The Absolute Stability and Mass Constraints of Strange Stars in the MIT Bag Model ^†

Abstract

The primary objective of this study is a comprehensive investigation of the self-bound properties of strange quark matter (SQM), which is hypothesized to represent the absolute ground state of superdense strongly interacting matter. An analysis is performed within the framework of the MIT bag model, including first-order perturbative QCD corrections and the finite strange quark mass. By systematically varying the vacuum pressure (bag constant, $B$) and the strong coupling constant (${\alpha }_c$) over a broad parameter space, while assuming a finite strange quark mass ($m_s\ne 0$), we explicitly compute the thermodynamic characteristics of the system including pressure, energy density, baryon number density, and the chemical potentials of quarks and charge-neutralizing electrons under conditions of $\beta$-equilibrium and global charge neutrality. Particular emphasis is placed on determining the minimum energy per baryon, which serves as the criterion for absolute stability. For parameter sets satisfying the self-binding condition, the integral properties of strange stars are derived via the numerical integration of the Tolman–Oppenheimer–Volkoff equations. The resulting mass–radius and mass–central density relations are analyzed, yielding the maximum stellar masses in the range $(1.9-2.4)M_{\odot }\hspace{0.17em}$. This study identifies the regions in the space of phenomenological parameters that allow for pure self-bound strange stars and demonstrates the sensitivity of stability and stellar properties to the underlying bag model parameters.

1. Introduction

The pioneering theoretical investigations by Landau [1], Baade and Zwicky [2], and Oppenheimer [3] into the structure of superdense stellar configurations initiated sustained interest in the properties of matter at densities exceeding nuclear saturation density. At sufficiently high densities, it is expected that hadronic matter undergoes deconfinement, leading to the liberation of quarks from baryonic confinement and the formation of quark matter [4,5].

Among the possible phases of deconfined matter, strange quark matter (SQM) composed of up (u), down (d), and strange (s) quarks in weak equilibrium has attracted particular attention. The hypothesis originally proposed by Bodmer [5] and later developed by Witten [4] suggests that, under certain conditions, three-flavor quark matter may possess a lower energy per baryon than the most stable atomic nucleus. In such a case, SQM could represent the absolute ground state of strongly interacting matter. If realized in nature, this would allow for the existence of self-bound compact configurations known as strange stars (SSs).

Observational advances in pulsar timing and gravitational-wave astronomy have significantly constrained the equation of state (EoS) of dense matter. Precisely measured heavy pulsars provide important benchmarks for theoretical models, including $PSR J1614-2230$, with $M/M_{⨀}=1.97\pm 0.04$ [6]; $PSR J0348+0432$, with $M/M_{⨀}=2.01\pm 0.04$ [7]; $PSR 0740+6620$, with $M/M_{⨀}=2.14$ [8]; $PSR J2215+5135$, with $M/M_{⨀}=2.27M_{⨀}$ [9]; and $PSR J0952-0607$, with $M/M_{⨀}=2.35\pm 0.17$ [10], as well as compact objects detected in binary mergers such as $\mathrm{G}\mathrm{W}190814$, with $M/M_{⨀}= 2.50-2.67$ [11]. Any viable EoS must be capable of supporting stellar masses of at least $2M_{⨀}$. To further constrain the model parameters, we compare the obtained mass–radius relations with observational limits from $PSR J0740+6620$ [8], $PSR J0030+0451$ [12], the $GW170817$ [13] merger event, and the compact object $HESS J1731-347$ [14]. The comparison shows that only a subset of parameter combinations ($B$ and ${\alpha }_c$) simultaneously satisfies both the maximum mass and radius constraints. These observational bounds, therefore, provide additional restrictions on the allowed MIT bag model parameter space.

The macroscopic properties of strange stars are determined entirely by the equation of state of SQM. Several theoretical frameworks have been developed to describe deconfined quark matter, including the Nambu–Jona-Lasinio (NJL) model [15], quark mass density-dependent models [16], and phenomenological approaches based on the MIT bag model [17,18]. Despite its simplicity, the MIT bag model remains a widely used and transparent framework for exploring the parametric dependence of SQM stability and stellar structure.

