On the Nature of the Mass-gap Object in the GW190814 Event

In 2019, GW190814 (Abbott et al. 2020) was detected as the result of a coalescence of a 25.6 M_⊙ black hole and a compact object of mass 2.5–2.67 M_⊙, which could either be the most massive neutron star or the least massive black hole ever seen, once it lies in the region known as the mass gap, in between 2.5 and 5 M_⊙.

The possibility of the mass-gap object being a neutron star has already been studied in previous works. For instance, in Wu et al. (2021) the influence of the symmetry energy was studied, and a pure nucleonic neutron star with mass around 2.75 M_⊙ was obtained. The possibility of dark matter admixed in a pure nucleonic neutron star was studied by Das et al. (2021a), and a maximum mass of 2.50 M_⊙ was shown to be possible. In Dexheimer et al. (2021) and Sedrakian et al. (2020), the authors discussed the possibility of hybrid stars and hadronic neutron stars with hyperons. They concluded that the mass-gap object could be a hybrid star only if the star is fast rotating (Dexheimer et al. 2021). Nevertheless, the presence of hyperons seems to be unlikely even in the Kepler frequency limit (Sedrakian et al. 2020; Dexheimer et al. 2021). Using the generic constant-sound-speed (CSS) parameterization, Han & Steiner (2019) show that a hybrid star with a maximum mass above 2.50 M_⊙ is possible, even in the static case.

On the other hand, Rather et al. (2021) rule out the possibility of quark-hadron phase transition, at least in the nonrotating case. But, according to a model-independent analysis based on sound velocity, quark cores are indeed expected inside massive stars (Annala et al. 2020). Δ-admixed neutron stars were studied by Li et al. (2020) and can account for the mass-gap object only in the Keplerian limit, ruling out static stars with Δ resonances. In these studies, no static neutron star with exotic matter fulfills the mass-gap object constraint.

Another possibility is the mass-gap object being a self-bound strange star satisfying the Bodmer–Witten conjecture(Bodmer 1971; Witten 1984). This possibility was studied in Bombaci et al. (2020) and Zhang & Mann (2021), where the authors considered a color superconducting quark matter, and also in Albino et al. (2021), where the authors considered a repulsive bag model with dynamically generated gluon mass. Moreover, using the CFL-NJL model, Lourenço et al. (2021) also produced quark stars with masses above 2.50 M_⊙. In an even earlier work (Horvath & Lugones 2004), the authors showed that self-bound stars made of CFL-paired quarks could reach a maximum mass of up to 4 M_⊙, and therefore, can explain the mass-gap object in the GW190814 event.

In this work, we study whether the mass gap object in the GW190814 event could be a purely nucleonic neutron star, a nucleonic admixed hyperon neutron star, a self-bound quark star satisfying the Bodmer–Witten conjecture (Bodmer 1971; Witten 1984), or a hybrid star with nucleons and a quark core or a hybrid star with nucleons, hyperons, and a quark core. This paper is organized as follows.

In Section 2, we discuss hadronic stars within the quantum hadrodynamics (QHD) model, which simulates the interaction between nucleons as well as hyperons. We discuss some parameterizations in light of both, the GW190814 event, as well as the necessary constraints that satisfy symmetric nuclear matter properties. We then discuss how hyperons affect the equation of state (EoS) and in what conditions can hyperonic neutron stars still reach at least 2.50 M_⊙. Constraints related to the radius and tidal deformability of the canonical 1.4 M_⊙ neutron stars are also discussed.

In Section 3, we discuss self-bound strange stars within the vector MIT bag model. Two parameterizations for the quark-quark interaction are discussed: a universal coupling, where the vector field of all the quarks has the same strength, and a coupling deduced from symmetry group arguments. The results are again compared with the constraints imposed by the radius and the tidal deformability of the canonical star.

In Section 4, we construct a hybrid branch stability window; i.e., we obtain values for the bag and for the vector field, where a stable hybrid star with a mass above 2.50 M_⊙ is possible. We also calculate the size and mass of the quark core of these hybrid stars. Finally, conclusions are drawn in Section 5.

Detailed discussions about the formalism and additional references are presented in the appendices.

To simulate the interaction between baryons in dense cold matter, we use an extended version of the QHD model, whose Lagrangian density reads as

$$\begin{eqnarray}&&\,\begin{array}{l}{{ \mathcal L }}_{{\rm{QHD}}}={\bar{\psi }}_{B}[{\gamma }^{\mu }(i{\partial }_{\mu }-{g}_{B\omega }{\omega }_{\mu }-{g}_{B\rho }\displaystyle \frac{1}{2}\vec{\tau }\cdot {\vec{\rho }}_{\mu })\\ \,-\,({M}_{B}-{g}_{B\sigma }\sigma )]{\psi }_{B}-U(\sigma )\\ \,+\,\displaystyle \frac{1}{2}\left({\partial }_{\mu }\sigma {\partial }^{\mu }\sigma -{m}_{s}^{2}{\sigma }^{2}\right)-\displaystyle \frac{1}{4}{{\rm{\Omega }}}^{\mu \nu }{{\rm{\Omega }}}_{\mu \nu }\\ \,+\,\displaystyle \frac{1}{2}{m}_{v}^{2}{\omega }_{\mu }{\omega }^{\mu }+\,\displaystyle \frac{1}{2}{m}_{\rho }^{2}{\vec{\rho }}_{\mu }\cdot {\vec{\rho }}^{\mu }\\ \,-\,\displaystyle \frac{1}{4}{{\boldsymbol{P}}}^{\mu \nu }\cdot {{\boldsymbol{P}}}_{\mu \nu }+{{ \mathcal L }}_{\omega \rho }+{{ \mathcal L }}_{\phi }.\end{array}\end{eqnarray} \tag{ 1 }$$

All details about the parameters and the relativistic mean field formalism used to obtain the EoS are presented in Appendix A and references therein.

Our knowledge of nuclear physics has taken a great leap forward in the last decade. From nuclear masses analysis (Wang et al. 2012), passing through nuclear resonances (Shlomo et al. 2006; Reinhard & Nazarewicz 2013), and heavy ion collisions (Filippo & Pagano 2014), we are able to constraint five parameters of the symmetric nuclear matter at the saturation point: the saturation density itself (n₀), the effective nucleon mass (${M}_{N}^{* }/{M}_{N}$), the incompressibility (K), the binding energy per baryon (B/A; Glendenning 2000), and the symmetry energy (S₀). The experimental values of these five parameters are taken from two extensive review articles, Dutra et al. (2014) and Oertel et al. (2017), and are presented in Table 1. Besides these five parameters, a sixth one is nowadays a matter of open discussion: the symmetry energy slope, L. For instance, an upper limit for L of 54.6, 61.9, and 66 MeV was presented in Paar et al. (2014), Lattimer & Lim (2013), and Lattimer & Steiner (2014a), respectively. These values are in strong contrast with recent measurements. For instance, an upper limit of 117.5 MeV was found in Estee et al. (2020), in a study about the spectra of pions in intermediate energy collisions, while PREX2 results not only presented an upper limit as high as 143 MeV, but also an inferior limit of only 69 MeV (Reed et al. 2020). Such a high inferior limit makes PREX2 constraint difficult to reconcile with the results obtained in Paar et al. (2014), Lattimer & Lim (2013), and Lattimer & Steiner (2014a). In order to maintain consistency with the other constraints, here we have opted to use 36 MeV <L < 86.8 MeV as a constraint, once it was pointed out in Oertel et al. (2017).

