{'fraction_non_alphanumeric': 0.03713049747656813, 'fraction_numerical': 0.012211607786589761, 'mean_word_length': 4.351579455027731, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 9, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Properties of \beta-stable neutron star matter with hyperons

We present results from many-body calculations for \beta-stable neutron star matter with nucleonic and hyperonic 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 emerging in \beta-stable matter up to total baryonic densities of 1.2 fm^{-3} are \Sigma^- and \Lambda. The corresponding equations of state are thence used to compute properties of neutron stars such as the masses, moments of inertia and radii. We also study the possibility of forming a hyperon superfluid and discuss its implications for neutron stars.


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 density ρ d ≈ 4·10 11 g/cm 3 at 'neutron drip', the density at which 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, ( with number density n 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 µ n = µ p + µ e , n p = n e , where µ i and n i refer to the chemical potential and number density in fm −3 of particle species i, respectively. At the saturation density of nuclear matter, n 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 − . We then develop a Fermi sea of degenerate negative muons, and we have to modify the charge balance according to n p = n e + n µ , and require 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 µ n equals its mass. 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 condition becomes, We have omitted isobars ∆, see the discussion below.
Hyperonic degrees of freedom have been considered by several authors, but 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. recently, see Ref. [6], in a many-body calculation in order to study where hyperons appear in neutron star matter. All these works show that hyperons appear at densities of the order of ∼ 2n 0 .
In Ref. [6] however, one was only able to fix the density where Σ − appears, since only a hyperon-nucleon interaction was employed. As soon as Σ − appears, one needs a hyperon-hyperon interaction in order to estimate e.g., the self-energy of Λ. The aim of this work is thus to present results from many-body calculations of hyperonic degrees of freedom for β-stable neutron star matter 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. [7]. This entails a microscopic description of matter starting from realistic nucleon-nucleon, hyperon-nucleon and hyperon-hyperon interactions.
In a recent work [8] 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 ). Results for various fractions of the above particles were also discussed.
Here we extend the calculations of Ref. [8] to studies of β-stable neutron star matter. Our results, together with a brief summary of the formalism discussed in Ref. [8], are presented in section II. There we discuss the equation of state (EoS) and the composition of β-stable matter with various baryon-baryon potentials. Based on the composition of matter we present also results for baryon superfluidity and discuss the possible neutron star structures.

II. EQUATION OF STATE AND COMPOSITION OF β-STABLE MATTER
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. [7]. 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 [9] 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. [7], since this model, together with the model NSC97f of Ref. [7], result in the best predicitions for hypernuclear observables [9]. For a discussion of other interaction models, see Refs. [7,10].