Within the bag model, the behavior of SQM is governed by phenomenological parameters, primarily the vacuum pressure (bag constant, $B$), the quark–gluon coupling constant ${\alpha }_c$, and the strange quark mass $m_s$ [19]. While the masses of the $u$ and $d$ quarks are non-zero, their small values lead only to minor quantitative corrections to macroscopic stellar properties. The realization of absolutely stable SQM depends sensitively on the interplay between ${B, \alpha }_c, \mathrm{a}\mathrm{n}\mathrm{d} m_s$.

Previous studies have examined the stability windows and stellar configurations arising within modified or extended bag model formulations [20,21,22,23,24,25,26,27]. In particular, in our recent work [28], stellar configurations compatible with heavy pulsars were investigated for selected parameter values. However, a comprehensive mapping of the multidimensional parameter space, including systematic variation in both $B$ and ${\alpha }_c$ with finite SQM, remains necessary to delineate the full domain of self-bound solutions.

The primary objective of the present work is therefore to identify and categorize the regions of the MIT bag model parameter space that allow for absolutely stable SQM, characterized by a negative binding energy per baryon, and to determine the corresponding integral properties of strange stars obtained from the numerical integration of the Tolman–Oppenheimer–Volkoff (TOV) equations. Particular attention is paid to the sensitivity of the stability condition and maximum stellar masses to variations in the phenomenological parameters. This approach enables a systematic assessment of the conditions under which pure self-bound strange stars may exist, rather than hybrid configurations involving phase coexistence [20,21]. In contrast to our previous study, which considered a density-dependent bag pressure $B\left(n_B\right)$, the present work focuses on the standard MIT bag model with constant vacuum pressure and systematically explores the dependence of SQM properties on the fundamental parameters ${B, \alpha }_c, \mathrm{a}\mathrm{n}\mathrm{d}{ m}_s$. This complementary approach allows us to isolate the role of perturbative QCD corrections and strange quark mass in determining the absolute stability window and the maximum masses of self-bound strange stars.

In the present study, we focus on the range $40\le B\le 70 \mathrm{M}\mathrm{e}\mathrm{V}/{\mathrm{f}\mathrm{m}}^3$, which corresponds to moderate binding energies and allows for a clearer assessment of the sensitivity of the EoS to variations in the bag constant $B$ and the strong coupling parameter ${\alpha }_c$. Smaller values of the bag constant ($B<40 \mathrm{M}\mathrm{e}\mathrm{V}/{\mathrm{f}\mathrm{m}}^3$) lead to very strongly bound SQM and an extremely stiff EoS, resulting in unusually large maximum masses of strange stars. Therefore, restricting the analysis to $B=40-70 \mathrm{M}\mathrm{e}\mathrm{V}/{\mathrm{f}\mathrm{m}}^3$ provides a more realistic parameter range for comparison with observational constraints on compact star properties.

2. Materials and Methods

In [22], the state of self-bound SQM was examined within the MIT bag model for specific values of its three phenomenological parameters, focusing on configurations for which the minimum energy per baryon satisfies the absolute stability criterion, ${\epsilon }_{min}\left(u,d,s\right)<0$. Consistent with the discussion in [4], it was demonstrated that the energy per baryon of three-flavor $(u,d,s$) matter, even in the approximation of vanishing quark masses, can be lower than that of non-strange $(u,d$) matter and the most tightly bound atomic nucleus ${Fe}_{56}$.

The resulting energy deficit, which remains of order $\left(50-80\right) \mathrm{M}\mathrm{e}\mathrm{V}$ even in the massless limit [19], supports the hypothesis that the introduction of the strange degree of freedom allows for a more efficient filling of Fermi levels due to the Pauli exclusion principle. Consequently, under appropriate conditions, the inequalities $\epsilon \left(u,d,s\right)<\epsilon (u,d)$ and $\epsilon \left(u,d,s\right)<\epsilon \left({Fe}_{56}\right)=930 \mathrm{M}\mathrm{e}\mathrm{V}$ may be fulfilled, indicating the possible absolute stability of SQM. Interest in the ground state of SQM has led to several extensions of the original MIT bag model. In the original formulation [1], quarks are treated as a non-interacting Fermi gas confined by a constant vacuum pressure (bag constant), and absolute stability is achieved when the energy per baryon satisfies $\epsilon \left(u,d,s\right)<\epsilon \left({Fe}_{56}\right)=930 \mathrm{M}\mathrm{e}\mathrm{V}$. Later, perturbative QCD corrections were introduced through the strong coupling parameter ${\alpha }_c$ [2], accounting for quark–gluon interactions in first-order approximation while keeping the bag constant fixed. Subsequent studies [23,24,25] considered a density-dependent bag parameter $B\left(n_B\right)$, often using a Gaussian parametrization, which modifies the stiffness of the equation of state and alters the SQM stability window. In addition, repulsive vector interactions between quarks were introduced in Refs. [26,27], leading to an additional contribution characterized by the coupling constant $G_V$. Such interactions stiffen the equation of state and allow for the description of massive strange stars with $M_{max}>2.1M_{⨀}$. The combined effects of strong interactions and density-dependent vacuum pressure were also explored in Ref. [25], where maintaining thermodynamic consistency is essential. These different approaches demonstrate that both interaction effects and the behavior of the bag parameter significantly influence the absolute stability of SQM and the corresponding mass constraints of strange stars.