It is very clear that we need a very stiff equation of state (EoS) in order to obtain at least a 2.50 M_⊙ star. There are a few QHD models that are able to produce such massive neutron stars and still be in agreement with the constraints presented in Table 1. We can highlight NL3 (Lalazissis et al. 1997), which although a little outdated, has still been used in very recent studies (Das et al. 2021a; Wu et al. 2021; Li et al. 2022), its updated version, the so-called NL3* (Lalazissis et al. 2009), NL-RA1 (Rashdan 2001), and the recent BigApple parameterization (Das et al. 2021b). But, of all of these parameterizations, only the BigApple has a symmetry energy slope consistent with the constraint presented in Oertel et al. (2017), 36 MeV < L < 86.8 MeV. Nevertheless, with the help of the nonlinear ω − ρ coupling given in Equation (A1), we can redefine the symmetry energy slope without affecting any of the other properties of the symmetric nuclear matter. A more serious issue is the incompressibility (K). Assuming that K lies between 240 ± 20, as pointed out in Oertel et al. (2017), only NL3* and the BigApple fulfill this constraint. As the BigApple produces a maximum star mass of 2.60 M_⊙ and NL3* has a maximum mass of 2.75 M_⊙, in this work we use a modified version of NL3*, which includes ω − ρ coupling. All the parameters are displayed in Table 1.

Once the parameterization of the nuclear matter is settled, we need to focus on the parameterization of the hyperon-meson coupling constants. There is little information about the hyperon interaction. One of the few well-known quantities is the Λ⁰ potential depth, U_Λ = −28 MeV.

The potential depths for Σ and the Ξ are known with less precision. In this work, we use the standard values U_Σ =+30 MeV and U_Ξ = −4 MeV, which was recently favored by lattice QCD calculations (Inoue 2019). Nevertheless, the knowledge of the hyperon potential depth only partially solves the problem once different combinations of the coupling constants can produce the same value for the potential depth. And, as pointed out in Glendenning (2000), different values of the hyperon-meson coupling constants can lead to a maximum mass difference of up to 100%. The situation becomes even worse, as the potential depth gives us no information about the hyperon-meson coupling constants for the ρ and ϕ mesons.

In this work, we use symmetry group arguments to fix the coupling constants of the hyperons with the vector mesons, while we use the potential depths to fix the coupling constants of the hyperons with the scalar meson. The calculated coupling constants for different values of α are presented in Table 2 while in Figure 1, we plot the particle population for each value of α. Notice that for α = 1 the symmetry of the SU(6) group is recovered. Further discussion is presented in Appendix B and references therein.

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We can see that for all values of α, the only hyperons present are Λ⁰ and t Ξ⁻. Also, for all values of α, Λ⁰ is the first hyperon to appear, at densities of around 0.33 fm⁻³. For the SU(6) parameterization, the potential depths are less repulsive at high density. Due to this fact, the hyperons are favored, resulting in two interesting features: Λ⁰ becomes the most populous particle at densities around 0.73 fm⁻³, matter is practically deleptonized, and electric charge neutrality is reached with equal proportions of protons and Ξ⁻ hyperons. For α = 0.75, the matter is also deleptonized, but at higher densities. For lower values, the depth of the potential is already too repulsive, and electrons and muons are always present.

A more clever way to understand the variation in the strangeness of the content particle is instead of looking at the individual hyperon population looking at the strangeness fraction, f_s , defined as

$$\begin{eqnarray}&&{f}_{s}=\displaystyle \frac{1}{3}\displaystyle \frac{\sum {n}_{i}| {s}_{i}| }{n},\end{eqnarray} \tag{ 2 }$$

where s_i is the strangeness of the ith baryon. The results are plotted in Figure 2. As can be seen, there is a direct relation between α and the strangeness fraction, as also pointed out in Lopes & Menezes (2014). When we move away from SU(6) we increase the repulsion of the hyperons, by increasing the Y − ω and the Y − ϕ coupling constants as shown in Table 2. This reduces the hyperon population at high densities, reducing the strangeness fraction.

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Now we use the EoS for different values of α as input to solve the TOV equations (Oppenheimer & Volkoff 1939) and obtain the mass–radius relations. The EoS as well as the mass–radius relation of the TOV solution are displayed in Figure 3, where we use the BPS EoS to simulate the neutron star crust (Baym et al. 1971). As can be seen, there is a clear relation between the value of α, the strangeness fraction, f_s , the EoS, and the maximum mass. The lower the value of α, the lower the value of f_s , which produces a stiffer EoS, as well as more massive stars. In light of the GW190814 event, we can see that this mass-gap object not only can be a pure nucleonic neutron star, but also a hyperon admixed star, if α < 0.75. For α ≥ 0.75, the maximum mass lies below 2.50 M_⊙. Also, as the onset of Λ⁰ happens around 0.33 fm⁻³ for all values of α, no hyperonic neutron stars with mass below 1.66 M_⊙ are possible.

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We can also discuss our results in light of some observational astronomical constraints. One of the hot topics in modern days is the radius of the canonical star, M = 1.40 M_⊙. However, as pointed out in Cavagnoli et al. (2011), the radius of the canonical star strongly depends on the symmetry energy slope, L. And, as discussed earlier, the high uncertainty about its value will obviously affect the uncertainty of the radius. For instance, in Ozel et al. (2016) a very small upper limit of only 11.1 km was presented. Another small value of 11.9 km was also pointed out in Capano et al. (2020), while in Lattimer & Steiner (2014b) an upper limit of 12.45 km was deduced. More recently, results obtained from Bayesian analysis indicate that the radius of the canonical star lies between 10.8 and 13.2 km (Li et al. 2021); and 11.3–13.5 km (Coughlin et al. 2019); while results coming from the NICER X-ray telescope points out that R_1.4 lies between 11.52 and 13.85 km in Riley et al. (2019) and between 11.96 and 14.26 km in Miller et al. (2019). State-of-the-art theoretical results at low and high baryon densities point to an upper limit of R_1.4 <13.6 km (Annala et al.2018). Finally, PREX2 results (Reed et al. 2020) indicate that the radius of the canonical star lies between 13.25 km <R_1.4 <14.26 km. This constraint is mutually exclusive with the result presented in Ozel et al. (2016), Capano et al. (2020), Lattimer & Steiner (2014b), and Li et al. (2021).

Notwithstanding, in this work we use as a constraint the radius of the canonical star between 12.2 km < R_1.4 < 13.7 km, as presented in Abbott et al. (2020). The reason we choose this value over all the others discussed above is that it is derived from the GW190814 event itself, the subject of the present work. As in our study, no hyperon is present in a 1.4 M_⊙ star, and its radius for all values of α is 13.38 km. Such value is in agreement with our main constraint from Abbott et al. (2020), as well as in agreement with NICER results (Riley et al. 2019; Miller et al. 2019), Bayesian analysis (Coughlin et al. 2019), PREX2 (Reed et al. 2020), and state-of-the-art theoretical results (Annala et al. 2018). However, it is in disagreement with the results presented in Ozel et al. (2016), Capano et al. (2020), Lattimer & Steiner (2014b), and Li et al. (2021).

Another important quantity and constraint is the so-called dimensionless tidal deformability parameter Λ. If we put an extended body in an inhomogeneous external field, it will experience different forces throughout its extent. The result is a tidal interaction. The tidal deformability of a compact object is a single parameter λ that quantifies how easily the object is deformed when subjected to an external tidal field. A larger tidal deformability indicates that the object is easily deformable. Conversely, a compact object with a small tidal deformability parameter is more compact and more difficult to deform. The tidal deformability is defined as the ratio between the induced quadrupole Q_ij and the perturbing tidal field ${{ \mathcal E }}_{{ij}}$ that causes the perturbation

$$\begin{eqnarray}&&\lambda =-\displaystyle \frac{{Q}_{{ij}}}{{{ \mathcal E }}_{{ij}}}.\end{eqnarray} \tag{ 3 }$$

However, in the literature the dimensionless tidal deformability parameter Λ is more commonly defined as

$$\begin{eqnarray}&&{\rm{\Lambda }}\,\equiv \,\displaystyle \frac{\lambda }{{M}^{5}}\,\equiv \,\displaystyle \frac{2{k}_{2}}{3{C}^{5}},\end{eqnarray} \tag{ 4 }$$

where M is the compact object mass and C = GM/R is its compactness. The parameter k₂ is called the second (order) Love number. Further discussion of the tidal deformability can be found in Appendix C and references therein.

