Massive relativistic compact stars from SU(3) symmetric quark models

Compact stars (CSs) provide unique laboratories to probe dense matter under extreme conditions which cannot be reproduced on Earth. The composition of the deep interiors of CS is not known. It represents the main uncertainty for the determination of the static and dynamic properties of CS. Various high-density compositions have been studied assuming different degrees of freedom, for example, compositions featuring purely nucleonic, heavy baryon-admixed, and/or deconfined quarks matter, for reviews see Refs. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. In particular, hyperons have been studied as an option as their nucleation may become energetically favorable above a threshold, which is distinct for each hyperon and is controlled by the conditions of $\beta$-equilibrium and charge neutrality among the baryons and leptons, see Fig. 5 of Ref. [18]. Hyperonization of dense matter then reduces the pressure of dense matter which has a significant impact on the maximum CS mass, for reviews see Refs. [14, 15, 16, 17].

Currently, the most rigorous constraints on the high-density behavior of the equation of state (EoS) come from the observations of a few massive pulsars with masses $\sim 2.0\,M_{\odot}$ [19, 20, 21, 22, 23]. These observations set a lower bound on the maximum mass of CS predicted by any model of dense matter. The long-awaited detection of gravitational waves (GWs) from a binary neutron star merger, the GW170817 event placed significant constraints on the tidal deformability of canonical-mass stars and thus provided additional constraints on the EoS of dense matter at intermediate densities [24, 25, 26]. The multi-messenger analyses of GW170817 event suggest that the maximum mass of static CS may not exceed $\sim 2.3\,M_{\odot}$ [27, 28, 29, 30, 31]. The X-ray pulse profile modeling of pulsars with data from the NICER observatories recently led to measurements of CS radii. The estimates for one of the most massive known pulsar, PSR J0740+6620 [32, 33], open prospects of constraining the properties of the EoS, in particular, the composition of matter at high densities. PSR J0740+6620 has a mass of about $\sim 2.1\,M_{\odot}$ and is thus about 50% more massive than PSR J0030+0451 [34, 35], yet current measurements do not indicate a significant difference in their sizes. This may indicate that the turning point, i.e., the maximum, of the mass-radius relation occurs above the mass of PSR J0740+6620 [36, 37]. Previous models of hyperonic stars were mainly constrained by the masses of massive pulsars [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 18, 51, 52, 53, 54].

Observational identification of neutron stars (black holes) as members of binary systems requires the knowledge of the upper (lower) limit on the gravitational mass of a neutron star (black hole). The GW190814 event [55], caused by the merger of two stellar objects with an extremely asymmetric mass ratio, contained a primary black hole with a mass of $23.2^{+1.1}_{-1.0}\,M_{\odot}$. The secondary’s mass was in the range of $2.59^{+0.08}_{-0.09}\,M_{\odot}$. The latter value of mass is within the hypothesized “mass-gap” between neutron stars and black holes, $2.5\lesssim M/M_{\odot}\lesssim 5$, where no compact object had ever been observed before. Whether the light companion is the most massive neutron star or the lightest black hole discovered so far is unclear yet [56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72]. Recently, the event GW200210 [73] was reported in which the components have masses of $24.1^{+7.5}_{-4.6}\,M_{\odot}$ and $2.83^{+0.47}_{-0.42}\,M_{\odot}$. In Refs. [57, 58] we suggested that the secondary component of GW190814 is more likely a black hole rather than a CS, by considering hyperonic EoS models where hyperonic couplings to vector mesons were based on the SU(6) quark model, while those to scalar mesons were fitted to the depth of their potential at nuclear saturation density.

The main motivation of this work is to extend the previous studies [57, 58] of massive hyperonic CS from SU(6) symmetry based vector meson couplings to those that arise within the more general SU(3) symmetry [74] which was implemented in the context of CSs within Hartree [75] and Hartree-Fock [18] based CDF models. This provides a more complete exploration of the parameter space that admits the existence of massive hyperonic stars. Indeed, the SU(6) model combines the flavor SU(3) and spin SU(2) symmetries, which is a special case of the more general SU(3) model [74]. Previously, several authors explored the effect of breaking of SU(6) symmetry down to SU(3) for selected nucleonic EoS in the vector-meson sector [41, 18], scalar-meson sector [42] and both [46]. In the scalar-meson sector the SU(3) relations do not hold after fixing the $\Lambda$-hyperon depth [50, 57] and the hyperonic couplings are fixed by the values of hyperonic potential depths. Therefore, the SU(3) relations are useful for the vector-meson sector only. These works demonstrated that within the SU(3) symmetric models for the vector-meson sector it is possible to construct massive hyperonic CSs with maximum masses as high as 2.2-$2.3\,M_{\odot}$. They used models with fixed properties of nucleonic component within the relativistic mean field models, which preclude by construction a study of the dependence of the results on continuous variations of nuclear matter characteristics such as symmetry energy, its slope as well as the skewness.