The renewed interest in SQM is further motivated by developments of the vector interaction MIT (vMIT) model, in which quark interactions are mediated by a repulsive vector field [26,27]. Within both the standard MIT framework [23,24] and its vector extensions [25], including formulations with density-dependent vacuum pressure, the characterization of SQM remains fundamentally based on the absolute stability hypothesis, defined through the minimization of the energy per baryon.

To perform a systematic parametric investigation, we consider discrete values of the vacuum pressure $B=40, 45, 50, 55, 60, 65, 70 \mathrm{M}\mathrm{e}\mathrm{V}/{\mathrm{f}\mathrm{m}}^3$ and include perturbative QCD corrections to the first order in the strong coupling constant, ${\alpha }_c=0.1, 0.2, 0.3, 0.4, 0.5, 0.6$. The constituent quark masses are taken as $m_u=3 \mathrm{M}\mathrm{e}\mathrm{V}$, $m_d=5.3 \mathrm{M}\mathrm{e}\mathrm{V}$, and $m_s=95 \mathrm{M}\mathrm{e}\mathrm{V}$, in accordance with phenomenological constraints [19].

Restricting the expansion to the first order in ${\alpha }_c$, the thermodynamic potential density for each quark flavor $f=u,d,s$ is written in the form derived in [22]:

$$\begin{gathered} {\mathsf{\Omega }}_f\left({\mu }_f\right)=-{\displaystyle \frac{1}{{4\pi }^2({\mathrm{\hslash }c)}^3}}({\mu }_f({\mu }_f^2-m_f^2{c}^4{)}^{1/2}({\mu }_f^2-{\displaystyle \frac{5}{2}}{m}_f^2{c}^4)+ \\ {\displaystyle \frac{3}{2}}{m}_f^4{c}^8{\displaystyle \ln (\frac{{\mu }_f+{\left({\mu }_f^2-m_f^2{c}^4\right)}^{\frac{1}{2}}}{m_f{c}^2})}-{\displaystyle \frac{2{\alpha }_c}{\pi }}\left(3{\left({\mu }_f{\left({\mu }_f^2-m_f^2{c}^4\right)}^{{\displaystyle \frac{1}{2}}}- \\ {m}_f^2{c}^4\ln \left({{\displaystyle \frac{{\mu }_f+\left({\mu }_f^2-m_f^2{c}^4\right)}{{\mu }_f}}}^{{\displaystyle \frac{1}{2}}}\right)\right)}^2-{2\left({\mu }_f^2-m_f^2{c}^4\right)}^2- 3m_f^4{c}^8{ln}^2\left({\displaystyle \frac{m_f{c}^2}{{\mu }_f}}\right)+ \\ 6\mathrm{l}\mathrm{n}\left({\displaystyle \frac{\tilde{\rho }}{{\mu }_f}}\right)\left({{\mu }_f{m}_f^2{c}^4\left({\mu }_f^2-m_f^2{c}^4\right)}^{{\displaystyle \frac{1}{2}}}-m_f^4{c}^8\ln ({\displaystyle \frac{{\mu }_f+{\left({\mu }_f^2-m_f^2{c}^4\right)}^{\frac{1}{2}}}{m_f{c}^2}})\right)\right), \end{gathered}$$

(1)

where $\tilde{\rho }=m_N{c}^2/3=313 \mathrm{M}\mathrm{e}\mathrm{V}$ represents the renormalization scale introduced in the perturbative treatment.