We display in Figure 4 the tidal deformability parameter Λ as a function of the star mass of pure nucleonic stars as well as for hyperonic stars with two values of α. Exactly as in the case of the radius, the dimensionless tidal parameter of the canonical star, Λ_1.4, can be used as a constraint. An upper limit of 860 was found in Coughlin et al. (2019). A close limit, Λ_1.4 < 800, was pointed out in Abbott et al. (2017). In Li et al. (2021), an upper limit of 686 was deduced from Bayesian analysis. On the other hand, two mutually exclusive constraints are presented in Abbott et al. (2018), which proposed a limit between 70 < Λ_1.4 < 580, and the PREX2 inferred values, whose limit lies between 642 <Λ_1.4 < 955 (Reed et al. 2020). Here, as done in the case of the radius, we use the values 458 < Λ_1.4 < 889 as the main constraint, once it was derived from the GW190814 event itself and presented in Abbott et al. (2020). As no hyperons are present in a 1.4M_⊙ star, with the choice of parameters used in this study, all canonical stars have the same value for Λ_1.4 = 644. This value is in agreement with the main constraint from Abbott et al. (2020), as well as with all the others, except for the one presented in Abbott et al. (2018).

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Another important quantity of dense cold nuclear matter is the speed of sound, defined as

$$\begin{eqnarray}&&{v}_{s}^{2}=\displaystyle \frac{\partial p}{\partial \epsilon }.\end{eqnarray} \tag{ 5 }$$

The speed of sound is related to the stiffness of the EoS, and can give us important insight into the internal composition of the neutron star. In general, for a pure nucleonic neutron star, the speed of sound grows monotonically. However, the onset of new degrees of freedom can produce a nontrivial behavior and the speed of sound can present maxima and minima. The speed of sound also acts as a constraint, once superluminal values, ${v}_{s}^{2}\gt 1$, violate Lorentz symmetry. Moreover, due to the conformal symmetry of the QCD, perturbative QCD (pQCD) predicts an upper limit ${v}_{s}^{2}\lt 1/3$ at very high densities, n > 40 n₀ (Bedaque & Steiner 2015). Such high density is far beyond those found in neutron star interiors.

The speed of sound may also be linked to the size of the quark core of a hybrid star. As recently pointed out in Annala et al. (2020), the speed of sound of quark matter is closely related to the mass and radius of the quark core in hybrid stars. The authors found that if the conformal bound (${v}_{s}^{2}\,\lt 1/3$ ) is not strongly violated, massive neutron stars are predicted to have sizable quark-matter cores. We show in Figure 5 the square of the speed of sound for pure nucleonic matter, as well as for hyperonic matter for different values of α.

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As can be seen, all of our models are causal, ${v}_{s}^{2}\lt 1$. This was expected, since we are dealing with a relativistic model. The onset of hyperons breaks the monotonic behavior and reduces the speed of sound. The higher the value of α, the lower the speed of the sound. Nevertheless, the conformal limit is always violated.

The last constraint for hadronic neutron stars presented in this work is the minimum mass that enables the direct Urca (DU) process. As pointed out in Klahn et al. (2006), any acceptable EoS shall not allow the DU process to occur in neutron stars with masses below 1.5 M_⊙. Therefore, we also use this value as a constraint. The trigger to the nucleonic DU channel is directly related to the leptonic fraction x_DU, defined as (Klahn et al. 2006; Cavagnoli et al. 2011)

$$\begin{eqnarray}&&{x}_{\mathrm{DU}}=\displaystyle \frac{1}{1+{\left(1+{x}_{e}^{1/3}\right)}^{3}},\end{eqnarray} \tag{ 6 }$$

where x_e = n_e /(n_e + n_μ ). For the models presented in this work, only pure nucleonic neutron stars enable the DU process, and at very high density: 0.51 fm⁻³, which imply a very massive star of 2.56 M_⊙ as an inferior limit for the DU process.

We now display in Table 3 some macroscopic and microscopic properties of nucleonic and hyperonic neutron stars. As we have already pointed out, all star families have the same radius and tidal deformability for the canonical mass. Also, we need α < 0.75 in order to reproduce a 2.50 M_⊙ star. From the microscopic point of view, we can infer some features from the speed of sound and the strangeness fraction. For instance, our study indicates that the central speed of sound should be ${v}_{s}^{2}\,\gt 0.62$ in order to produce at least 2.50 M_⊙. Such a high value is almost twice the conformal limit imposed by pQCD (Bedaque & Steiner 2015). In the same sense, we also need f_s < 18% at the core of the neutron star. When we look at the more well-established 2.0 M_⊙, we see that a strangeness fraction of only 3.6% is enough to prevent the formation of a 2.50 M_⊙.

Table 3. Nucleonic and Hyperonic Neutron Star Properties for Different Values of α and Some Astrophysical Constraints

Model	${M}_{\max }/{M}_{0}$	R (km)	nc (fm−3)	fsc	vsc 2	MDU/ M⊙	R1.4 (km)	Λ1.4	fs2.0
SU(6)	2.36	12.56	0.777	0.210	0.56	⋯	13.38	644	0.056
α = 0.75	2.46	12.59	0.762	0.181	0.62	⋯	13.38	644	0.036
α = 0.50	2.57	12.70	0.736	0.133	0.68	⋯	13.38	644	0.025
α = 0.25	2.64	12.75	0.720	0.093	0.74	⋯	13.38	644	0.014
Nucl.	2.75	12.87	0.699	⋯	0.79	2.56	13.38	644	⋯
Constraints	>2.50	⋯	⋯	⋯	<1.0	>1.50	12.2–13.7	458–889	⋯

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Let us now consider the possibility of the mass-gap object of GW190814 being a stable quark star, sometimes called a strange star. This idea is based on the so-called Bodmer–Witten hypothesis (Bodmer 1971; Witten 1984). According to it, the matter composed of protons and neutrons may be only metastable. The true ground state of strongly interacting matter would therefore consist of strange matter (SM), which in turn is composed of deconfined up, down, and strange quarks. For the SM hypothesis to be true, the energy per baryon of the deconfined phase (for p = 0 and T = 0) is lower than the nonstrange infinite baryonic matter. Or explicitly(Bodmer 1971; Witten 1984; Lopes et al. 2021),

$$\begin{eqnarray}&&{E}_{({uds})}/A\lt 930\,{\rm{MeV}},\end{eqnarray} \tag{ 7 }$$

at the same time, the nonstrange matter still needs to have an energy per baryon higher than the one of nonstrange infinite baryonic matter; otherwise, protons and neutrons would decay into u and d quarks:

$$\begin{eqnarray}&&{E}_{({ud})}/A\gt 930\,{\rm{MeV}}.\end{eqnarray} \tag{ 8 }$$

Therefore, both Equations (7) and (8) must simultaneously be true.

One of the simplest models to study quark matter is the so-called MIT bag model (Chodos et al. 1974). This model considers that each baryon is composed of three noninteracting quarks inside a bag. The bag, in turn, corresponds to an infinity potential that confines the quarks. In this simple model, the quarks are free inside the bag and are forbidden to reach its exterior. All the information about the strong force relies on the bag pressure value, which mimics the vacuum pressure. However, the maximum mass of a stable quark star in the original MIT bag model is way below 2.50 M_⊙.

We can overcome this issue by using a modified MIT bag model that adds a massive repulsive vector field, analogous to the ω meson of the QHD. We follow this path here. The Lagrangian of the modified MIT bag model reads as (Lopes et al. 2021)

$$\begin{eqnarray}&&\begin{array}{l}{{ \mathcal L }}_{v\mathrm{MIT}}=\{{\bar{\psi }}_{q}[{\gamma }^{\mu }(i{\partial }_{\mu }-{g}_{{qV}}{V}_{\mu })-{m}_{q}]{\psi }_{q}\\ \,+\,\displaystyle \frac{1}{2}{m}_{V}^{2}{V}_{\mu }{V}^{\mu }-B\}{\rm{\Theta }}({\bar{\psi }}_{q}{\psi }_{q}).\end{array}\end{eqnarray} \tag{ 9 }$$

The formalism and the parameters used in the present work are discussed in Appendix D and references therein.

