1. Introduction

The equation of state (EOS) for neutron star matter and infinite nuclear matter has been intensively studied for many years (see, for examples, Refs [1-5]). A correct description of the EOS would have far reaching consequences for topics ranging from the cooling of neutron stars [1, 2] to the heavy ion collisions physics [3]. Furthermore, experiments of radioactive ion beam [4, 5] have provided new information on the structure of unstable nuclei far from equilibrium. The latter may open the possibility of extracting information on the EOS for asymmetric matter and the density dependence of the nuclear symmetry energy. In the present work we do not discuss heavy-ion collisions.

Moreover, Neutron stars are macroscopic objects where the stability is guaranteed by the Pauli principle of nucleons (together with a repulsive short-range interaction). Therefore the structure of a neutron star is dictated by the strong interaction (and, of course, gravity). The key ingredient that enters the stability condition is precisely the equation of state [6].

Recently, high-quality observational data of neutron stars set new stringent constraints for the EOS of cold and dense matter, otherwise inaccessible by experiment. The masses of two heavy pulsars have been determined with high precision. One of those is the radio pulsar (PSR) J1614−2230 with a mass M = (1.97 ± 0.04) M_ʘ [7] and the other PSR J0348+0432 accurately determined mass with M = (2.01 ± 0.04) M_ʘ [8], where M_ʘ denotes the mass of the sun. Only a sufficiently stiff EOS can support such neutron stars against gravitational collapse. Whereas neutron star radii are much less accurately known, the combination of available data makes these objects nonetheless an indispensable tool to constrain possible EOS [9].

Tews et al. [10] have done perturbation-theory calculations for pure neutron matter (PNM) with nuclear interactions derived from chiral perturbation theory including the full next-to-next-to-next-to-leading order (N³LO) contribution, with three- and four-body forces. Compared to calculations with three-body interactions defined only to next to- next-to-leading order (NNLO), the inclusion of all N³LO diagrams was found to be very important for nuclear structure and reactions. Their results provide constraints for the nuclear equation of state and for neutron-rich matter in astrophysics. While, in the present work, the N³LO NN potential complemented by phenomenological Urbana TBFs [11], instead of chiral three-nucleon forces, have been applied in calculations to neutron matter.

In this work we will derive the EOS of PNM by many-body theory, derived from different realistic nucleon-nucleon (NN) interactions such as the CD-Bonn potential [12], the N³LO potential [13] and the Argonne V₁₈ potential [14]. The many-body approaches that are used, the Brueckner-Hartree-Fock (BHF) approach with the inclusion of three-body force [11] or contact-term interaction to give more repulsive EOS [15].

The plan of the paper is the following. The main features of BHF approach used in this work including two types of three-body forces are shown in the following section. The results for the neutron matter equation of state are presented and discussed in Sec. 3. In Sec. 4 neutron star matter is examined and discussed in view of the new observational constraints. Finally, the conclusions and perspectives for the present work are given in Sec. 5.

2. BHF Approach

One main feature of the BHF approach to nuclear matter is that the binding energy and related quantities such as the self-energy, i.e. mass operator, can be developed according to the so-called hole-line expansion. Its expression is

(1)

where the subscript A means that the G-matrix has to be antisymmetrized.

The auxiliary single-particle energy e(k) is defined below according to the scheme of the iterative solution of the Bethe-Goldstone equation. If one chooses only U (k) =Re Σ(k,e) as auxiliary potential, then single-particle energy and has the form:

(2)

First we solve the Bethe-Goldstone equation

(3)

where V is the bare nucleon-nucleon interaction, ω = e (k) +e (k`) is the starting energy, and Q is the Pauli operator, which prevents two particles to scatter inside the Fermi sphere. Using the continuous choice [16], the auxiliary potential U(k) has been self-consistently evaluated along with the G-matrix from Eq. 3. Furthermore, within BHF approach with exact Pauli operator [17], one can easily evaluate the binding energy per nucleon for nuclear matter.

(4)

Where k_F is the Fermi momentum.

We use two methods to make the EOS for pure neutron matter be more repulsion. First method is done by adding the three-body-force. The phenomenological TBF model consists of an attractive term due to two-pion exchange with the excitation of an intermediate Δ-resonance, and a repulsive phenomenological term

(5)

The first term is the so-called two-pion exchange contribution. It is a cyclic sum over the nucleon indices i, j, k of products of anticommutator {,} and commutator [,] terms

(6)

Where

(7)

is the one-pion exchange operator, σ and τ are known as the Pauli spin and isospin operators, and

is called the tensor operator. Both Y(r) and T(r) are the Yukawa and tensor functions associated to the one-pion exchange as in the two-body potential. The repulsive term is written as

(8)

The strengths A and U are parameters that in the present work are adjusted to reproduce the exact saturation point of symmetric nuclear matter.