For electrons, they are treated as an ultra-relativistic free Fermi gas

$${\mathsf{\Omega }}_e{\displaystyle \left({\mu }_e\right)}=-\frac{{\mu }_{e^4}}{{12\pi }^2{\left(\mathrm{\hslash }c\right)}^3}.$$

(2)

The particle number densities follow from $n_f=-\partial {\mathsf{\Omega }}_f/\partial {\mu }_f, n_e=-\partial {\mathsf{\Omega }}_e/\partial {\mu }_e$. Chemical equilibrium under weak interactions ($\beta$-equilibrium) requires ${\mu }_d={\mu }_u+{\mu }_e$ and ${\mu }_d={\mu }_s\equiv \mu$, while global charge neutrality is imposed through $2n_u-n_d-n_s=3n_e$.

Using a numerical C⁺⁺ framework and adopting the up quark chemical potential ${\mu }_u$ as the independent variable, the thermodynamic potentials are evaluated for each selected parameter set. For a given bag constant $B$, the total pressure, energy density, baryon number density, and baryon chemical potential are calculated as $P\left(\mu \right)=-\sum _i{\mathsf{\Omega }}_i\left(\mu \right)-B$, $=\sum \left({\mathsf{\Omega }}_i+{\mu }_i{n}_i\right)+B$ $n=(n_u+n_d+n_s)/3$, and ${\mu }_Q\left(P\right)=(\rho \left(P\right)c^2+P)/n\left(P\right)$, respectively.

At zero pressure, the baryon chemical potential equals the energy per baryon, ${\mu }_Q\left(P=0\right)=(\rho \left(P\right)c^2+P)/n\left(P\right)$. Defining $\epsilon =\rho c^2/n-930 \mathrm{M}\mathrm{e}\mathrm{V}$, the self-binding condition is satisfied if ${\epsilon }_{min}<0$, meaning that the minimum energy per baryon in SQM is lower than that of ${Fe}_{56}$ (approximately $930 \mathrm{M}\mathrm{e}\mathrm{V}$). The aim of this study is to numerically determine the regions of the multidimensional parameter space (${B, a}_c, m_s$) that satisfy this absolute stability criterion.

3. Results and Discussion

For the selected values of the vacuum pressure and quark–gluon coupling constant, calculations were previously performed in [28] for ${\alpha }_c=0.01, 0.05$ and strange quark masses $m_s=95 \mathrm{M}\mathrm{e}\mathrm{V},150 \mathrm{M}\mathrm{e}\mathrm{V}$, treating the up and down quark masses as zero. In these cases, the minimum energy per baryon satisfied the self-binding condition (${\epsilon }_{min}<0$), and the resulting strange star configurations exhibited maximum masses comparable to those of observed pulsars.

In the present work, we extend the analysis to a broader parameter space defined by ${\alpha }_c=0.1, 0.2, 0.3, 0.4, 0.5, 0.6$ and $B=40,45,50,55,60,65,70 \mathrm{M}\mathrm{e}\mathrm{V}/{\mathrm{f}\mathrm{m}}^3$ for a fixed strange quark mass $m_s=95 \mathrm{M}\mathrm{e}\mathrm{V}$. The resulting integral parameters of the corresponding strange star configurations, obtained from the numerical integration of the TOV equations, are summarized in Table 1. The calculated quantities include:

Table 1. Integral parameters of SQM EoS and SS configurations, when $m_s=95 \mathrm{M}\mathrm{e}\mathrm{V}$.