The interaction of the vector field with different quark flavors can follow two different prescriptions. In the first one we have a universal coupling, i.e., the strength of the interaction is the same for all three quarks. This is a more conventional approach, and was done, for instance, in Gomes et al. (2019b, 2019a). In this case, g_uV = g_dV = g_sV . Another possibility explored in Lopes et al. (2021) is that the vector field is not only analogous to the ω meson, but it is the ω meson itself. In this approach, we can use symmetry group arguments and construct an invariant Lagrangian. With this approach we have g_uV = g_dV and g_sV = 0.4g_uV . We now follow Lopes et al. (2021) and define two quantities:

$$\begin{eqnarray}&&{G}_{V}\,\equiv \,{\left(\displaystyle \frac{{g}_{{uV}}}{{m}_{V}}\right)}^{2}\quad {\rm{and}}\quad {X}_{V}\,\equiv \,\displaystyle \frac{{g}_{{sV}}}{{g}_{{uV}}},\end{eqnarray} \tag{ 10 }$$

G_V is related to the absolute strength of the vector field itself and X_V is related to the relative strength of the vector field. If X_V = 1.0, we are dealing with a universal coupling, while X_V = 0.4 implies symmetry group arguments. We study how different values of G_V and X_V affect the macroscopic properties of strange stars.

Now, for chosen values of G_V and X_V , the bag pressure B is not arbitrary. To predict the existence of strange stars, B must be chosen in order to satisfy both Equation (7) and Equation (8). The set of values that satisfies both equations simultaneously is used to construct the stability window(Bodmer 1971; Witten 1984). We display the stability window for 0.30 fm² < G_V < 0.40 fm² in Figure 6 and the numerical results are shown in Table 4.

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Table 4. Stability Windows Obtained with the Vector MIT Bag Model

GV (fm2)	XV	${B}_{\mathrm{Min}}^{1/4}$ (MeV)	${B}_{\mathrm{Max}}^{1/4}$ (MeV)
0.30	0.4	139	150
0.32	0.4	138	149
0.36	0.4	138	148
0.40	0.4	137	148
0.30	1.0	139	146
0.32	1.0	138	145
0.36	1.0	138	144
0.40	1.0	137	143

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We can see that the lower limit of the stability window is independent of the value of X_V . This is expected, once it is related to the stability of the two flavored quark matter, expressed in Equation (8). On the other hand, the stability of three flavored quark matter depends on X_V . The stability window is always wider for X_V = 0.4, once the repulsion in the strange quark is lower, reducing the energy per baryon. We now show in Figure 7 the EoS and the mass–radius relation for strange stars with X_V = 1.0 and 0.4. For X_V = 1.0, we plot the EoS and the TOV solution for both the minimum and maximum values of the bag as presented in Table 4. As can be seen, for a given bag value, the higher the value of G_V , the stiffer the EoS is and consequently, the higher the maximum mass is. For X_V = 1.0, a maximum mass above 2.50 M_⊙ is reached for the minimum bag value for all three values of G_V presented in this work. In the case of the maximum bag value in Table 4, we see that only G_V > 0.32 can reach at least 2.50 M_⊙. For X_V = 0.4, we only plot the results for the minimum bag value. We see that the EoS is softer when compared with X_V = 1.0, and therefore produces lower values of the maximum mass, and only values G_V > 0.32 fm² can reach at least 2.50 M_⊙.

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Another important constraint is the radius of the 1.4 M_⊙ star. Considering only the results that reach at least 2.50 M_⊙, the radii shown in Figure 7 lie between 11.59 and 12.58 km. We see that several radii agree with the NICER results (Riley et al. 2019; Miller et al. 2019), Bayesian analysis (Coughlin et al. 2019; Li et al. 2021), as well as with Annala et al. (2020). However, as we use here the results coming from Abbott et al. (2020) as the main constraint, not all radius values are in the range between 12.2 and 13.7 km. Indeed, neither stars produced with the maximum bag value nor stars with G_V = 0.32 fm² have a radius above 12.2 km.

Now we check the values of the dimensionless tidal parameter Λ. It is important to note that the discontinuity present at the surface of the strange star must be taken into account (see Equation (C7) in Appendix C).

The results for X_V = 1.0 and 0.4 are plotted in Figure 8:

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As can be seen for X_V = 0.4, only G_V = 0.32 fm² fulfills the constraint for Λ_1.4 in Abbott et al. (2020). However, since its maximum mass is below 2.50 M_⊙, we can rule out the possibility of the mass-gap object of the GW190814 event being a strange star with X_V = 0.4. In the case of X_V = 1.0, only the values of the maximum allowed bag fulfill the constraint. However, as pointed out earlier, these values present low radii, in disagreement with Abbott et al. (2020). We cannot completely rule out the possibility of the mass-gap object of GW190814 being a strange star, since the radii values still fulfill NICER constraints. Nevertheless, in order to maintain internal coherence, in light of the constraints presented in Abbott et al. (2020), we assert that the probability of the mass-gap object being a strange star is significantly lower than that of being a hadronic star.

In Figure 9, we display the speed of sound. As can be seen, the causality ${v}_{s}^{2}\lt 1$ is always satisfied. However, the conformal limit ${v}_{s}^{2}\lt 1/3$ is violated, even at low densities. Nevertheless, we are far below the pQCD limit of n > 40 n₀. We finish this section by displaying some macroscopic and microscopic properties of the strange stars in Table 5. We only show results for X_V = 1.0 as discussed in the text because for X_V = 0.4 no parameterization is able to reach M > 2.50 M_⊙ and Λ_1.4 < 889 simultaneously.

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Table 5. Some Strange Star Properties of X_V = 1.0 and Some Astrophysical Constraints

GV (fm 2)	B1/4 (MeV)	${M}_{\max }/{M}_{0}$	R (km)	nc (fm−3)	ΔS (MeV/fm3)	vsc 2	R1.4 (km)	Λ1.4
0.32	138	2.64	13.14	0.699	180	0.50	12.17	983
0.36	138	2.70	13.30	0.671	174	0.51	12.31	1023
0.40	137	2.81	13.78	0.632	166	0.52	12.58	1072
0.32	145	2.44	11.96	0.794	216	0.51	11.40	634
0.36	144	2.51	12.26	0.759	208	0.52	11.59	729
0.40	143	2.58	12.57	0.725	198	0.53	11.80	817
Constraints	stability window	>2.50	⋯	⋯	⋯	<1.0	12.2–13.7	458–889

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As can be seen, for the minimum bag value, the tidal parameter Λ_1.4 is close to or even higher than 1000. We highlight here that for B^1/4 = 144 MeV and G_V = 0.36, we can produce a massive star with M > 2.50 M_⊙ with Λ_1.4 < 800, in agreement with Abbott et al. (2018), while the radius agrees with Bayesian analysis (Coughlin et al. 2019; Li et al. 2021) and the result presented in Annala et al. (2020). Unlike the hadronic neutron star, we see that the central speed of the sound is only weakly linked to the stiffness of the EoS, being almost constant for all models.

We now investigate the possibility of the GW190814 event being a dynamically stable hybrid star. There are some questions we try to answer: Is it possible that the mass-gap object in the GW190814 event is a hybrid star? How do hyperons affect the presence of quarks in the neutron star core? What are the values of the chemical potential at the phase transition point? What are the maximum and minimum values of G_V that produce a dynamically stable hybrid star? What is the influence of the bag value? How does the factor X_V influence the macroscopic properties? What is the stellar minimum mass that presents a quark core? What are the size and mass of the quark core in a stable hybrid star?

The possibility that the mass-gap object of the GW190814 event is a hybrid star has already been studied by Han & Steiner (2019). In that paper, the authors use the generic constant-sound-speed (CSS) parameterization, with no information about the quark interaction nor the chemical composition of the matter and the EoS simple reads as (Zdunik & Haensel 2013)

$$\begin{eqnarray}&&p(\epsilon )=a(\epsilon -{\epsilon }_{0}),\end{eqnarray} \tag{ 11 }$$

where a and ₀ are constants related to the speed of the sound and the energy density at zero pressure, respectively. Here, we use a more physical, Lagrangian density-based model described in Equation (9).