This paper is organized as follows. In Sec. 2 we briefly outline the key features of the CDF model for hyperonic matter. Section 3 discusses the bulk properties of hyperonic stars predicted by our CDF approach for a broad range of variations of the parameters. We discuss the implications of these models for the interpretation of GWs produced in binary stellar collisions involving massive secondaries whose masses lies in the “mass-gap”. Finally, a summary of our results is provided in Sec. 4.

We use in this work the standard CDF theory with density-dependent meson-baryon couplings for a many-body nuclear system whose interaction Lagrangian is given by [76, 77, 18, 78]

where $\psi_{B}$ stands for the Dirac spinors. The index $B$ labels the $J^{P}=\frac{1}{2}^{+}$ baryonic octet with the member masses denoted by $m_{B}$. The explicit form of the free Lagrangian can be found in Ref. [76, 77, 18]. The octet of baryons interacts via exchanges of $\sigma,\,\omega$, and $\rho$ mesons, which comprise the minimal set necessary for a quantitative description of nuclear phenomena [79, 80]. We consider further two hidden-strangeness mesons, $\sigma^{\ast}$ and $\phi$, which describe interactions between hyperons [75, 47, 18, 81]. The mesons couple to baryons with coupling constants $g_{mB}$, which are functions of the baryonic density, $g_{mB}=g_{mB}(\rho_{\rm sat})f_{m}(r)$, where $r=\rho/\rho_{\rm sat}$ with $\rho_{\rm sat}$ being the nuclear saturation density. For the explicit form of the functions $f_{m}(r)$ see Refs. [77, 18]. The interaction Lagrangian (2) is fixed by first assigning the baryons and mesons their masses in the vacuum. Next, one fixes the three coupling constants ($g_{\sigma N},g_{\omega N},g_{\rho N}$) in the nucleonic sector and the four parameters that enter the functions $f_{m}(r)$. Then ground state properties of infinite nuclear matter and finite nuclei can be computed uniquely in terms of the above seven adjustable parameters. Note that the constraint conditions on the $f_{m}(r)$ function reduce the eight parameters for $\sigma$ and $\omega$-mesons to three [77, 18]. In addition, there is one parameter for the $\rho$-meson. There are in total four parameters that enter the functions $f_{m}(r)$.

The EoS of isospin asymmetric nuclear matter can be expanded around nuclear saturation and the isospin symmetric limit in power series [82, 78]

where $n=(\rho-\rho_{\rm sat})/3\rho_{\rm sat}$ and $\delta=(\rho_{\rm n}-\rho_{\rm p})/\rho$. The coefficients of the density-expansion in the first line of Eq. (2) are the characteristic coefficients of nuclear matter in the isoscalar channel, specifically, the saturation energy $E_{\rm sat}$, incompressibility $K_{\rm sat}$, and skewness $Q_{\rm sat}$. The coefficients associated with the expansion away from the symmetric limit in the second line are the characteristic parameters in the isovector channel, i.e., the symmetry energy $E_{\rm sym}$ and its slope parameter $L_{\rm sym}$. The quantities which arise at a higher order of the expansion, specifically $Q_{\text{sat}}$ and $L_{\text{sym}}$, are only weakly constrained by the conventional fitting protocol used in constructing the density functionals, i.e., the procedure which involves usually fits to nuclear masses and radii. However, the value of $Q_{\text{sat}}$ controls the high-density behavior of the nucleonic energy density, while the value of $L_{\text{sym}}$ determines the intermediate-density behavior of the nucleonic energy density according to Eq. (2).

It is interesting to examine the potentials of the baryons in pure neutron matter, given by

where the meson fields are replaced by their respective expectation values in the Hartree mean-field approximation [76, 77, 42, 18], and $\Sigma_{R}$ denotes the rearrangement term that comes from the density-dependence of the meson-baryon coupling constants [76, 77, 42, 18].