$\mathit{B}, \mathbf{M}\mathbf{e}\mathbf{V}/{\mathbf{f}\mathbf{m}}^3$	${\mathit{\epsilon }}_{\mathit{m}\mathit{i}\mathit{n}}$	${\mathit{n}}_{\mathit{m}\mathit{i}\mathit{n}}/{\mathit{n}}_0$	${\mathit{M}}_{\mathit{m}\mathit{a}\mathit{x}}/{\mathit{M}}_{⨀}$	$\mathit{R}, \mathbf{k}\mathbf{m}$	${\mathit{Z}}_{\mathit{s}}$
${\alpha }_c=0.1$
40	−138.6	1.424	2.304	12.76	0.462
45	−116.2	1.554	2.178	11.94	0.471
50	−95.59	1.679	2.069	11.44	0.463
55	−76.44	1.802	1.978	10.85	0.471
60	−58.54	1.923	1.896	10.47	0.465
65	−41.72	2.04	1.825	10.01	0.470
70	−25.83	2.155	1.761	9.603	0.475
${\alpha }_c=0.2$
40	−125.3	1.403	2.303	12.75	0.463
45	−102.6	1.530	2.178	11.93	0.471
50	−81.65	1.654	2.069	11.44	0.464
55	−62.21	1.775	1.979	10.85	0.471
60	−44.03	1.893	1.897	10.33	0.476
65	−26.94	2.008	1.825	10.01	0.471
70	−10.80	2.155	1.756	9.499	0.482
${\mathsf{\alpha }}_{\mathrm{c}}=0.3$
40	−110.8	1.380	2.303	12.74	0.463
45	−87.76	1.506	2.178	11.93	0.471
50	−66.47	1.627	2.071	11.44	0.464
55	−46.71	1.746	1.979	10.84	0.471
60	−28.22	1.862	1.897	10.33	0.477
65	−10.84	1.975	1.827	10.01	0.471
70	5.572	2.086	1.762	9.604	0.476
${\alpha }_c=0.4$
40	−95.02	1.356	2.303	12.74	0.463
45	−71.50	1.479	2.172	11.73	0.484
50	−49.83	1.598	2.071	11.43	0.465
55	−29.71	1.715	1.976	10.69	0.482
60	−10.89	1.828	1.898	10.33	0.477
65	6.805	1.939	1.828	10.01	0.472
70	23.52	2.048	1.763	9.61	0.476
${\alpha }_c=0.5$
40	−77.52	1.331	2.303	12.73	0.463
45	−53.55	1.451	2.181	11.93	0.454
50	−31.46	1.567	2.072	11.43	0.465
55	−10.94	1.682	1.982	10.85	0.472
60	8.251	1.793	1.899	10.34	0.477
65	26.30	1.901	1.829	10.01	0.459
70	43.36	2.007	1.765	9.611	0.477
${\alpha }_c=0.6$
40	−58.04	1.304	2.304	12.73	0.464
45	−33.56	1.421	2.181	11.93	0.473
50	−10.99	1.535	2.074	11.26	0.479
55	9.970	1.646	1.983	10.85	0.473
60	26.58	1.754	1.902	10.34	0.478
65	48.03	1.860	1.831	10.02	0.473
70	65.47	1.964	1.766	9.725	0.468

• The minimum energy per baryon ${\epsilon }_{min}$.
• The minimum baryon number density $n_{min}$.
• The maximum stellar mass $M_{max}$.
• The corresponding radius $R$.
• The gravitational surface redshift $Z_s$.

Figure 1 shows the dependence of the energy per baryon on pressure for representative values of $B$ at fixed ${\alpha }_c=0.1$. The calculations show that self-bound SQM is not realized for all parameter combinations. The condition ${\epsilon }_{min}<0$ depends sensitively on both $B$ and ${\mathsf{\alpha }}_{\mathrm{c}}$. For small coupling constants (${\mathsf{\alpha }}_{\mathrm{c}}=0.1$), negative values of ${\epsilon }_{min}$ are obtained for the entire considered range of $B$ (see Figure 1).

Figure 1. Energy per baryon as a function of pressure for $B=40, 50, 60, 70 \mathrm{M}\mathrm{e}\mathrm{V}/{\mathrm{f}\mathrm{m}}^3$ and ${\alpha }_c=0.1$ with $m_s=95 \mathrm{M}\mathrm{e}\mathrm{V}$.

For example,

At $B=70 \mathrm{M}\mathrm{e}\mathrm{V}/{\mathrm{f}\mathrm{m}}^3$: ${\epsilon }_{min}=-25.83 \mathrm{M}\mathrm{e}\mathrm{V}$, $n_{min}/n_0=2.155$, $M_{max}/M_{⨀}=1.761$, and $R=9.603 \mathrm{k}\mathrm{m}$.

At $B=40 \mathrm{M}\mathrm{e}\mathrm{V}/{\mathrm{f}\mathrm{m}}^3$: ${\epsilon }_{min}=-138.6 \mathrm{M}\mathrm{e}\mathrm{V}$, $n_{min}/n_0=1.424$, $M_{max}/M_{⨀}=2.304$, and $R=12.76 \mathrm{k}\mathrm{m}$.