Another difference between the present work and the study presented in Han & Steiner (2019) is the quark-hadron phase transition criteria. In Han & Steiner (2019), the transition pressure is treated as a free parameter. We use the so-called Maxwell construction, and the transition pressure is the one where the Gibbs free energy per baryon G/n_B of both phases intersect, being the energetically preferred phase the one with lower G/n_B (Chamel et al. 2013). The Gibbs free energy per baryon coincides with the baryon chemical potential; therefore, we call the intersection point the critical pressure and critical chemical potential. The Maxwell criteria read as

$$\begin{eqnarray}&&{\mu }_{H}={\mu }_{Q}={\mu }_{C},\quad {\rm{at}}\quad {p}_{H}={p}_{Q}={p}_{C},\end{eqnarray} \tag{ 12 }$$

where the subscript H indicates a hadronic phase, Q indicates a quark phase, and C indicates the critical values.

It is worth pointing out that there is yet another criterion to construct a hybrid star based on the Gibbs condition that favors a mixed phase where the quarks and hadrons can coexist. Some authors (Maruyama et al. 2007; Paoli & Menezes 2010) performed studies on hybrid stars with both, Maxwell and Gibbs construction and concluded that there is no significant difference in the macroscopic properties of the hybrid stars. Moreover, a recent study (Lugones & Grunfeld 2021) has shown that vector interactions inhibit quark-hadron mixed phases in neutron stars. Due to these facts, here we only use the Maxwell construction.

We begin by reanalyzing the stability window shown in Figure 6. To construct strange stars, we imposed that the energy per baryon of the deconfined phase is lower than the nonstrange infinite baryonic matter. But here we need the opposite. To produce a stable hybrid star, the strange matter must be unstable, otherwise, as soon as the core of the star converts into the quark phase, the entire star will convert into a quark star in a finite amount of time (Olinto 1987; Marquez & Menezes 2017). Therefore, we fix our bag values between 150 MeV < B^1/4 < 160 MeV to ensure an unstable strange matter. We study the possibility of a hybrid star with nucleons and quarks, and a hybrid star with nucleons, hyperons, and quarks. To obtain very massive hybrid stars with nucleons and hyperons, we use α = 0.25. Now, for a chosen value of X_V , we solve the TOV equations for different values of G_V and of the bag. The values of G_V that produce a stable hybrid star with M > 2.50 M_⊙ form what we call the hybrid branch stability window.

We start by constructing the hybrid branch stability window for pure nucleonic (NN) and hyperonic (NH) EoSs for the hadronic phase with X_V = 1.0 and 0.4. The results are presented in Figure 10. We can see that for X_V = 1.0 and 150 MeV < B^1/4 < 160 MeV, the values of G_V that produce dynamical stable hybrid stars, with M > 2.50 M_⊙ lie between 0.39 fm² < G_V < 0.48 fm² for a hybrid star with nucleons, hyperons, and quarks, and 0.39 fm ² < G_V < 0.52 fm² for a hybrid star with only nucleons and quarks. With X_V = 0.4 the results are very different. The values of G_V now lie between 0.66 fm² < G_V < 0.85 fm² for a hybrid star with nucleons, hyperons, and quarks and 0.66 fm² < G_V < 0.92 fm² for a hybrid star with only nucleons and quarks. We can also see that the hybrid branch stability window for X_V = 0.4 is significantly broader when compared with X_V = 1.0. The consequences of using a G_V below or above the values of the hybrid branch stability window are also different. If G_V is too low, stable hybrid stars still exist; however, the maximum mass is lower than 2.50 M_⊙. If G_V is too high, there is no dynamically stable hybrid star, yet the maximum mass is above 2.50 M_⊙, but it is purely hadronic.

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We also estimate the mass and size of the quark cores present in the most massive hybrid star of each model presented. To accomplish that, we follow Lopes & Menezes (2021) and Lopes et al. (2022), and solve the TOV equations (Oppenheimer & Volkoff 1939) for the quark EoS from the density corresponding to the central density at the maximum mass neutron stars and stop at the density corresponding to the critical chemical potential. The EoS and the TOV solution with X_V = 1.0 and 0.4 for the extreme values of G_V that satisfy the hybrid branch stability window are shown in Figure 11, while the macroscopic and microscopic properties are displayed in Table 6.

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Table 6. Maximum Mass, Radius, Central Density, Critical Chemical Potential, M_min, Mass and Radius of the Quark Core and the Speed of Sound at the Critical Chemical Potential for the Extreme Values of G_V That Allow Dynamical Stable Hybrid Stars

XV	B1/4 (MeV)	Type	GV (fm 2)	${M}_{\max }/\,{M}_{\odot }$	R (km)	nc (fm−3)	μC (MeV)	${M}_{\min }/{M}_{\odot }$	MQ (M⊙)	RQ (km)	vSC 2
1.0	150	NH	0.44	2.53	12.90	0.719	1336	2.28	0.874	7.23	0.50
1.0	150	NN	0.44	2.53	12.90	0.720	1280	2.18	1.064	7.89	0.49
1.0	150	NH	0.48	2.63	13.02	0.710	1707	2.62	0.029	2.08	0.55
1.0	150	NN	0.52	2.73	13.21	0.689	1736	2.72	0.019	1.82	0.57
1.0	160	NH	0.39	2.53	13.31	0.665	1499	2.51	0.125	3.55	0.51
1.0	160	NN	0.39	2.54	13.47	0.624	1445	2.50	0.148	3.87	0.51
1.0	160	NH	0.43	2.62	13.13	0.718	1680	2.62	0.009	1.35	0.54
1.0	160	NN	0.46	2.71	13.32	0.718	1683	2.70	0.043	2.36	0.55
0.4	150	NH	0.75	2.50	13.00	0.712	1373	2.35	0.650	6.44	0.48
0.4	150	NN	0.75	2.50	13.01	0.713	1321	2.27	0.817	7.08	0.47
0.4	150	NH	0.85	2.63	13.08	0.715	1695	2.62	0.015	1.66	0.51
0.4	150	NN	0.92	2.71	13.29	0.689	1703	2.71	0.011	1.50	0.51
0.4	160	NH	0.66	2.52	13.40	0.661	1490	2.51	0.087	3.14	0.48
0.4	160	NN	0.66	2.53	13.57	0.632	1444	2.51	0.115	3.53	0.48
0.4	160	NH	0.76	2.62	13.11	0.740	1689	2.62	0.002	0.85	0.50
0.4	160	NN	0.82	2.71	13.33	0.715	1695	2.71	0.004	1.11	0.51

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Several different comparisons can be performed from the results presented in Table 6. For instance, for a fixed value of B, X_V , and G_V we can see the influence of hyperons in hybrid stars. In this case, the presence of hyperons does not affect the maximum mass of a hybrid star. Indeed, hyperons soften the EoS and make the hadron-quark phase transition more difficult. A hybrid star family with only nucleons presents a lower critical chemical potential, and therefore a lower M_min, which is the minimum star that presents a quark core, and at the same time presents a larger quark core when compared with the case with hyperons.

By increasing G_V , we can see an increase in the maximum mass. However, as it also increases the critical chemical potential, M_min becomes very close to the maximum mass. For the maximum allowed value of G_V , the mass and radius of the quark core are very small. Moreover, we see that nucleonic hybrid stars allow a higher value of G_V , and therefore produce a higher maximum mass.

For the correct choice of G_V , the maximum mass does not depend on the bag value. However, the mass and radius of the quark core do. Higher values of the bag produce lower sizes of the quark core, even for the lower value of G_V in the hybrid branch stability window.

The mass and size of the quark core strongly depend on the value of the critical chemical potential. Higher values of μ_C produce low values of the quark core. Massive quark cores can be obtained with low values of the bag and low values of G_V . We can also see that the critical chemical potential can be as high as 1700 MeV. This value is significantly higher than those around 1200 MeV presented in other studies on hybrid stars (Klahn et al. 2017; Ayriyan et al. 2018; Lopes et al. 2022).

For the correct choice of G_V , the hybrid star's maximum mass is independent of X_V . However, X_V = 0.4 seems to produce a slightly lower value of the mass and radius of the quark core when compared with the ones obtained with X_V = 1.0.

