Deconfinement Confined to the Heaviest Neutron Stars: Strange Quark Matter from a Flavor-Dressed Contact Interaction

Introduction

The equation of state (EoS) of strongly interacting matter at several times the nuclear saturation density $n_{0} \simeq0.16\,\mathrm{fm}^{-3}$ remains one of the central open problems of nuclear and particle physics. It controls the structure of neutron stars, the dynamics of core-collapse supernovae, and the waveform of gravitational radiation emitted by merging compact binaries [1, 2]. At these densities perturbative QCD does not apply, and lattice QCD, which establishes a chiral crossover at $T_{pc} \approx155\,\mathrm{MeV}$ for vanishing baryon chemical potential [3, 4], runs into the fermion sign problem as soon as the real baryon chemical potential becomes large [5]. Cold and dense matter is therefore described with effective models, and the observable properties of neutron stars provide the most direct test of those descriptions.

The observational situation has changed considerably over the last decade. Pulsar timing has established masses close to or above $(2M_{\odot})$ for PSR J1614−2230 [6, 7], PSR J0348+0432 [8], and PSR J0740+6620, whose mass is $2.08 \pm0.07\,M_{\odot}$ [9]. The NICER mission has added joint mass and radius estimates for PSR J0030+0451 and PSR J0740+6620 [10, 11, 12], with central radius estimates between about 12.4 and 13.7 km. Analyses of the larger NICER data set now available for PSR J0740+6620 give $12.49^{+1.28}_{-0.88}$ km [13] and $12.92^{+2.09}_{-1.13}$ km [14], consistent with the earlier estimates. The gravitational-wave event GW170817 constrains the tidal deformability of a $1.4\,M_{\odot}$ star to $\Lambda_{1.4} = 190^{+390}_{-120}$ at 90% credibility [15]. Taken together, these data demand an EoS that is stiff enough to support $2M_{\odot}$ yet soft enough at intermediate densities to keep $\Lambda_{1.4}$ below about 580. Whether quark matter plays any role in meeting both requirements is still an open question [16, 17].

Continuum approaches to QCD offer a way to model quark matter while keeping contact with the dynamics of dynamical chiral symmetry breaking. Within the framework of Schwinger–Dyson equations [18, 19], the contact interaction (CI) replaces the momentum-dependent gluon exchange by a symmetry-preserving, ultra-local kernel regularized with a proper-time scheme. Despite its simplicity, the CI reproduces a broad set of light and strange hadron properties once its infrared coupling and regulators are fixed in vacuum [20, 21, 22, 23], and it has been applied to the QCD phase diagram at finite temperature, chemical potential, and external fields [24, 25, 26, 27, 28]. A refinement relevant for the present work is the flavor-dressed CI, in which the effective coupling decreases with the number of active flavors $N_{f}$ so as to reproduce the suppression of chiral symmetry breaking expected near the conformal window [29, 30]. For two light flavors this dressing reduces to the identity. With a strange quark present it does not, and its effect on dense matter can be quantified directly.

After the incorporation of the $s$-quark into the game, two questions follow naturally. The first is whether strange quark matter described by the CI can form self-bound stars or support the observed maximum masses on its own. The Bodmer–Witten hypothesis allows, in principle, for three-flavor quark matter to be the absolute ground state of strong interactions [31, 32], and effective models with vector repulsion have been used to build $2M_{\odot}$ quark stars [33, 34]. The second question concerns hybrid stars, in which a quark core forms inside a hadronic envelope. Here the location of the deconfinement transition relative to the central density of the heaviest stable configuration decides everything, and models of the Nambu–Jona-Lasinio type have often found that quark matter either appears only near the maximum mass or destabilizes the star as soon as it appears [16, 35, 36, 37].