Thus, decreasing $B$ leads to a stronger binding energy, lower minimum baryon concentration, and larger maximum mass and radius, as illustrated in Figure 2.

Figure 2. The maximum stellar mass $M_{max}$ as a function of central density ${\rho }_c$ for $B=40, 50, 60, 70 \mathrm{M}\mathrm{e}\mathrm{V}/{\mathrm{f}\mathrm{m}}^3$, calculated for $m_s=95 \mathrm{M}\mathrm{e}\mathrm{V}$ and ${\alpha }_c=0.1$.

To further constrain the model parameters, we compare the resulting mass–radius relations with observational limits from $PSR J0740+6620$, $PSR J0030+0451$, and the $GW170817$ binary neutron star merger. The corresponding confidence regions are shown individually in Figure 3a. For $PSR J0740+6620$, the inferred mass and radius are $M=2.14_{-0.09}^{+0.10}{M}_{\odot }$ and $R=12.39_{-0.98}^{+1.30}$ km [29]. For $PSR J0030+0451$, the measurements yield $M=1.44_{-0.14}^{+0.15}{M}_{\odot }$ and $R=13.02_{-1.06}^{+1.24}$ km [12]. The $GW170817$ binary neutron star merger constrains the total mass to $M\approx \left(2.73\text{–}2.74\right)M_{\odot }$ and suggests typical stellar radii in the range $R\approx 11\text{–}11.9$ km [13]. These observational limits provide simultaneous constraints on both mass and radius.

Figure 3. (a) Mass–radius relations for strange quark stars with $m_s=95 \mathrm{M}\mathrm{e}\mathrm{V}$, $B=40, 50, 60, 70 \mathrm{M}\mathrm{e}\mathrm{V}/{\mathrm{f}\mathrm{m}}^3$, and ${\alpha }_c=0.1$. The shaded regions indicate the observational mass–radius constraints for light (light gray) and heavy (dark gray) compact stars [6,7,8,9,10,11,12,13]. (b) Dimensionless tidal deformability as a function of stellar mass for the same parameter sets. The vertical dashed line marks the reference mass $M/M_{⨀}=1.4 (\mathrm{gray} \mathrm{region} \mathrm{is} \mathrm{for} 70<{\mathsf{\Lambda }}_{1.4}<580)$. The inset shows the dependence of ${\mathsf{\Lambda }}_{1.4}$ on the bag constant $B$.

The comparison shown in Figure 3a indicates that only a subset of parameter combinations ($B$ and ${\alpha }_c$) satisfies these constraints simultaneously. In particular, parameter sets with $B=40-70 \mathrm{M}\mathrm{e}\mathrm{V}/{\mathrm{f}\mathrm{m}}^3$ and ${\alpha }_c=0.1, 0.2$ are consistent with the observational mass–radius regions, whereas the case $B=40 \mathrm{M}\mathrm{e}\mathrm{V}/{\mathrm{f}\mathrm{m}}^3$ predicts larger radii and is consistent with the preferred constraints. These astrophysical bounds therefore provide additional restrictions on the allowed MIT bag model parameter space.

The compact object $HESS J1731-347$ has been proposed as a low-mass candidate ($M\approx 0.77M_{\odot }$), which lies significantly below the typical mass range of stable strange star configurations obtained within the present parameter space. For this reason, it is not included in Figure 3a, which focuses on the mass range relevant to the TOV solutions considered here. The existence of such a low-mass object may indicate either a different formation channel or the need to extend the model, such as by including density-dependent effects or additional interactions, which are beyond the scope of the present study.