The mass and radius of the quark core are strongly model dependent. The mass can vary from only 0.002 M_⊙ to values larger than 1 M_⊙. The quark core can vary from less than 1 km to almost 8 km. Also, we do not discuss the tidal deformability or the radius of the canonical mass, as in our model all hybrid stars have a mass of at least 2.18 M_⊙.

It is also worth pointing out that we have only analyzed stars that satisfy the criterion ∂M/∂₀ > 0. This is needed in all EoSs discussed in this work. However, had we assumed a slow phase transition, this criterion would neither be necessary nor sufficient. A complete discussion of this subject is far beyond the scope of this work, but we refer the interested reader to Pereira et al. (2018) and references therein.

We finish our analysis by discussing the speed of sound at the critical chemical potential in light of the results presented in Annala et al. (2020). The authors claim that the speed of sound of the quark matter is closely related to the mass and radius of the quark core in hybrid stars. Moreover, they assert that if the conformal bound (${v}_{s}^{2}\,\lt$ 1/3) is not strongly violated, massive neutron stars are predicted to have sizable quark-matter cores. As can be seen, we do not find such a correlation between the size of the quark core and the speed of sound. We are even able to produce a quark core with mass above 1 M_⊙, with ${v}_{s}^{2}\,=$ 0.50, far above the conformal limit. Also, the most massive quark core is not those with the lower speed of sound.

In this work, we discuss the possibility of the mass-gap object in the GW190814 event being a degenerate object instead of a black hole. Our main conclusions are as follows:

• 1.  
  We start by looking at whether there is a parameterization of the QHD that fulfills all symmetric nuclear matter constraints, as discussed in two review works (Dutra et al.2014; Oertel et al. 2017), and at the same time, produce stars that can reach at least 2.50 M_⊙. We choose a modified version of NL3* to accomplish this task.
• 2.  
  We then study under what conditions hyperons can be present in such massive neutron stars. We show that a hyperonic neutron star with mass above 2.50 M_⊙ is possible if we use symmetry group arguments to fix the hyperon-meson coupling constant and use α < 0.75.
• 3.  
  We show that the maximum mass, as well as the strangeness fraction are strongly linked to the choice of α. The lower the value of α, the lower the strangeness fraction, and the higher the maximum mass.
• 4.  
  The maximum mass is also linked with the speed of sound at the core of the neutron star. The lower the α, the higher the speed of sound, and consequently the higher the maximum mass. The conformal limit (${v}_{s}^{2}\,\lt$ 1/3) (Bedaque & Steiner 2015) is always violated.
• 5.  
  We discuss the minimum mass that enables the DU process, in light of the constraint M_DU > 1.50 M_⊙ (Klahn et al. 2006). We see that only pure nucleonic neutron stars are able to undergo a DU process and only stars with M > 2.56 M_⊙.
• 6.  
  We also analyze the constraints related to the canonical M = 1.40 M_⊙ star. As hyperons are only present in stars with M > 1.66 M_⊙, all hadronic models present the same R_1.4 = 13.38 km and Λ_1.4 = 644. These values agree with the main constraint from Abbott et al. (2020), i.e., 12.2 km < R_1.4 < 13.7 and 458 < Λ_1.4 < 889.
• 7.  
  We then study the possibility of the mass-gap object being a self-bound quark star, satisfying the Bodmer–Witten conjecture (Bodmer 1971; Witten 1984). We construct a stability window for two values of X_V and show that for X_V = 0.4, no strange star is able to simultaneously fulfill M > 2.50 M_⊙ and Λ_1.4 < 889.
• 8.  
  For X_V = 1.0, we are able to simultaneously fulfill M > 2.50 M_⊙ and Λ_1.4 < 889, but with R_1.4 < 12.2 km. This is in disagreement with the main constraint, but still in agreement with other results found in the literature(Coughlin et al. 2019; Riley et al. 2019; Li et al. 2021); therefore, this possibility cannot be completely ruled out. Nevertheless, we found it less probable than a hadronic or a hybrid star.
• 9.  
  We finally analyze the possibility that the mass-gap object is a hybrid star. We discuss whether such a hybrid star could be composed of nucleons, hyperons, and quarks, or nucleons only and quarks. We show that for the correct choice of G_V , both possibilities can be satisfied. Also, we are able to produce hybrid stars for all values of the bag lying between 150 and 160 MeV, for both X_V = 1.0 and 0.4
• 10.  
  For a fixed G_V , the presence of hyperons reduces the mass and radius of the quark core, but has little effect on the maximum mass of the hybrid star.
• 11.  
  The size and the mass of the quark core are strongly model dependent, and its mass varies from values lower than 0.01 M_⊙ to values larger than 1.0 M_⊙.
• 12.  
  We did not find a correlation between the speed of sound of the quark matter and the size of the quark core, as suggested in Annala et al. (2020). Nevertheless, we are able to produce a very massive quark core M > 1 M_⊙, even though the conformal limit is violated.

Before concluding, we would like to mention that we just became aware of a new possible massive neutron star, PSR J0952-0607, with a mass calculated as 2.35 ± 0.17 M_⊙ (Romani et al. 2022). If confirmed, this is indeed the most massive neutron star confirmed so far and even some of our results that were excluded for not being able to reproduce the mass-gap object can comfortably describe it.

This work is a part of the project INCT-FNA Proc. No. 464898/2014-5. D.P.M. is partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq/Brazil) under grant 301155.2017-8.

Let us start with the Lagrangian density presented in Equation (1) (Serot 1992; Fattoyev et al. 2010):

$$\begin{eqnarray*}&&\begin{array}{l}{{ \mathcal L }}_{{\rm{QHD}}}={\bar{\psi }}_{B}[{\gamma }^{\mu }(i{\partial }_{\mu }-{g}_{B\omega }{\omega }_{\mu }-{g}_{B\rho }\displaystyle \frac{1}{2}\vec{\tau }\cdot {\vec{\rho }}_{\mu })\\ \,-\,({M}_{B}-{g}_{B\sigma }\sigma )]{\psi }_{B}-U(\sigma )\\ \,+\,\displaystyle \frac{1}{2}({\partial }_{\mu }\sigma {\partial }^{\mu }\sigma -{m}_{s}^{2}{\sigma }^{2})-\displaystyle \frac{1}{4}{{\rm{\Omega }}}^{\mu \nu }{{\rm{\Omega }}}_{\mu \nu }\\ \,+\,\displaystyle \frac{1}{2}{m}_{v}^{2}{\omega }_{\mu }{\omega }^{\mu }+\displaystyle \frac{1}{2}{m}_{\rho }^{2}{\vec{\rho }}_{\mu }\cdot {\vec{\rho }}^{\mu }\\ \,-\,\displaystyle \frac{1}{4}{{\boldsymbol{P}}}^{\mu \nu }\cdot {{\boldsymbol{P}}}_{\mu \nu }+{{ \mathcal L }}_{\omega \rho }+{{ \mathcal L }}_{\phi },\end{array}\end{eqnarray*}$$

in natural units. ψ_B is the baryonic Dirac field, where B can stand either for nucleons only (N) or can run over nucleons (N) and hyperons (H). σ, ω_μ , and ${\vec{\rho }}_{\mu }$ are the mesonic fields, while $\vec{\tau }$ are the Pauli matrices. The g's are the Yukawa coupling constants that simulate the strong interaction, M_B is the baryon mass, m_s , m_v , and m_ρ are the masses of the σ, ω, and ρ mesons, respectively. The U(σ) is the self-interaction term introduced in Boguta & Bodmer (1977):

$$\begin{eqnarray*}&&U(\sigma )=\displaystyle \frac{1}{3}\lambda {\sigma }^{3}+\displaystyle \frac{1}{4}\kappa {\sigma }^{4},\end{eqnarray*}$$

and ${{ \mathcal L }}_{\omega \rho }$ is a nonlinear ω-ρ coupling interaction as described in Fattoyev et al. (2010):

