Hyperon-hyperon interactions and properties of neutron matter

We present results from Brueckner-Hartree-Fock calculatons for beta stable neutron star matter with nucleonic and hyperonic degress degrees of freedom, employing the most recent parametrizations of the baryon-baryon interaction of the Nijmegen group. It is found that the only strange baryons emergin in beta stable matter up to total barionic densities of 1.2 fm^-3 are $\Sigma^-$ and $\Lambda$. The corresponding equations of state are then used to compute properties of neutron stars such as masses and radii.


I. INTRODUCTION
The physics of compact objects like neutron stars offers an intriguing interplay between nuclear processes and astrophysical observables. Neutron stars exhibit conditions far from those encountered on earth; typically, expected densities ρ of a neutron star interior are of the order of 10 3 or more times the 'neutron drip' density ≈ 4 · 10 11 g/cm 3 , where nuclei begin to dissolve and merge together. Thus, the determination of an equation of state (EoS) for dense matter is essential to calculations of neutron star properties. The EoS determines properties such as the mass range, the mass-radius relationship, the crust thickness and the cooling rate. The same EoS is also crucial in calculating the energy released in a supernova explosion.
At densities near to the saturation density of nuclear matter, (ρ 0 = 0.16 fm −3 ), we expect the matter to be composed of mainly neutrons, protons and electrons in β-equilibrium, since neutrinos have on average a mean free path larger than the radius of the neutron star. The equilibrium conditions can then be summarized as where µ i and ρ i refer to the chemical potential and density of particle species i, respectively.
At the saturation density of nuclear matter, ρ 0 , the electron chemical potential is of the order ∼ 100 MeV. Once the rest mass of the muon is exceeded, it becomes energetically favorable for an electron at the top of the e − Fermi surface to decay into a µ − . A Fermi sea of degenerate negative muons starts then to develop and, consequently, the charge balance needs to be changed according to ρ p = ρ e + ρ µ as well as requiring that µ e = µ µ .
As the density increases, new hadronic degrees of freedom may appear in addition to neutrons and protons. One such degree of freedom is hyperons, baryons with a strangeness content. Contrary to terrestrial conditions where hyperons are unstable and decay into nucleons through the weak interaction, the equilibrium conditions in neutron stars can make the inverse process happen, so that the formation of hyperons becomes energetically favorable. As soon as the chemical potential of the neutron becomes sufficiently large, energetic neutrons can decay via weak strangeness non-conserving interactions into Λ hyperons leading to a Λ Fermi sea with µ Λ = µ n . However, one expects Σ − to appear via at lower densities than the Λ, even though Σ − is more massive. The negatively charged hyperons appear in the ground state of matter when their masses equal µ e + µ n , while the neutral hyperon Λ appears when its mass equals µ n . Since the electron chemical potential in matter is larger than the mass difference m Σ − − m Λ = 81.76 MeV, Σ − will appear at lower densities than Λ. For matter with hyperons as well the chemical equilibrium conditions become Hyperonic degrees of freedom have been considered by several authors, mainly within the framework of relativistic mean field models [1][2][3] or parametrized effective interactions [4], see also Balberg et al. [5] for a recent update. Realistic hyperon-nucleon interactions were employed by Schulze et al. [6], in a many-body calculation in order to study the onset of hyperon formation in neutron star matter. In a recent work [7], they extend their work to study the properties of neutron stars with hyperons, paying special attention to the role played by three-body nucleon forces. All these works show that hyperons appear at densities of the order of ∼ 2ρ 0 .
In Refs. [6,7] the hyperon-hyperon interaction was not included. However, it is clear that as soon as the Σ − hyperon appears, one needs to consider the interaction between hyperon pairs since it will influence the single-particle energy of hyperons, hence affecting the equilibrium conditions from that density on and the density where other hyperons (e.g. the Λ) appear. The aim of this work is thus to present results of many-body calculations for β-stable neutron star matter with hyperonic degrees of freedom, employing interactions which also account for strangeness S < −1. To achieve this goal, our many-body scheme starts with the most recent parametrization of the free baryon-baryon potentials for the complete baryon octet as defined by Stoks and Rijken in Ref. [8]. This entails a microscopic description of matter starting from realistic nucleon-nucleon, hyperon-nucleon and hyperonhyperon interactions. In a recent work [9] we have developed a formalism for microscopic Brueckner-type calculations of dense nuclear matter that includes all types of baryon-baryon interactions and allows to treat any asymmetry on the fractions of the different species (n, p, Λ, Σ − , Σ 0 , Σ + , Ξ − and Ξ 0 ).