In addition to the mass–radius constraints, we also calculated the dimensionless tidal deformability for the same parameter sets shown in Figure 3a. The results are presented in Figure 3b, where $\mathsf{\Lambda }$ is plotted as a function of stellar mass for $B=40, 50, 60, 70 \mathrm{M}\mathrm{e}\mathrm{V}/{\mathrm{f}\mathrm{m}}^3$, with $m_s=95 \mathrm{M}\mathrm{e}\mathrm{V} \mathrm{a}\mathrm{n}\mathrm{d} {\alpha }_c=0.1$. The observational constraint derived from the $GW170817$ event, ${\mathsf{\Lambda }}_{1.4}={190}_{-120}^{+390}$ (corresponding approximately to ${\mathsf{\Lambda }}_{1.4}<580$), is shown as a shaded region in Figure 3b [30]. For the parameter sets $B=50, 60, 70 \mathrm{M}\mathrm{e}\mathrm{V}/{\mathrm{f}\mathrm{m}}^3$, the obtained values are ${\mathsf{\Lambda }}_{1.4}=477, 350, 213$, respectively, which satisfy the observational bound. The corresponding radii are $R_{1.4}=11.39, 10.54, 9.93 \mathrm{k}\mathrm{m}$. In contrast, for $B=40 \mathrm{M}\mathrm{e}\mathrm{V}/{\mathrm{f}\mathrm{m}}^3$, the calculated value ${\mathsf{\Lambda }}_{1.4}=840$ exceeds the allowed limit, with a corresponding radius $R_{1.4}=12.38 \mathrm{k}\mathrm{m}$. Therefore, this parameter set is disfavored by the tidal deformability constraint. The remaining parameter sets satisfy the tidal deformability bound and remain consistent with the mass–radius constraints shown in Figure 3a. These results show that the inclusion of tidal deformability further restricts the allowed parameter space of the MIT bag model. In particular, the range $B=50-70 \mathrm{M}\mathrm{e}\mathrm{V}/{\mathrm{f}\mathrm{m}}^3$ ($m_s=95 \mathrm{MeV}\mathrm{and} {\alpha }_c=0.1$) simultaneously satisfies both the mass–radius constraints shown in Figure 3a and the tidal deformability bound derived from $GW170817$.

However, as the coupling constant increases, the stability window narrows. This behavior is clearly evident in Figure 4, where the minimum energy per baryon is plotted as a function of the vacuum pressure for different values of ${\alpha }_c$.

Figure 4. The minimum energy per baryon as a function of vacuum pressure $B$ for ${\alpha }_c=0.1, 0.2, 0.3, 0.4, 0.5, 0.6$ and $m_s=95 \mathrm{M}\mathrm{e}\mathrm{V}$. The horizontal dashed line ${\epsilon }_{min}=0$ separates self-bound (${\epsilon }_{min }<0$) and non-self-bound (${\epsilon }_{min }>0$) configurations. The right vertical axis shows the corresponding dependence of the maximum stellar mass on $B$.

For example,

For ${\alpha }_c=0.6:$ $-58.04 \mathrm{M}\mathrm{e}\mathrm{V}\le {\epsilon }_{min }\le 65.47 \mathrm{M}\mathrm{e}\mathrm{V}$ within $40 \mathrm{M}\mathrm{e}\mathrm{V}/{\mathrm{f}\mathrm{m}}^3 \le \mathrm{B}\le 70 \mathrm{M}\mathrm{e}\mathrm{V}/{\mathrm{f}\mathrm{m}}^3$, and ${\epsilon }_{min }>0$ for $B\ge 55 \mathrm{M}\mathrm{e}\mathrm{V}/{\mathrm{f}\mathrm{m}}^3$.

For ${\alpha }_c=0.5, 0.4, 0.3$: ${\epsilon }_{min}$ becomes positive at $B=60, 65, 70 \mathrm{M}\mathrm{e}\mathrm{V}/{\mathrm{f}\mathrm{m}}^3$, respectively.

In contrast, for ${\alpha }_c=0.2$, self-boundness is maintained for all considered values of $B$.

The mass curves for different values of ${\alpha }_c=0.1-0.6$ nearly coincide over the considered parameter range, indicating a weak sensitivity of $M_{max}/M_{⨀}$ to ${\alpha }_c$ compared to its stronger dependence on $B$ (see Figure 4).

These results indicate that both parameters act in opposite directions: (i) increasing $B$ increases the vacuum energy contribution, reducing stability, and (ii) increasing ${\alpha }_c$ enhances perturbative interaction effects, also increasing the energy per baryon. Therefore, only a restricted region of the (${B, \alpha }_c$) parameter space allows for absolutely stable SQM.

The maximum stellar masses obtained in the explored parameter space lie within ${1.761\le M}_{max}/M_{⨀}\le 2.305$, with corresponding central densities in the range $1.38\times {10}^{15} \mathrm{g}/{\mathrm{c}\mathrm{m}}^3\le {\rho }_c\le 2.67\times {10}^{15} \mathrm{g}/{\mathrm{c}\mathrm{m}}^3$, and $40 \mathrm{M}\mathrm{e}\mathrm{V}/{\mathrm{f}\mathrm{m}}^3\le B\le 70 \mathrm{M}\mathrm{e}\mathrm{V}/{\mathrm{f}\mathrm{m}}^3$.

