saturation density. The parameters of the effective chiral mean-field model are

It shows that self-consistency and chiral-symmetry are essentially related to saturation mechanism and confined by physical quantities, such as self-energies, effective masses of baryons and mesons, effective meson- nucleon coupling constants and properties of neutron stars.

In the calculation, nuclear symmetry energy,

where

the conserving effective DHF

The stability criterion for neutron stars is given by the derivative of neutron star mass with respect to central energy:

The MIT-bag model of QCD is employed to the equation of state for quark-phase calculations. The equation of state generated by MIT-bag model is connected to the hadronic phase (assumed the first-order phase transition) to calculate hadron-quark neutron stars [

As hadron-quark (H-Q) stars are 2-phase compact stars (i.e., the quark phase for a star’s core and hadron phase for a mantle), the stability of H-Q stars should be considered carefully. One can see that the quark-core (monotonically increasing dotted line in

Hence, the stability criterion,

The

The total mass of H-Q stars decreases at high densities (

star-core with quark particles by way of hadron-quark phase transition, resulting in the development of a stable quark-core.

Therefore, the H-Q star is stable in the sense that a stable quark-core is developing, although the total mass of the star decreases. The decreasing mass of hadron-mantle or equivalently the decreasing energy of hadron-man- tle could be used to convert hadrons to produce free quark particles in the deconfined vacuum relative to the confined vacuum, which may be the concept of the bag-constant,

By comparing stable energy densities of the quark phase with those of H-Q stars, the central energy density of stable H-Q stars is found to be extended at higher densities than that of single phase stars. This suggests that compact stars consisting of a mantle and a high density core are more stable at extremely high densities than stars in a homogeneous single phase structure [

When the QCD coupling constant,

and (

Although neutron stars and high density objects have been discussed in a rigorous fashion, they are still in a state of conjecture because of unknown, uncertain observables and parameters of theories and experiments. The interactions of related fields are important for the progress of science and understanding of nuclear astrophysics.

The mean-field approximations by replacing meson quantum fields with classical fields in nuclear hadronic models are all equivalent to the Hartree approximation when nonlinear interactions are properly renormalized [

The hadronic model employed in Section 4 yields reasonable results for properties of nuclear and neutron stars. However, the scalar

The nuclear physics and astrophysics are interrelated on the fundamental level, and hence, discoveries of each field would generate remarkable progresses on theories and technologies for both fields. The knowledge of nuclear physics is important for applications to diverse fields of science. Nuclear energy has been investigated to maintain the energy requirement for our societies. Radioactive materials are very useful for studies and applications of diverse fields. The progress of technologies to control energies is therefore essential, and we have to understand nuclear energies as clearly as possible both in theoretical structures and technological applications in order to control them for the purpose of our societies and environments.

Hiroshi Uechi, (2015) The Effective Chiral Model of Quantum Hadrodynamics Applied to Nuclear Matter and Neutron Stars. Journal of Applied Mathematics and Physics,03,114-123. doi: 10.4236/jamp.2015.32017