Here we extend the calculations of Ref. [9] to studies of β-stable neutron star matter. A brief summary of the formalism discussed in Ref. [9] is presented in section II. Our results are shown in section III. In III A we discuss the equation of state (EoS) and the composition of β-stable matter with strangeness, using various nucleonic contributions to the EoS. Based on the composition of matter we discuss in section III B the possible neutron star structures.
Our conclusions are given in section IV.

II. FORMALISM
Our many-body scheme starts with the most recent parametrization of the free baryonbaryon potentials for the complete baryon octet as defined by Stoks and Rijken in Ref. [8].
This potential model, which aims at describing all interaction channels with strangeness from S = 0 to S = −4, is based on SU(3) extensions of the Nijmegen potential models [10] for the S = 0 and S = −1 channels, which are fitted to the available body of experimental data and constrain all free parameters in the model. In our discussion we employ the interaction version NSC97e of Ref. [8], since this model, together with the model NSC97f of Ref. [8], results in the best predictions for hypernuclear observables [10]. For a discussion of other interaction models, see Refs. [8,11].
With a given interaction model, the next step is to introduce effects from the nuclear medium. Here we will construct the so-called G-matrix, which takes into account short-range correlations for all strangeness sectors, and solve the equations for the single-particle energies of the various baryons self-consistently. The G-matrix is formally given by Here B i represents all possible baryons n, p, Λ, Σ − , Σ 0 , Σ + , Ξ − and Ξ 0 and their quantum numbers such as spin, isospin, strangeness, linear momenta and orbital momenta.
Q is the Pauli operator which allows only intermediate states B 5 B 6 compatible with the Pauli principle, and the energy variable ω is the starting energy defined by the singleparticle energies of the incoming external particles B 3 B 4 . The G-matrix is solved using relative and centre-of-mass coordinates, see e.g., Refs. [9,11] for computational details. The single-particle energies are given by where T B i is the kinetic energy and M B i the mass of baryon B i . We note that the G-matrix, Eq. (4), has been solved using the standard prescription (i.e. E B = T B + M B ) for the intermediate states B 5 B 6 . The single-particle potential U B i is defined by where the linear momentum of the intermediate single-particle state B j is limited by the size of the Fermi surface F j for particle species B j . The matrix element in Eq. hyperons. Detailed expressions for the single-particle energies and the G-matrices involved can be found in Ref. [9].
The total non-relativistic energy density, ε, measured with respect to the nucleon mass, is obtained by adding the non-interacting leptonic contribution, ε l , and the baryonic contribution, ε b , the latter being obtained from the baryon single-particle potentials The total binding energy per baryon, E, is then given by where ρ is the total baryonic density. The density of a given fermion species is given by where x i is the fraction of particle species i and ρ T = ρ + ρ l is the total density which includes the baryonic (ρ) and the leptonic one (ρ l ).
In order to satisfy the equations for β-stable matter summarized in Eq. (3), we need to know the chemical potentials of the particles involved. In Brueckner-Hartree-Fock (BHF) theory the chemical potential is taken as the single particle energy at the Fermi momentum of the baryon, k F , which at the lowest order reads where, in the last equality, the baryon single-particle potential U B has been split into a contribution, U N B , coming from the nucleonic Fermi seas (p, n) and a contribution, U Y B , coming from the hyperonic ones (Σ − , Λ, . . .). From calculations in pure nucleonic matter it is well known that the nucleon chemical potential obtained from of Eq. (10) differs considerably from the thermodynamic definition Therefore, for the nucleons, we replace the nucleonic contribution to the chemical potential is the nucleonic contribution to the baryonic energy density including only the interaction between NN pairs. For the hyperons, we keep the prescription of Eq. (10). As shown in Ref. [7], these approximations amount to ignoring the weak dependence of U N N , U N Y on the hyperon fractions and of U Y N , U Y Y on the nucleon ones, and are good enough as long as the proton and hyperon fractions keep moderately small. Using the parabolic approximation for ε N N one obtains [7] where β = 1 − 2ρ p /ρ N is the asymmetry parameter, with ρ N = ρ n + ρ p . The symmetry energy E sym can be expressed as the difference of the energy per nucleon,Ẽ, between pure neutron (β = 1) and symmetric nuclear (β = 0) matter: whereẼ is the nucleonic contribution to the total energy per nucleon,Ẽ = ε N N /ρ N , and µ p,n (ρ N , β = 0) is given by The many-body approach outlined above is the lowest-order BHF method extended to the hyperon sector. This means also that we consider only two-body interactions. However, it is well-known from studies of nuclear matter and neutron star matter with nucleonic degrees of freedom only that three-body forces are important in order to reproduce the saturation properties of nuclear matter, see e.g., Ref. [12] for the most recent approach. The effect of nucleon three-body forces on the properties of β-stable matter with hyperons has been studied in Refs. [6,7]. It is found that the repulsion induced by the three-body force at high densities enhances substantially the hyperon population and produces a strong softening of the EoS. In order to include such effects, we will alternatively use for the nucleonic sector, the EoS of Ref. [12] (hereafter referred to as APR98), which is obtained from a variational calculation using the Argonne V 18 nucleon-nucleon interaction [13] with relativistic boost corrections and a fitted three-body interaction model.