For ${\alpha }_c=0.2$ and $40 \mathrm{M}\mathrm{e}\mathrm{V}/{\mathrm{f}\mathrm{m}}^3\le B\le 70 \mathrm{M}\mathrm{e}\mathrm{V}/{\mathrm{f}\mathrm{m}}^3$, ${1.756\le M}_{max}/M_{⨀}\le 2.303$, and $9.499 \mathrm{k}\mathrm{m}\le R\le 12.75 \mathrm{k}\mathrm{m}$.

A clear correlation is observed: as $B$ decreases, both the maximum mass and radius increase (Figure 2 and Figure 3). This reflects the reduced effective confinement and the corresponding stiffening of the equation of state at high densities.

The shaded region in Figure 3 corresponds to the observational mass interval $M=(1.97-2.4)M_{⨀}$, associated with heavy pulsars [6,7,8,9,10,11,12,13,14]. Only a subset of the parameter space satisfies both the self-binding condition and the observational lower bound of approximately $2M_{⨀}$.

For larger values of (${\alpha }_c>0.6$), the condition ${\epsilon }_{min}>0$ is realized over most of the parameter space (Figure 4). In such cases, SQM cannot exist as a self-bound phase and would instead appear in equilibrium with hadronic matter, leading to hybrid star configurations with a quark core, consistent with the Maxwell construction scenario [20,21].

The presented results are obtained within the first-order perturbative approximation $O\left({\alpha }_c\right)$. Extension to higher orders, $O\left({\alpha }_c^2\right)$ [31], and the inclusion of repulsive vector interactions (vMIT model) [23,24] may further modify the stability window and generally increase the maximum supported masses. Therefore, the explored range of vacuum pressure values remains relevant for future refinements of the model.

4. Conclusions

In this work, the equations of state of SQM were investigated within the framework of the MIT bag model. The energy per baryon, minimum baryon concentration, and chemical potentials of quarks were calculated over a wide range of phenomenological parameters $B$ and ${\mathsf{\alpha }}_{\mathrm{c}}$, for a fixed strange quark mass $m_s=95 \mathrm{M}\mathrm{e}\mathrm{V}$. The integral parameters of the corresponding stellar configurations, maximum masses, radii, and central densities, were obtained through the numerical integration of the TOV equations.

It is noteworthy that the considered ranges of the parameters $B$ and ${\mathsf{\alpha }}_{\mathrm{c}}$ predominantly yield self-bound strange quark matter characterized by a negative minimum energy per baryon. The resulting maximum stellar masses lie in the range ${1.756\le M}_{max}/M_{⨀}\le 2.31$, which is comparable to the accurately measured masses of heavy pulsars, including:

• $PSRJ0952-0607$($M=(2.35\pm 0.17)M_{⨀}$).
• $PSRJ2215+5135$ $(M=2.27M_{⨀}$).
• $PSRJ0740+6620 (M={2.14 \pm }_{0.09}^{0.10}{M}_{⨀}$).
• $PSRJ0348+0432$ ($M=(2.01\pm 0.04$)$M_{⨀}$).
• The companion of $R J0514-4002\mathrm{E}$ $(M=(2.09-2.71)M_{⨀}$).
• $PSRJ0030+0451$ ($M={1.44 \pm }_{0.14}^{0.15}{M}_{⨀}$).
• Compact object $HESSJ1731-347$ ($M=0.77{ \pm }_{0.17}^{0.20}{M}_{⨀}$).
• Binary merger event $GW170817$ ($M={2.74 \pm }_{0.01}^{0.04}{M}_{⨀}$).

These results demonstrate that, within the first-order perturbative approximation $O\left({\alpha }_c\right)$, the MIT bag model can produce equations of state compatible with the observed mass range of compact stars.

From this perspective, it is natural to consider extensions of the present analysis to higher-order perturbative corrections $O\left({\alpha }_c^2\right)$ and to formulations incorporating repulsive vector interactions (vMIT model). Such refinements may further modify the stability window and increase the maximum supported masses. The explored ranges of the phenomenological parameters, therefore, provide a meaningful basis for future investigations of dense quark matter in compact stars.