In the SU(3) model three parameters are describing the deviation from SU(6) flavor-spin symmetry. Considering the vector meson sector, the parameter $\alpha_{\rm v}$ is the weight factor for the contributions of the symmetric and antisymmetric couplings. Its SU(6) value is $\alpha_{\rm v}=1$. Another parameter is the mixing angle  $\theta_{\rm v}$ which relates the physical mesons to their pure octet and singlet counterparts. And, finally, the third parameter $z$ is the ratio of the meson octet and singlet couplings [83, 84, 41, 18].

The roles played by the parameters $Q_{\text{sat}}$ and $L_{\text{sym}}$ for the single-particle potentials of baryons are shown in Fig. 1, panel (a), where the nucleonic potentials are shown for models with $L_{\text{sym}}\in[40,100]$ MeV and $Q_{\text{sat}}\in[-600,1000]$ MeV, and in panels (b) and (c) where the hyperonic potentials are shown for the cases of SU(6) and extreme SU(3) with $z=0$.

Given the five macroscopic coefficients in Eq. (2) together with the preassigned values of $\rho_{\rm sat}$ and Dirac mass $M^{\ast}_{D}$ [18], we could determine uniquely the seven adjustable parameters of the Lagrangian (2). In Ref. [78] it has been suggested that one can generate a set of nucleonic CDF models by varying only one coefficient in Eq. (2) while keeping the others fixed. Having this in mind, we map the nucleonic EoS given by the Lagrangian (2) for each set of parameters $Q_{\rm sat}$ and $L_{\rm sym}$. For our analysis below we adopt the lower-order coefficients in Eq. (2), i.e., $E_{\rm sat}=-16.14$, $K_{\rm sat}=251.15$ MeV, and $E_{\rm sym}=32.31$ MeV, as those inferred from the DDME2 parametrization [77, 78], which was adjusted to the properties of finite nuclei.

The determination of the meson-hyperon couplings $g_{mY}$ represents a long-standing theoretical challenge due to the lack of sufficiently abundant and accurate experimental data. In the present work, we restrict our attention only to the three lightest quark flavors and adopt the flavor SU(3) symmetric model [74, 75, 18]. To explore the parameter space associate with the SU(3) model we proceed by assuming an ideal mixing value of $\theta_{\rm v}=\rm tan^{-1}(1/\sqrt{2})$ [85]. This fixation of the mixing angle $\theta$, which describes the mixing between the singlet and the octet members of a physical isoscalar vector mesons, is motivated by the fact that the mixing between nonstrange and strange quark wave functions in the $\omega$ and $\phi$-mesons is ideal, i.e., the mixing angle assumes the ideal mixing value quoted above. In addition, from the quadratic mass formula for mesons, one obtains $\theta\approx 40^{\circ}$ [85], a value that is a very close to the ideal mixing angle $\theta\approx 35.3^{\circ}$. Thus, it is reasonable to keep the condition of “ideal mixing” for the isoscalar vector mesons. The dependence on the remaining parameters, $\alpha_{\rm v}$ and $z$, can be explored by fixing one of them and varying the other. This has been done previously in Ref. [18] (see their Figs. 11 and 12) showing that reducing the value of either $\alpha_{\rm v}$ or $z$ from their SU(6) values at fixed value of the other parameter yields qualitatively similar modifications of the EoS and the particle fractions. We thus choose to vary only one of them, i.e., $z$, while keeping $\alpha_{\rm v}=1$ fixed at its SU(6) value.

Then we are left with a single free parameter $z$ to quantify the effects of the SU(3) symmetric model. In this case, the hyperonic coupling constants are defined as [41, 18]

where in each equation the last relation shows the $z\to 0$ asymptotes neglecting terms $\mathcal{O}(z^{2})$. These asymptotic values can be compared with the SU(6) values of the coupling constants,

It is seen that the $z=0$ limit of the SU(3) model implies a much stronger repulsive interaction among hyperons due to $\omega$-exchange.

In the SU(6) symmetric model, the $\phi$-meson has a vanishing $\phi$-$N$ coupling, whereas it does couple to the nucleon in SU(3) symmetric model in terms of

where the last relation is the $z\to 0$ asymptote as above.