In the discussions below we will thus present two sets of results for β-stable matter, one where the nucleonic contributions to the self-energy of nucleons are derived from the baryonbaryon potential model of Stoks and Rijken [8] and one where the nucleonic contributions to the neutron and proton chemical potentials are calculated from the parametrization of the APR98 EoS discussed in Eq. (49) of Ref. [14]. We note that replacing the nucleon-nucleon part of the interaction model of Ref. [8] with that from the V 18 nucleon-nucleon interaction [13], does not introduce large differences at the BHF level. However, the inclusion of threebody forces as done in Ref. [12] is important. Hyperonic contributions will however all be calculated with the baryon-baryon interaction of Stoks and Rijken [8]. We emphasize that, in the present work, Y N as well as Y Y interactions are taken into account. As can be seen by comparing the solid lines in both panels in Fig. 2, the composition of β-stable neutron star matter has a strong dependence on the model used to describe the nonstrange sector. In both cases, Σ − is the first hyperon to appear due to its negative charge.
Since the APR98 EoS yields a stiffer pure nucleonic matter EoS than the corresponding one for NSC97e, the onset of Σ − for the APR98 case occurs at a smaller density (ρ = 0.27 fm −3 ) than for the NSC97e case (ρ = 0.34 fm −3 ). In both cases, as soon as the Σ − hyperon appears, leptons tend to disappear, totally in the APR98 case (the electron chemical potential changes sign at ρ = 1.01 fm −3 , signaling the appearance of positrons), whereas in the NSC97e case only muons disappear. The onset of Λ formation takes place at higher density. Recalling the condition for the appearance of Λ, µ Λ = µ n = µ p + µ e − , and that the APR98 EoS from Σ − hole states to the neutron self-energy are attractive, see e.g., Ref. [9] for a detailed account of these aspects of the interaction model. We note that the isospin-dependent component (Lane term) of the Σ − single-particle potential for the new Nijmegen interaction [8] is strongly attractive, as opposed to what is found [17] for other interactions, including the old Nijmegen one [16]. This in turn implies a strong attraction for Σ − n (T = 3/2) pairs, which is 10 times that obtained for the old Nijmegen potential at saturation density.
These differences become more noticeable as density increases: while the Σ − n pairs become increasingly more attractive with the new Nijmegen potentials (see e.g. Fig. 5 in [9]), they turn out to be strongly repulsive for the old one (see e.g. Figs. 1 and 2 in [7]). This is why in Ref. [7] the onset density for the appearance of Σ − is larger than that for free hyperons, whereas the reverse situation is found here (compare the Σ − onset point in Fig.   2 with what would be extracted from Fig. 3). Within our many-body approach, no other hyperons appear at densities below ρ = 1. In order to examine the role of the hyperon-hyperon interaction on the composition of β-stable neutron star matter, the dashed lines in the lower panel of Fig. 2 show the results of a calculation in which only the NN and Y N interactions are taken into account. When the Y Y interaction is switched off, the scenario described above changes only quantitatively.
The onset point of Σ − does not change, because Σ − is the first hyperon to appear and therefore the Y Y interaction plays no role for densities below this point. We note that the reduction of the Σ − fraction compared with the case which includes the Y Y interaction, is a consequence of neglecting the strongly attractive Σ − Σ − interaction [8], which allows the energy balance (nn ↔ Σ − p) to be fulfilled with a smaller Σ − Fermi sea. In turn, the reduction of the Σ − fraction yields a moderate increase of the leptonic content in order to keep charge neutrality (in fact only muons disappear now). On the other hand, a smaller amount of Σ − 's implies less Σ − n pairs. Recalling that the Σ − n interaction is attractive in this model (see e.g. Fig. 7 of Ref. [9]), this means that the chemical potential of the neutrons becomes now less attractive. As a consequence, the Λ hyperons appear at a smaller density (ρ = 0.65 fm −3 ) and have a larger relative fraction.
As it has been mentioned, the composition of β-stable matter depends strongly on the model used to describe the nucleonic sector. In order to study this dependence, Fig. 3 shows the chemical potential of the neutron and the sum µ n + µ e − for β-stable matter composed of nucleons and free hyperons for three different NN interaction models. The solid lines correspond to the variational calculation denoted by APR98, which uses the Argonne V 18 interaction and includes three-body forces and relativistic boost corrections. The dashed and dot-dashed lines correspond to lowest-order BHF calculations using, respectively, the NSC97e and the Argonne V 18 potentials, the latter extracted from the results of Ref. [7].