We introduced the Urbana three-nucleon model within the BHF approach. To incorporate the TBF in the Brueckner scheme we followed the method of Lejeune et al. [18]. The TBF is reduced to an effective two-body force by averaging on the position of the third particle, assuming that the probability of having two particles at a given distance is reduced according to the two-body correlation function. The resulting effective two-body force is of course density dependent. We have adjusted the parameters A and U in order to reproduce the correct saturation point of symmetric nuclear matter and the EOS become much more repulsive at high density, since the higher density region needed in neutron star studies.

The second method to achieve saturation in nuclear matter one has to add the effective interaction or the self-energy of BHF calculations by a simple contact interaction, which we have chosen following the notation of the Skyrme interaction to be in the form [19].

(9)

where ρ is the density and t₀, t₃ and α are free parameters. For symmetric nuclear matter α = 0.5, we have fitted t₀ and t₃ in such a way that BHF calculations plus the contact term (CT) yield the empirical saturation point for symmetric nuclear matter. In Table I, we show the values of t₀ and t₃ for CD-Bonn potential, N³LO potential and Argonne V₁₈ potential.

Table 1. The values of the parameters t0 and t3 that are used to fit the saturation point Parameters CD-Bonn N3LO Arg. V18 t0 -136.2609 -131.8981 -181.4014 t3 2440.734 2538.8274 2689.352

3. EOS for Pure Neutron Matter

The energy per nucleon of neutron matter is the only input required for determining the mass-radius relationship for neutron stars, and hence the ranges of physically allowable neutron star masses. In Figure 1. we show the results for the energy per nucleon in the case of a system composed of neutrons only. For three choices of the NN potential, curves of EOS with two types TBF are displayed supplemented to BHF approach. When phenomenological TBF are used, the parameters A and U are fixed by the calculation of saturation properties in the symmetric case.

Figure 1. The equation of state E/A in MeV for pure neutron matter as a function of density ρ in fm-3 for different potentials. The potentials employed are the CD-Bonn +TBF (solid line), the CD-Bonn + CT (dotted line), + TBF (short dashed line), + CT (dot-dashed line), the Argonne V18 potential + TBF (double dot-dashed line) and Argonne V18 potential + CT (double dashed dot line)

The introduction of TBF does not change qualitatively the density dependence of E/A but makes the EOS more stiff. The same behavior is noticed when contact interaction is added to the effective interaction to yield the empirical saturation point for symmetric nuclear matter with the same parameters defined in table 1.

We find agreement at low densities between all interactions extend to very high densities especially for CD-Bonn + CT and N³LO+ CT presented by dotted and dot-dashed lines and they are non-local potentials.

In contrast to EOS results with supplemented phenomenological terms, there are agreements at low densities between all interactions because the strong tensor effect in the ³S₁-³D₁ channel is absent. While, there are noticeable differences at very high densities for all interactions. The EOS for the Argonne V₁₈ + CT appears the most repulsive than the other potentials. Chiral N³LO + TBF EOS seem to provide a very soft pure neutron matter EOS. Moreover, there is another indication one can see it from Figure 1. We observe that the EOS for pure neutron matter takes only positive values this means that it is unbound. The energy per nucleon is rising approximately monotonically with increasing density, which is in agreement with the many-body calculations which it has been done by Kohno [20].

4. Neutron Star Structure

The structure of a neutron star is characterized by its mass and radius. Additional parameters of interest are the moment of inertia and the crust thickness; these are important for the dynamics and transport properties of pulsars. In the present work, we assume that a neutron star is a spherically symmetric distribution of mass in hydrostatic equilibrium. The effects of rotations can be neglected; assume there is no magnetic field. Then the equilibrium configurations are simply obtained by solving the Tolman – Oppenheimer - Volkoff (TOV) equations [21, 22]:

(10)

(11)

where P(r) and ρ are the pressure and density respectively at radius r, m(r) is the gravitational mass inside r, G = 1.327 × 10¹¹ km³/(M_ʘ.s²) is the universal constant of gravitation. The TOV equations are first order differential equations, which can be integrated by the following boundary conditions:

(12)

The first condition means that the density ρ and the pressure P(r) are finite at the center of the neutron star, the second condition determine the pressure at the surface of the neutron star. Starting with a central density ρ(r=0) ≡ ρ_c the solutions of these equations are obtained by integrating them out from the neutron-star center till its edge where P is zero. This gives the stellar radius R and the gravitational mass is then

(13)

It turns out that the mass of the neutron star has a maximum value as a function of radius (or central density), above which the star is unstable against collapse to a black hole. The value of the maximum mass depends on the nuclear EOS [23].