To ensure this new coupling scheme does not spoil the fits to the purely nuclear data, we make the replacement [18]

where the $\tilde{g}_{\omega N}$ denotes the coupling for the case of $g_{\phi N}=0$. For such a scheme, it has been shown in Ref. [18] that the EoS of purely nucleonic matter is (almost) independent of the appearance of $\phi$-meson. For the isovector meson $\rho$, one has [41, 18]

The isoscalar-scalar meson-hyperon couplings are then determined by fitting them to certain preselected properties of hypernuclear systems. We fix the coupling constants $g_{\sigma Y}$ using the following hyperon potentials in symmetric nucleonic matter at saturation density, $\rho_{\rm sat}$, extracted from hypernuclear phenomena [86, 87]:

Finally, we use the estimate $U^{(\Lambda)}_{\Lambda}(\rho_{\rm{sat}}/5)=-0.67~{}\text{MeV}$, which is extracted from the $\Lambda\Lambda$ bond energy [88], to fix the coupling constant $g_{\sigma^{\ast}\Lambda}$. The couplings of the remaining hyperons $\Xi$ and $\Sigma$ to the $\sigma^{\ast}$-meson are determined by the relation $g_{\sigma^{\ast}Y}/g_{\phi Y}=g_{\sigma^{\ast}\Lambda}/g_{\phi\Lambda}$. In this manner, we assume that the hyperon potentials scale with density as the nucleonic potential, therefore their high-density behavior is inferred from that of the nucleons. In Fig. 1 (b) we show the potentials of hyperons in pure neutron matter for two limiting cases, $z=1/\sqrt{6}$ and 0. The former corresponds to the SU(6) model while the latter is the extreme case of the SU(3) model. It is seen that the hyperon potential depths (9), indeed, determine only the EoS region around the saturation density. In contrast, the meson coupling constants (4) affect largely the high-density regime of the EoS and consequently the degree of stiffness of the EoS, which is closely linked to the inner core composition of CSs.

For any set of coupling constants the EoS of the core of a CS is determined by applying the conditions of weak equilibrium and change neutrality. This EoS is then matched (interpolated) smoothly to the EoS of the crustal matter given by Refs. [89, 90] at the core-crust transition density $\sim\rho_{\rm sat}/2$. The details of core-crust matching procedure and the model of the crust EoS affect to some extent the value of the radius and, therefore, the tidal deformability for light CSs [91, 92], but the uncertainties are negligible for massive stars of interest herein.

In this section, we will explore the gross parameters of hyperonic stars for a selected set of parameters that control the stiffness of the EoS. In the nucleonic sector we vary the characteristics of nuclear matter $Q_{\text{sat}}$ and $L_{\text{sym}}$. In the hyperonic sector, we vary the parameter $z$ associated with the breaking of the SU(3) symmetry. Our choice of parameters that describe the EoS of hypernuclear matter is as follows:

• 1.

  (I) Soft EoS in the nucleonic sector with $L_{\text{sym}}=40$ MeV, which is close to the lower values of the 90% confidence ranges of the PREX-2 neutron skin measurement [93, 94, 95]. Our EoS predicts for $1.4\,M_{\odot}$ mass star a radius and tidal deformability in the ranges of $11.8\lesssim R_{1.4}\lesssim 13.2$ km and $280\lesssim\Lambda_{1.4}\lesssim 750$ when the remaining parameters $Q_{\text{sat}}$ and $z$ are varied. These values are within the range derived for the multimessenger GW170817 event [25, 26].

• 2.

  (II) Stiff EoS in the nucleonic sector with $L_{\text{sym}}=100$ MeV, which is close to the central value of the PREX-2 analysis [94]. In this case, we find that the radius and deformability are larger, their values for a $1.4M_{\odot}$ mass star being in the range of $12.8\lesssim R_{1.4}\lesssim 14.3$ km and $450\lesssim\Lambda_{1.4}\lesssim 1200$. These values are in agreement with the mass and radius inferences from the NICER experiment for PSR J0030+0451 [34, 35], but are outside of the range deduced from the GW170817 event [25, 26]. Exceptions to this are models with $Q_{\text{sat}}\lesssim-400$ MeV.