Dotted lines denote the Λ and Σ − masses. In this case, the onset conditions for Λ and Σ − are, respectively, µ n = m Λ and µ n + µ e − = m Σ − . As can bee seen from the figure, the onset points of both hyperons are different depending on the NN interaction model employed. In the APR98 model both the Σ − and Λ hyperons appear at lower densities than in the lowestorder BHF models using the Argonne V 18 or the NSC97e interactions. This is a consequence of the different stiffness of the EoS generated by the three NN interaction models. In fact, the NSC97e interaction gives the softest EoS and it is not even able to produce Λ hyperons in the range of densities explored. Note that the hyperon onset points determined from At present, the APR98 EoS represents perhaps the most sophisticated many-body approach to nuclear matter. Therefore, in what follows, we restrict our results to this NN interaction model supplemented with the NSC97e one for the hyperonic sector.
In Fig. 4 we show the chemical potentials in β-stable neutron star matter for different baryons. We note that neither the Σ 0 nor the Σ + do appear since, as seen from the figure, the respective stability criteria of Eq. (3) are not fulfilled. This is due, partly, to the fact that none of the Σ 0 -baryon and the Σ + -baryon self-energies are attractive enough. A similar argument applies to Ξ 0 and Ξ − . In the latter case the mass of the particle is ∼ 1315 MeV and an attraction of around 200 MeV would be needed to fulfill the condition µ Λ = µ Ξ 0 = µ n at the highest density explored in this work. From the figure we see, however, that the Σ 0 hyperon could appear at densities close to 1.3 fm −3 .  [19]. Several X-ray binary masses have been measured of which the heaviest are Vela X-1 with M = (1.9 ± 0.2)M ⊙ [20] and Cygnus X-2 with M = (1.78 ± 0.2)M ⊙ [21].
The recent discovery of high-frequency brightness oscillations in low-mass X-ray binaries provides a promising new method for determining masses and radii of neutron stars, see Ref. [22]. In order to obtain the radius and mass of a neutron star, we have solved the Tolman-Oppenheimer-Volkov equation with and without rotational corrections, following the approach of Hartle [25], see also Ref. [14].
Following the discussion of Figs. 3, 4 and 5, we use the pure nucleonic matter EoS of Akmal et al. [12], including both relativistic boost corrections and three-body interactions.
Relating to the discussion of the previous subsection, we study two additional cases as well.
One case where we just include the hyperon-nucleon interaction as done in e.g., Refs. [7] and another case where we include both the hyperon-nucleon and the hyperon-hyperon interactions.
In Fig. 6 we show the resulting mass as function of these equations of state without rotational corrections. In Fig. 7 we include rotational corrections to the mass. The massradius relations for the same equations of state are shown in Fig. 8. The reader should however note that our calculation of hyperon degrees freedom is based on a non-relativistic BHF approach. Although the nucleonic part extracted from Ref. [12], including three-body forces and relativistic boost corrections, is to be considered as a benchmark calculation for nucleonic degrees of freedom, relativistic effects in the hyperonic calculation could result in a stiffer EoS and thereby larger mass. However, relativistic mean field calculations with parameters which result in a similar composition of matter as shown in Fig. 2, result in similar masses as those reported in Fig. 6. Although we have only considered the formation of hyperons in neutron stars, transitions to other degrees of freedom such as quark matter, kaon condensation and pion condensation may or may not take place in neutron star matter. We would however like to emphasize that the hyperon formation mechanism is perhaps the most robust one and is likely to occur in the interior of a neutron star, unless the hyperon self-energies are strongly repulsive due to repulsive hyperon-nucleon and hyperon-hyperon interactions, a repulsion which would contradict present data on hypernuclei [26]. The EoS with hyperons yields however neutron star masses without rotational corrections which are even below ∼ 1.4M ⊙ . This means that our EoS with hyperons needs to be stiffer, a fact which may in turn imply that more complicated many-body terms not included in our calculations, such as three-body forces between nucleons and hyperons and/or relativistic effects, are needed. Our novel result is that a further softening of the EoS is obtained when including the effect of the Y Y interaction since it is attractive over the whole density range explored. This is mainly due to the ΣΣ interaction which is strongly enough to develop a bound state [8]. We note that the ΛΛ attraction produced by this model is only mild, not being able to reproduce the experimental 2Λ separation energy of ∆B ΛΛ ∼ 4 − 5 MeV [27].
Wether this additional softening is realistic or not will depend on the details of the Y Y interaction that is, unfortunatelly, not well constrained at present. New data in the S = −2 sector, either from double-Λ hypernuclei or from Ξ − -atoms, are very much awaited for.