In this paper we address both questions with a CI that includes the $u$, $d$, and $s$ quarks, repulsive vector interactions of strength $G_V = \eta G_S$, electric charge neutrality, and $\beta$ equilibrium with electrons and muons. All vacuum parameters are fixed beforehand; with a current strange mass $m_s = 170$ MeV the model yields a dressed strange mass $M_s = 533$ MeV, in line with earlier CI studies of strange hadrons [23]. We scan $\eta$ between 0 and 3/2 and compare the results with and without flavor dressing. For hybrid stars we describe the hadronic phase with a relativistic mean-field model fitted to the GM1 saturation properties [38], join it to the quark phase through both a Maxwell and a Gibbs construction [39], and allow for a deconfinement bag shift $B_{\mathrm{dc}}$ in the spirit of the vBag model [33]. Requiring that the pressure remain continuous in the baryon chemical potential sets an upper bound on $B_{\mathrm{dc}}$ for each value of $\eta$, which removes most of the freedom that such a shift would otherwise introduce.

The outcome is unambiguous within this setup. Pure CI quark matter is not self-bound, and no value of $\eta$ reconciles $2M_{\odot}$ with the tidal constraint from GW170817. When a hadronic envelope is added, the quark phase appears inside stable stars only for $\eta\lesssim1/2$, and never below about $4n_0$. Stable quark cores then exist in stars between $2.2$ and $2.35M_{\odot}$, over a mass window never wider than a few hundredths of a solar mass. Stars of intermediate mass remain purely hadronic, so their radii and tidal deformabilities carry no imprint of the quark phase. The vector strength that pure quark stars need to reach $2M_{\odot}$ is precisely the one that prevents deconfinement inside stable hybrid stars. For the presentation of ideas, we organized the remaining of the paper as follows. § 2 presents the flavor-dressed CI with three flavors and its thermodynamic potential. § 3 constructs neutral quark matter in $\beta$ equilibrium, and § 4 discusses pure quark stars. The hadronic model and the phase transition are introduced in § 5, while § 6 presents hybrid stars and confronts them with observations. Our conclusions are drawn in § 7, and the numerical checks are collected in Appendix § 8.

Flavor-dressed contact interaction with three flavors

Gap equation and flavor dressing

In Euclidean space the dressed propagator of a quark of flavor $f$ obeys the Schwinger–Dyson equation

where $m_f$ is the current mass, $4/3$ is the color factor for $N_c = 3$, $D_{\mu\nu}$ is the dressed gluon propagator, and $\Gamma_\nu$ is the dressed quark-gluon vertex [18, 19]. The contact interaction keeps the rainbow vertex, $\Gamma_\nu\to\gamma_\nu$, and replaces the gluon exchange by its infrared limit, in which the gluon carries a dynamically generated mass $m_g$ [20, 22, 40],

We use $\alpha_{\rm ir} = 0.93\pi$ and $m_g = 0.8\ \mathrm{GeV}$, the values fixed by vacuum meson phenomenology [22, 23], which give $\alpha_{\rm eff} = 57.4\ \mathrm{GeV}^{-2}$. With a momentum-independent kernel the propagator takes the form $S_f^{-1} = i\gamma\cdot p + M_f$, and the dressed mass $M_f$ is a constant determined by

The effect of additional light flavors on dynamical chiral symmetry breaking is modeled through a flavor-dependent coupling. Following Refs. [27, 29, 30] we write

with $\varsigma= 2.9$ chosen so that chiral symmetry is restored at $N_f^c \approx8$ in the chiral limit. The coupling equals $\alpha_{\rm eff}$ for $N_f = 2$. For the three-flavor matter studied here it is reduced by a factor 0.953. Since the vacuum parameters were calibrated with two flavors, this reduction changes the vacuum masses, and we therefore present every result both with the dressing ($N_f = 3$ in Eq. (4)) and without it ($\alpha_{\rm eff}$ unchanged). The two sets of results bracket the uncertainty associated with this choice.