$$\begin{eqnarray}&&\begin{array}{l}{{ \mathcal L }}_{\omega \rho }={{\rm{\Lambda }}}_{\omega \rho }\left({g}_{N\rho }^{2}\vec{{\rho }^{\mu }}\cdot \vec{{\rho }_{\mu }}\right)\left({g}_{N\omega }^{2}{\omega }^{\mu }{\omega }_{\mu }\right),\end{array}\end{eqnarray} \tag{ A1 }$$

which is necessary to correct the slope of the symmetry energy (L) and has a strong influence on the radii and tidal deformation of neutron stars (Cavagnoli et al. 2011; Dexheimer et al. 2019); ${{ \mathcal L }}_{\phi }$ is related to the strangeness hidden ϕ vector meson, which couples only with the hyperons (H), while not affecting the properties of symmetric nuclear matter:

$$\begin{eqnarray}&&\begin{array}{l}{{ \mathcal L }}_{\phi }={g}_{H\phi }{\bar{\psi }}_{H}\left({\gamma }^{\mu }{\phi }_{\mu }\right){\psi }_{H}\\ \,\,\,+\,\displaystyle \frac{1}{2}{m}_{\phi }^{2}{\phi }_{\mu }{\phi }^{\mu }-\displaystyle \frac{1}{4}{{\rm{\Phi }}}^{\mu \nu }{{\rm{\Phi }}}_{\mu \nu }.\end{array}\end{eqnarray} \tag{ A2 }$$

As pointed out in Lopes & Menezes (2020) and Weissenborn et al. (2012), this vector channel is crucial to obtaining massive hyperonic neutron stars.

As neutron stars are stable macroscopic objects, we need to describe a neutral, chemically stable matter, and hence, leptons are added as free Fermi gases, whose Lagrangian density is the usual one:

$$\begin{eqnarray*}&&\begin{array}{l}{{ \mathcal L }}_{l}={\bar{\psi }}_{l}\left[{\gamma }^{\mu }\left(i{\partial }_{\mu }-{m}_{l}\right)\right]{\psi }_{l}.\end{array}\end{eqnarray*}$$

To solve the equations of motion, we use the mean field approximation (MFA), where the meson fields are replaced by their expected values. Applying the Euler–Lagrange formalism, and using the quantization rules (E = ∂⁰, k = i∂^j ) we can easily obtain the eigenvalue for the energy

$$\begin{eqnarray}&&{E}_{B}=\sqrt{{k}^{2}+{M}_{B}^{* 2}}+{g}_{B\omega }{\omega }_{0}+{g}_{B\phi }{\phi }_{0}+\displaystyle \frac{{\tau }_{3B}}{2}{g}_{B\rho }{\rho }_{0},\end{eqnarray} \tag{ A3 }$$

where ${M}_{B}^{* }\,\equiv \,{M}_{B}-{g}_{B\sigma }{\sigma }_{0}$ is the effective baryon mass and τ_3B assumes the value of +1 for p, Σ⁺, and Ξ⁰; zero for Λ⁰ and Σ⁰; and -1 for n, Σ⁻ and Ξ⁻. For the leptons, we have

$$\begin{eqnarray}&&{E}_{l}=\sqrt{{k}^{2}+{m}_{l}^{2}},\end{eqnarray} \tag{ A4 }$$

and the mesonic fields in MFA are given by

$$\begin{eqnarray}\begin{array}{rcl}{m}_{\sigma }^{2}{\sigma }_{0}+\lambda {\sigma }_{0}^{2}+\kappa {\sigma }_{0}^{3} & = & \displaystyle \sum _{B}{g}_{B\sigma }{n}_{B}^{s},\\ ({m}_{\omega }^{2}+2{{\rm{\Lambda }}}_{v}{\rho }_{0}^{2})\omega & = & \displaystyle \sum _{B}{g}_{B\omega }{n}_{B},\\ {m}_{\phi }^{2}{\phi }_{0} & = & \displaystyle \sum _{B}{g}_{B\phi }{n}_{B},\\ ({m}_{\rho }^{2}+2{{\rm{\Lambda }}}_{v}{\omega }_{0}^{2}){\rho }_{0} & = & \sum {g}_{B\rho }\frac{{\tau }_{3B}}{2}{n}_{B},\end{array}\end{eqnarray} \tag{ A5 }$$

where ${{\rm{\Lambda }}}_{v}\,\equiv \,{{\rm{\Lambda }}}_{\omega \rho }{g}_{N\omega }^{2}{g}_{N\rho }^{2}$, and n_B ^s and n_B are, respectively, the scalar density and the number density of the baryon B. Finally, applying Fermi–Dirac statistics to baryons and leptons and with the help of Equation (A5), we can write the total energy density as (Miyatsu & Cheoun 2013)

$$\begin{eqnarray}&&\begin{array}{l}\epsilon =\displaystyle \sum _{B}\displaystyle \frac{1}{{\pi }^{2}}{\displaystyle \int }_{0}^{{k}_{{Bf}}}{{dkk}}^{2}\sqrt{{k}^{2}+{M}_{B}^{* 2}}+\displaystyle \frac{1}{2}{m}_{\sigma }^{2}{\sigma }_{0}^{2}\\ \,+\,\displaystyle \frac{1}{2}{m}_{\omega }^{2}{\omega }_{0}^{2}+\displaystyle \frac{1}{2}{m}_{\phi }^{2}{\phi }_{0}^{2}+\displaystyle \frac{1}{2}{m}_{\rho }^{2}{\rho }_{0}^{2}\\ \,+\,U({\sigma }_{0})+3{{\rm{\Lambda }}}_{v}{\omega }_{0}^{2}{\rho }_{0}^{2}+\displaystyle \sum _{l}\displaystyle \frac{1}{{\pi }^{2}}{\displaystyle \int }_{0}^{{k}_{{lf}}}{{dkk}}^{2}\sqrt{{k}^{2}+{m}_{l}^{2}},\end{array}\end{eqnarray} \tag{ A6 }$$

and the pressure is easily obtained by thermodynamic relations p = ∑_f μ_f n_f − , where the sum runs over all the fermions and μ_f is the corresponding chemical potential. Now, to determine each particle population, we impose that the matter is β stable and has a total electric net charge equal to zero.

Once we knew all the relevant quantum numbers (multiplet dimension, hypercharge, isospin, and isospin projection (Lopes & Menezes 2014)) for all the baryons and the vector mesons, we could use symmetry group arguments to calculate the relative strength of their coupling constants. For instance, all coupling constants for the vector mesons can be fixed simultaneously by applying SU(6) symmetry group(Weissenborn et al. 2012; Lopes & Menezes 2020). However, in general, SU(6) symmetry group produces neutron stars not as massive as desired in the present investigation. We can solve this problem by breaking the SU(6) symmetry group into a more general SU(3) flavor symmetry group. In this case, the coupling constant for the vector meson depends on a single parameter, α. Therefore we have for the ω meson (Miyatsu & Cheoun 2013; Lopes & Menezes 2014, 2021):

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{g}_{{\rm{\Lambda }}\omega }}{{g}_{N\omega }}=\displaystyle \frac{4+2\alpha }{5+4\alpha },\,\displaystyle \frac{{g}_{{\rm{\Sigma }}\omega }}{{g}_{N\omega }}=\displaystyle \frac{8-2\alpha }{5+4\alpha },\quad \displaystyle \frac{{g}_{{\rm{\Xi }}\omega }}{{g}_{N\omega }}=\displaystyle \frac{5-2\alpha }{5+4\alpha },\end{array}\end{eqnarray} \tag{ B1 }$$

for the ϕ meson:

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{g}_{{\rm{\Lambda }}\phi }}{{g}_{N\omega }}=\sqrt{2}\left(\displaystyle \frac{2\alpha -5}{5+4\alpha }\right),\,\displaystyle \frac{{g}_{{\rm{\Sigma }}\phi }}{{g}_{N\omega }}=-\sqrt{2}\left(\displaystyle \frac{2{\alpha }_{v}+1}{5+4\alpha }\right),\\ \,\displaystyle \frac{{g}_{{\rm{\Xi }}\phi }}{{g}_{N\omega }}=-\sqrt{2}\left(\displaystyle \frac{2\alpha +4}{5+4\alpha }\right),\,\displaystyle \frac{{g}_{N\phi }}{{g}_{N\omega }}=0,\end{array}\end{eqnarray} \tag{ B2 }$$

and finally for the ρ meson:

$$\begin{eqnarray}&&\displaystyle \frac{{g}_{{\rm{\Sigma }}\rho }}{{g}_{N\rho }}=2\alpha ,\quad \displaystyle \frac{{g}_{{\rm{\Xi }}\rho }}{{g}_{N\rho }}=2{\alpha }_{v}-1,\,\displaystyle \frac{{g}_{{\rm{\Lambda }}\rho }}{{g}_{N\rho }}=0.\end{eqnarray} \tag{ B3 }$$

We next choose some arbitrary values for α and calculate the vector meson coupling constants, while the scalar meson coupling constants are fixed in order to reproduce realistic values of the potential depth, as mentioned in the text: U_Λ = −28, U_Σ = = +30, and U_Ξ = −4 MeV, where the hyperon potential depth is defined as (Glendenning 2000)

$$\begin{eqnarray}&&{U}_{H}={g}_{H\omega }{\omega }_{0}-{g}_{H\sigma }{\sigma }_{0}.\end{eqnarray} \tag{ B4 }$$

To obtain the mass–radius relation, we use the EoS as an input to the TOV equations (Oppenheimer & Volkoff 1939):

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{dp}}{{dr}}=\displaystyle \frac{-{GM}(r)\epsilon (r)}{{r}^{2}}\left[1+\displaystyle \frac{p(r}{\epsilon (r)}\right]\left[1+\displaystyle \frac{4\pi p(r){r}^{3}}{M(r)}\right]\\ \,\times \,{\left[1-\displaystyle \frac{2{GM}(r)}{r}\right]}^{-1},\displaystyle \frac{{dM}}{{dr}}=4\pi {r}^{2}\epsilon (r).\end{array}\end{eqnarray} \tag{ C1 }$$

On the other hand, the dimensionless tidal deformability parameter Λ is defined as in Equation (4):

$$\begin{eqnarray}&&{\rm{\Lambda }}\,\equiv \,\displaystyle \frac{\lambda }{{M}^{5}}\,\equiv \,\displaystyle \frac{2{k}_{2}}{3{C}^{5}},\end{eqnarray} \tag{ C2 }$$

where M is the compact object mass and C = GM/R is its compactness. The Love number k₂ is defined as

$$\begin{eqnarray}&&\begin{array}{l}{k}_{2}=\displaystyle \frac{8{C}^{5}}{5}{\left(1-2C\right)}^{2}\left[2+2C({y}_{R}-1)-{y}_{R}\right]\\ \times \,\left\{2C\left[6-3{y}_{R}+3C(5{y}_{R}-8)\right]\right.\\ +\,4{C}^{3}[13-11{y}_{R}+C(3{y}_{R}-2)+2{C}^{2}(1+{y}_{R})]\\ {\left.+\,3{\left(1-2C\right)}^{2}\left[2-{y}_{R}+2C({y}_{R}-1))\right]\mathrm{ln}(1-2C)\right\}}^{-1},\end{array}\end{eqnarray} \tag{ C3 }$$

where y_R = y(r = R) and y(r) are obtained by solving

$$\begin{eqnarray}&&y\displaystyle \frac{{dy}}{{dr}}+{y}^{2}+{yF}(r)+{r}^{2}Q(r)=0.\end{eqnarray} \tag{ C4 }$$

Equation (C4) must be solved coupled with the TOV equations, Equation (C1). The coefficients F(r) and Q(r) are given by

$$\begin{eqnarray}&&F(r)=\displaystyle \frac{1-4\pi {{Gr}}^{2}[\epsilon (r)-p(r)]}{E(r)},\end{eqnarray} \tag{ C5 }$$

$$\begin{eqnarray}&&\begin{array}{l}Q(r)=\displaystyle \frac{4\pi G}{E(r)}\left[5\epsilon (r)+9p(r)+\displaystyle \frac{\epsilon (r)+p(r)}{\partial p/\partial \epsilon }-\displaystyle \frac{6}{4\pi {{Gr}}^{2}}\right]\\ \,-\,4{\left[\displaystyle \frac{G[M(r)+4\pi {r}^{3}p(r)]}{{r}^{2}E(r)}\right]}^{2},\end{array}\end{eqnarray} \tag{ C6 }$$

where E(r) = (1 − 2GM(r)/r).

These equations are valid if the EoS presents no discontinuity. When discontinuity is present (for instance, in the case of self-bounded strange stars), the value of y_R calculated in Equation (C4) must be corrected:

$$\begin{eqnarray}&&{y}_{R}\to {y}_{R}-\displaystyle \frac{4\pi {R}^{3}{\rm{\Delta }}{\epsilon }_{S}}{M},\end{eqnarray} \tag{ C7 }$$

where R and M are the star radius and mass, respectively, and Δ_S is the difference between the energy density at the surface (p = 0) and the exterior of the star (which implies = 0).

Further discussion of the theory of tidal deformability, as well as the tidal Love numbers are beyond the scope of this work and can be found in Lourenço et al. (2021), Abbott et al. (2018), Flores et al. (2020), Chatziioannou (2020), Abbott et al. (2017), Postnikov et al. (2010), and references therein.

The Lagrangian density of the vector MIT bag model reads as (Lopes et al. 2021, 2022)

$$\begin{eqnarray}&&\begin{array}{l}{{ \mathcal L }}_{v\mathrm{MIT}}=\{{\bar{\psi }}_{q}[{\gamma }^{\mu }(i{\partial }_{\mu }-{g}_{{qV}}{V}_{\mu })-{m}_{q}]{\psi }_{q}\\ \,+\,\displaystyle \frac{1}{2}{m}_{V}^{2}{V}_{\mu }{V}^{\mu }-B\}{\rm{\Theta }}({\bar{\psi }}_{q}{\psi }_{q}),\end{array}\end{eqnarray} \tag{ D1 }$$

where m_q is the mass of the quark q of flavor u, d, or s. Here, we follow Lopes et al. (2021) and use (m_u = m_d = 4 MeV, m_s = 95 MeV); ψ_q is the Dirac quark field, B is the constant vacuum pressure, and ${\rm{\Theta }}({\bar{\psi }}_{q}{\psi }_{q})$ is the Heaviside step function to assure that the quarks exist only confined to the bag. The quark interaction is mediated by the massive vector channel V_μ analogous to the ω meson in QHD (Serot 1992). Besides, leptons are added to account for β stable matter. Imposing MFA, and applying the Euler–Lagrange formalism to Equation (D1), we obtain the energy eigenvalue for the quark, as well as the expected value for the vector field.

$$\begin{eqnarray}&&E=\sqrt{{m}_{q}^{2}+{k}^{2}}+{g}_{{qV}}{V}_{0},\end{eqnarray} \tag{ D2 }$$

$$\begin{eqnarray}&&{m}_{V}^{2}{V}_{0}=\sum _{q}{g}_{{qV}}{n}_{q},\end{eqnarray} \tag{ D3 }$$

where n_q is the number density of the quark q. Now, applying Fermi–Dirac statistics, the energy density is analogous to the QHD plus the bag term:

$$\begin{eqnarray}&&\begin{array}{l}\epsilon =\displaystyle \sum _{q}\displaystyle \frac{{N}_{c}}{{\pi }^{2}}{\displaystyle \int }_{0}^{{k}_{f}}{{dkk}}^{2}\sqrt{{k}^{2}+{m}_{q}^{2}}+\displaystyle \frac{1}{2}{m}_{V}^{2}{V}_{0}^{2}\\ \,+\,B+\displaystyle \sum _{l}\displaystyle \frac{1}{{\pi }^{2}}{\displaystyle \int }_{0}^{{k}_{f}}{{dkk}}^{2}\sqrt{{k}^{2}+{m}_{l}^{2}},\end{array}\end{eqnarray} \tag{ D4 }$$

where N_c is the number of colors and p = μ n − .