A. Formalism
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 shortrange 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. The intermediate states B 5 B 6 are those which are allowed by the Pauli principle, and the energy variable ω is the starting energy defined by the single-particle 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. [8,10] 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 . The single-particle potential u B i is defined by The linear momentum of the intermediate single-particle state B j is limited by the size of the The total non-relativistic energy density, ε, and the total binding energy per baryon, E, can be evaluated from the baryon single-particle potentials in the following way where n is the total baryonic density. The density of a given baryon species is given by where x B = n B /n is the fraction of baryon B, which is of course constrained by Detailed expressions for the single-particle energies and the G-matrices involved can be found in Ref. [8]. In order to satisfy the equations for β-stable matter summarized in Eq.
(3), we need to solve Eqs. (4) and (5) to obtain the single-particle energies of the particles involved at the corresponding Fermi momenta. Typically, for every total baryonic density n = n N + n Y , the density of nucleons plus hyperons, Eqs. (4) and (5)  The many-body approach outlined above is the lowest-order Brueckner-Hartree-Fock (BHF) method extended to the hyperon sector. This means also that we consider only twobody 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. [11] for the most recent approach. In order to include such effects, we replace the contributions to the proton and neutron self-energies arising from intermediate nucleonic states only, see diagram (a) of Fig. 1, with those derived from Ref. [11] (hereafter APR98) where the Argonne V 18 nucleon-nucleon interaction [12] is used with relativistic boost corrections and a fitted threebody interaction, model. The calculations of Ref. [11] represent at present perhaps the most sophisticated many-body approach to dense matter. 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 is derived from the baryon-baryon potential model of Stoks and Rijken [7] and one where the nucleonic contributions are replaced with the results from Ref. [11] following the parametrization discussed in Eq. (49) of Ref. [13]. Replacing the nucleon-nucleon part of the interaction model of Ref. [7] with that from the V 18 nucleonnucleon interaction [12], does not introduce large differences at the BHF level. However, the inclusion of three-body forces as done in Ref. [11] is important. Hyperonic contributions will however all be calculated with the baryon-baryon interaction of Stoks and Rijken [7].
B. β-stable neutron star matter The above models for the pure nucleonic part combined with the hyperon contribution yield the composition of β-stable matter, up to total baryonic number density n = 1. These results are in fair agreement with results obtained from mean field calculations, see e.g., Refs. [1][2][3]. The introduction of hyperons leads to a considerable softening of the EoS.
Moreover, as soon as hyperons appear, the leptons tend to disappear, totally in the APR98 case whereas in the BHF calculation only muons disappear. For the APR98 case, positrons appear at higher densities, i.e., n = 1.18 fm −3 . This result is related to the fact that Λ does not appear at the densities considered here for the BHF EoS. For the APR98 EoS, Λ appears at a density n = 0.67 fm −3 . Recalling µ Λ = µ n = µ p + µ e and that the APR98 EoS is stiffer due to the inclusion of three-body forces, this clearly enhances the possibility of creating a Λ with the APR98 EoS. However, the fact that Λ does not appear in the BHF calculation can also, in addition to the softer EoS, be retraced to a delicate balance between the nucleonic and hyperonic hole state contributions (and thereby to features of the baryon-baryon interaction) to the self-energy of the baryons considered here, see diagrams (a) and (b) in Fig. 1. Stated differently, the contributions from Σ − , proton and neutron hole states to the Λ chemical potential are not attractive enough to lower the chemical potential of the Λ so that it equals that of the neutron. Furthermore, the chemical potential of the neutron does not increase enough since contributions from Σ − hole states to the neutron self-energy are attractive, see e.g., Ref. [8] for a detailed account of these aspects of the interaction model.
We illustrate the role played by the two different choices for nucleonic EoS in Fig. 4 in terms of the chemical potentials for various baryons for matter in β-equilibrium. We also note that, using the criteria in Eq. (3), neither the Σ 0 nor Σ + do appear for both the BHF and the APR98 equations of state. This is due to the fact that none of the Σ 0 -baryon and Σ + -baryon interactions 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 almost 200 MeV in attraction is needed in order to fullfil e.g., the condition µ Λ = µ Ξ 0 = µ n . This has also been checked by us [14] in studies of the self-energy of Ξ − in finite nuclei, using the recipe outlined in Ref. [15].
For both light and medium heavy nuclei, Ξ − is unbound with the present hyperon-hyperon interactions, except for version NSC97f of Ref. [7]. The latter results in a weakly bound Ξ − , in agreement with the recent studies of Batty et al. [16]. From the bottom panel of Fig. 4 we see however that Σ 0 could appear at densities close to 1.2 fm −3 . Thus, for the present densities, which would be within the range of energies for where the interaction model has been fitted, the only hyperons which can appear are Σ − and Λ.
In summary, using the realistic EoS of Akmal et al. [11] for the nucleonic sector and including hyperons through the most recent model for the baryon-baryon interaction of the Nijmegen group [7], we find through a many-body calculation for matter in β-equilibrium