For the outer part of the neutron star (ρ < 0.001 fm^-3) we have used the equations of state by Baym et al. [24] whereas for the middle-density regime (0.001 fm^-3< ρ < 0.08 fm^-3), the results of Lorenz et al. [25] are used. Furthermore the present equations of states for pure neutron matter with the high-density EOSs (ρ > 0.08 fm^-3) employed to describe the core of the neutron star.

Figure 2. show the calculated mass-radius relations of neutron stars. Consistent with the EOSs in Figure 1, the maximum mass of the neutron star and the corresponding radius given by N³LO+TBF are the smallest, and those obtained by Argonne V₁₈ + CT are the largest. The results for CD-Bonn +CT are similar to N³LO+CT but its calculations for both mass and radius are a little smaller than V₁₈ + CT. In contrast, when the EOS results are supplemented by phenomenological TBF terms, there are differences observed in the maximum values of mass.

Figure 2. Predicted mass-radius relations of neutron stars are shown as a function of the stellar radius in km for different potentials. The potentials employed are the CD-Bonn +TBF (solid line), the CD-Bonn + CT (dotted line), + TBF (short dashed line), + CT (dot-dashed line), the Argonne V18 potential + TBF (double dot-dashed line) and Argonne V18 potential + CT (double dashed dot line)

Figure 3. displays the neutron star masses as a function of the central density ρ_c of the star, with several effective interactions mentioned above. From Figure 3, it can be seen that the N³LO+ TBF interaction exhibits the smallest maximum mass at the largest central density ρ_max. Table II shows the maximum mass limits M_max, the radius R and the corresponding central densities ρ_max extracted from Figures 2 and 3.

Figure 3. Neutron star gravitational masses MG (in units of Mʘ) as a function of the central density ρc in units of gm/cm3 for different potentials. The notations are the same as Figure 2

Table 2. The maximum mass, radius and ρc/ ρ0 of neutron star for different models Model Mmax (Mʘ) Radius (km) ρc/ ρ0 CD-Bonn +TBF 1.674 6.611 25 CD-Bonn + CT 1.977 9.469 11 N3LO+ TBF 1.605 6.271 28 N3LO + CT 1.959 9.559 11 Arg. V18 + TBF 1.741 6.908 22.5 Arg. V18 + CT 2.102 9.481 11

From Table 2, One finds a maximum mass of neutron star

at a central density ρ_c/ ρ₀=25 with a radius R = 6.611 km for CD-Bonn + TBF model. Also we find a maximum mass of neutron star

at a central density ρ_c/ ρ₀=11 with a radius R = 9.469 km for CD-Bonn + CT model. In the case of N³LO+ TBF, one notes that a maximum mass of

at a central density ρ_c/ ρ₀ = 28 with a radius R = 6.271 km. If we use N³LO+ CT, we find a maximum mass of

at a central density ρ_c/ ρ₀=11 with a radius R = 9.559 km. In the case of Argonne V₁₈ + TBF, we find a maximum mass of neutron star

at a central density ρ_c/ ρ₀=22.5 with a radius R = 6.908 km. Lastly, in the case of Argonne V₁₈ + CT a maximum mass of neutron star has the value

at a central density ρ_c/ ρ₀=11 with a radius R = 9.481 km.

Finally one can see from the two figures (2 and 3) that neutron stars properties calculated with a stiff equation of state (e.g., CD-Bonn + CT, N³LO + CT and Argonne V₁₈+ CT) have greater maximum masses than neutron stars derived from a soft equation of state. Furthermore neutron stars derived from a stiff equation of state have a lower central density and larger radius than do neutron stars of the same mass computed from a soft equation of state. The present results are more compatible with recent microscopic calculations of neutron matter based on nuclear interactions derived from chiral effective field theory [26]. They have constructed three representative EOSs (soft, intermediate, and stiff), consistent with the constraints from nuclear physics and observation. To one obtain more repulsive EOS, one can go beyond the present approach [27].

5. Conclusions

We studied neutron star properties, in particular NS’s maximum masses and radius using the BHF approximation with exact Pauli’s operator with the inclusion of two types of three body forces. The three body forces that are used, the phenomenological TBF and a simple contact term added to the effective interaction. It is found that neutron star properties are sensitive to microscopic model calculations, this means that the maximum masses and radii depend on the stiffness of the suggested EOS. Neutron star properties calculated with a stiff EOS have a lower central density, a larger radius than stars of the same mass computed from a soft EOS. In general, simple construction of neutron stars from the EOS of pure neutron matter is not enough for a consistency check between the theoretical neutron star maximum mass and the observed neutron star masses. If it is possible to duly deduce the information on the EOS of nuclear matter by ignoring other constituents than nucleons, it would be reasonable to focus on low mass neutron stars (see, e.g., Sotani et al. [28]).