Figure 2 shows a collection of EoS that cover the relevant range of parameters both for the nucleonic and hyperonic sectors. The model EoS is distinguished by (a) the values of $L_{\text{sym}}=40$ and 100 MeV which control the intermediate-density stiffness in the nucleonic sector; (b) the values of $Q_{\text{sat}}$ which control the high-density stiffness of the nucleonic sector and are drawn from the interval $[-600,1000]$ MeV with a step size of 100 MeV; (c) the values of the $z$-parameter which takes on two values: $z=1/\sqrt{6}$ for the SU(6) model and $z=0$, which is an extreme case of the SU(3) model. As can be seen, the intermediate-density soft models show a delay in the appearance of hyperons as the density is increased. As a consequence, the EoS is stiffer at high densities once the hyperons are admixed with the nucleonic matter. Note the different ordering of the thresholds of the appearance of hyperons in the SU(3) and SU(6) models. In the SU(6) case, $\Lambda$ hyperons appear first and are followed by $\Xi^{-}$, then $\Xi^{0}$ hyperons as the density is increased. In the SU(3) model, $\Lambda$’s are followed by the $\Sigma^{-}$ hyperons and the onset of $\Xi^{0}$ is shifted to densities that are not relevant for stable CSs. Note that here and below we will keep occasionally the EoS models with maximum masses below the $2.0\,M_{\odot}$ mass limit to account for the possibility of two families of CSs, in which case stars with masses of $2.0\,M_{\odot}$ and higher are strange stars [96].

In this work, we constructed EoS models within CDF theory with degrees of freedom that include the full baryon octet. The meson-hyperon coupling constants are chosen to break the SU(6) spin-flavor symmetry down to SU(3). The hyperon potentials were further fitted to the most reliable values of their potentials at nuclear saturation density extracted from hypernuclear data. Because of the more general SU(3) symmetry, the hyperonic couplings depend on additional parameters, among which the $z$-parameter (defined above) is most suitable for exploring the impact of symmetry breaking. The density-dependences of the nucleonic and hyperonic couplings is modeled using the same parameters. The nucleonic sector of the CDF was modeled phenomenologically at high density by varying the slope coefficient $L_{\text{sym}}$ and skewness coefficient $Q_{\text{sat}}$, while maintaining the low-density features predicted by the DDME2 parametrization.

With this input, we investigated the mass and radius of non-rotating as well as rapidly rotating stellar configurations. Our EoS models can accommodate static CSs as massive as $M\simeq 2.3$-$2.5\,M_{\odot}$ in the large-$Q_{\text{sat}}$ and small-$z$ domain. However, the hyperon content in this regime drops to several percent and, therefore, cannot significantly influence the properties of CS. Thus, one may conclude that the highly massive stellar models obtained with the SU(3) symmetric models for the EoS are essentially nucleonic stars. The global parameters of these stars are consistent with the parameters of stars based on purely nucleonic EoS models [58, 60, 61, 62, 63, 64, 65] (we exclude here the models calling for quark deconfinement [66, 67, 68, 69, 70, 71, 13, 72]). This also confirms that genuinely hyperonic stars with a significant hyperonic fraction of 10-20% are confined to lower masses [57, 58, 66, 53, 54].

We further constructed the rotating counterparts of our static stellar models, including stars rotating at the Keplerian limit, in which case the maximum mass of the nearly nucleonic models can reach values up to $3.0\,M_{\odot}$. Our modeling allows us to estimate the ratio of the Keplerian to static maximum mass, $\eta$, and to show that it is constant only for stars with $M^{\rm max}_{\rm TOV}\geq 2.0\,M_{\odot}$. We find a linear dependence of $\eta$ on $M^{\rm max}_{\rm TOV}$. Note that sub-two-solar-mass stars are not excluded if there are two families of CS in which the massive stars are strange.

We have also determined the minimum frequencies required to explain the secondary stellar objects in the gravitational events GW190814 and GW200210. We found that the most extreme models from the large-$Q_{\text{sat}}$ and small-$z$ domain produce masses in the required range. This domain is minimal for non-rotating stars and increases as rotation is allowed. It is maximal for the Keplerian case which allows for very fast rotation at frequencies $f\sim 1500$ Hz. We stress again that even though our CDF study includes the full baryon octet, the highly massive CS models turn out to contain only a very small amount of hyperons (strangeness fractions of typical $\lesssim 3\%$). These stars can therefore be considered as nucleonic stars.

The research of H.F. and J.L. is supported by the National Natural Science Foundation of China (Grant No. 12105232), the Venture & Innovation Support Program for Chongqing Overseas Returnees (Grant No. CX2021007), and by the “Fundamental Research Funds for the Central Universities” (Grant No. SWU-020021). The research of A.S. is funded by Deutsche Forschungsgemeinschaft Grant No. SE 1836/5-2 and the Polish NCN Grant No. 2020/37/B/ST9/01937 at Wrocław University. F.W. acknowledges support by the U.S. National Science Foundation under Grant PHY-2012152.