C. Baryon superfluidity in β-stable matter
A generic feature of fermion systems with attractive interactions is that they may be superfluid in a region of the density-temperature plane. The 1 S 0 wave of the nucleonnucleon interaction is the best known and most investigated case in neutron stars, and the results indicate that one may expect a neutron superfluid in the inner crust of the star and a proton superfluid in the quantum liquid interior, both with energy gaps of the order of 1 MeV [17][18][19][20][21][22]. Furthermore, neutrons in the quantum liquid interior may form a superfluid due to the attractive 3 P 2 -3 F 2 wave of the nucleon-nucleon interaction [23]. Baryon superfluidity has important consequences for a number of neutron star phenomena, including glitches [24] and cooling [25]. If hyperons appear in neutron stars, they may also form superfluids if their interactions are sufficiently attractive. The case of Λ superfluidity has been investigated by Balberg and Barnea [26] using parametrized effective Λ-Λ interactions. Results for Λ and Σ − -pairing using bare hyperon-hyperon interaction models have recently been presented by Takatsuka and Tamagaki [27]. The result of both groups indicate the presence of a Λ superfluid for baryon densities in the range of 2-4n 0 . The latter authors also suggest that the formation of a Σ − superfluid may be more likely than Λ-superfluidity. Along the lines followed by these authors we will here present results for hyperon superfluidity within our model.
The crucial quantity in determining the onset of superfluidity is the energy gap function ∆(k). The value of this function at the Fermi surface is proportional to the critical temperature of the superfluid, and by determining ∆ we therefore map out the region of the density-temperature plane where the superfluid may exist. When the 1 S 0 interaction is the driving cause of the superfluidity, the gap function becomes isotropic and depends on the magnitude of k only. It can be determined by solving the BCS gap equation In this equation, ǫ k is the momentum-dependent single particle energy in the medium for the particle species in question, µ is the corresponding chemical potential, andṼ1 S 0 is the effective pairing interaction. At this point we emphasize that using parametrized effective interactions in the gap equation can lead to errors. The gap equation includes diagrams also found in the G-matrix, and one therefore needs to calculateṼ systematically from microscopic manybody theory to avoid double counting of ladder contributions. The expansion forṼ can be found in e.g. Migdal [28], and to lowest orderṼ = V , the free-space two-particle interaction.
Higher order terms include contributions from e.g. density-and spin-density fluctuations. In this first exploratory calculation we will follow Ref. [27] and use the bare hyperon-hyperon interaction in Eq. (11). The relevant hyperon fractions and single-particle energies are taken from the BHF calculations described earlier in this paper. Details of the numerical solution of the gap equation can be found in Ref. [22]. ) as a function of the total baryon density for Σ − hyperons in β-stable matter for the NSC97E model. Although Λ may appear at higher densities, the 1 S 0 Λ-Λ matrix elements of the NSC97E interaction are all repulsive, and therefore the energy gap for Λ hyperons would (to lowest order) have been zero at all densities, i.e. these particles would not have formed a superfluid. This is it at variance with the results of Ref. [26], however, as remarked earlier this work employs an effective, parametrized interaction to drive the gap equation and therefore overestimates the Λ energy gap. Our Σ − results are comparable to those of Ref. [27] which were obtained with a gaussian soft core parametrization of the bare Σ − -Σ − interaction.
If taken at face value these results have implications for neutron star cooling. Since at low densities Σ − is the only hyperon species that is present in our calculation, the most important contribution to the neutrino cooling rate at such densities comes from the reaction Σ − → n + e − + ν e . According to Ref. [29] the threshold density for this reaction to occur is at around 2.4n 0 . If the Σ − s are superfluid with energy gaps similar to what we found here, a sizeable reduction of the order of exp(−∆ F /kT ) may be expected in the reaction rate. If neutron stars were to cool through direct Urca processes, their surface temperatures would be barely detectable within less than 100 yr of the star's birth. This is at askance with present observations. Thus, the formation of a hyperon superfluid will clearly suppress the hyperon direct Urca process and cooling will most likely proceed through less efficient processes and bring the results closer to experimental surface temperatures.

D. Structure of neutron stars
We end this section with a discussion on neutron star properties with the above equations of state. In the following, we display the results for mass and radius using the equations of state discussed above. 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 [37], see also Ref. [13]. Our results are shown in in Figs. 6 and 7.
The equations of state we have used are those for 1. β-stable pn-matter with the parametrization of the results from Akmal et al. [11] made in Ref. [13]. This EoS is rather stiff compared with the EoS obtained with hyperons, see 2. The other EoS employed is that which combines the nucleonic part of Ref. [11] with the computed hyperon contribution. As can be seen from Fig. 6, the softening of the EoS due to additional binding from hyperons leads to a reduction of the total mass. Without rotational corrections, we obtain a maximum mass M ∼ 1.3M ⊙ whilst the rotational correction increases the mass to M ∼ 1.4M ⊙ . The size of the reduction, ∆M ∼ 0.6 − 0.7M ⊙ , and the obtained neutron star masses due to hyperons are comparable to those reported by Balberg et al. [5].
There are other features as well to be noted from Fig. 6. The EoS with hyperons reaches a maximum mass at a central density n c ∼ 1.2 − 1.3 fm −3 . In Fig. 2 we showed that the only hyperons which can appear at these densities are Λ and Σ − . If other hyperons were to appear at higher densities, this would most likely lead to a further softening of the EoS, and thereby smaller neutron star masses. Furthermore, the softer EoS yields also a smaller moment of inertia, as seen in Fig. 8.
The reader should however note that our calculation of hyperon degrees freedom is based on a non-relativistic Brueckner-Hartree-Fock approach. Although the nucleonic part extracted from Ref. [11], including three-body forces and relativistic boost corrections, is to be 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 mechanisms 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 [38]. 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.

III. CONCLUSIONS
Employing the recent parametrization of the free baryon-baryon potentials for the com- In addition, since Σ − hyperons appear already at total baryonic densities ∼ n = 0.27 fm −3 ), we have also considered the possibility of forming a hyperon superfluid. The latter would in turn quench the increased emission of neutrinos due to the presence of hyperons.
Within our many-body approach, we find that Σ − forms a superfluid in the 1 S 0 wave, whereas the Λ − Λ interaction for the same partial wave leads to a vanishing gap for the potential model employed here.
We are much indebted to H. Heiselberg  with the potential of Stoks and Rijken [7]. In the lower panel the nucleonic part of the self-energy of the nucleons has been replaced with the EoS of Ref. [11]. For the latter, (not shown in the figure) positrons appear at a density 1.18 fm −3 .