Regularization and vacuum

The integral in Eq. (3) is regularized with the proper-time scheme of Refs. [20, 22],

with $\tau_{\rm uv} = 1/\Lambda_{\rm uv}$, $\Lambda_{\rm uv} = 0.905\ \mathrm{GeV}$, and $\tau_{\rm ir} = 1/\Lambda_{\rm ir}$, $\Lambda_{\rm ir} = 0.240\ \mathrm{GeV}$. The infrared cutoff removes quark production thresholds and so implements a simple form of confinement (see, for example, a recent comparative studies on confinement from various model of low energy QCD in [41]), while the ultraviolet cutoff sets the hadronic scale of the model. The momentum integration can then be done in closed form, and Eq. (3) becomes

where

and $\operatorname{Ei}(x)$ is the exponential integral. For the vacuum energy we also need

which also has a closed form in terms of exponentials and $\operatorname{Ei}(x)$. With $m_u = m_d = 7$ MeV the vacuum solution is $M_u = M_d = 367.5$ MeV, and the condensate $-\langle\bar{q}q\rangle= 9(M_u - m_u)/(4\alpha_{\mathrm{eff}})$ gives $(-\langle\bar{q}q\rangle)^{1/3} = 242$ MeV. For the strange quark we take $m_s = 170$ MeV, the value used in CI studies of strange hadrons [23], and obtain $M_s = 533$ MeV. With flavor dressing these masses become $M_u = 343.2$ MeV and $M_s = 518.5$ MeV. Table 1 collects the vacuum quantities.

Thermodynamic potential at finite density

After a Fierz rearrangement of the contact vertex, the rainbow gap equation coincides with the mean-field gap equation of a four-fermion model with scalar coupling $G_S = 2\alpha_{\mathrm{eff}}/9$ in each flavor channel [33, 35]. The same rearrangement produces a repulsive vector coupling $G_V = \alpha_{\mathrm{eff}}/9$, that is, $G_V/G_S = 1/2$. Because the vector channel of the full QCD kernel is not fixed by the vacuum phenomenology that calibrates the CI, we treat its strength as a parameter and write $G_V = \eta G_S$, with $\eta= 1/2$ as the reference value. The rainbow kernel does not mix flavors, so at zero temperature the thermodynamic potential is a sum of independent flavor contributions, $\Omega= \sum_f \Omega_f$, with

Here $\mu_f$ is the chemical potential of flavor $f$, $\tilde{\mu}_f$ the chemical potential shifted by the vector mean field, and

with $k_F = \sqrt{\mu^2 - M^2}$, is the pressure of a degenerate gas of quasiparticles of mass $M$ with spin and color degeneracy $2N_c$. The medium contribution is ultraviolet finite and is left unregularized, as is customary. The stationarity conditions $\partial\Omega_f/\partial M_f=\partial\Omega_f/\partial\widetilde{\mu}_f=0$ give

where

both evaluated at $\widetilde{\mu}_f$. Equation (11) reduces to Eq. (6) in vacuum, and the medium term, through Pauli blocking, is what drives the restoration of chiral symmetry. Equation (12) expresses the vector repulsion: at fixed $\mu_f$ it lowers the Fermi momentum and therefore the density.

When Eqs. (11) and (12) admit several solutions, the physical one is the solution with the lowest $\Omega_f$. The pressure is measured with respect to the physical vacuum,

and the quark number densities follow from $n_f=\partial P_q/\partial\mu_f$. We checked numerically that the density given by Eq. (13) equals the derivative of Eq. (14) to a relative accuracy of $10^{-5}$ across the whole range of chemical potentials used below. This check matters. With an incorrect relative normalization between the condensate term and the vacuum energy in Eq. (9), the gap equation can still be satisfied while the pressure is off by a constant factor, and the resulting stellar masses and radii are then meaningless.

Chiral restoration at finite density

Figure 1 shows the dressed masses as functions of the quark chemical potential for several values of $\eta$. Without vector repulsion the light-quark transition is of first order, with a jump in $M_u$ at $\mu=397\ \mathrm{MeV}$, or at $378\ \mathrm{MeV}$ with flavor dressing. Vector repulsion softens the transition and moves it to larger $\mu$. At $\eta=1/4$ it is already a continuous crossover in both cases, although a steep one, particularly without dressing, where $M_u$ drops by about $180\ \mathrm{MeV}$ per MeV of chemical potential at its steepest point. We locate the light-quark transition at the value $\mu_\chi$ where $-dM_u/d\mu$ is largest; with flavor dressing we find $\mu_\chi=401$, $427$, $481$, and $535\ \mathrm{MeV}$ for $\eta=1/4$, $1/2$, $1$, and $3/2$. Above $\mu_\chi$ the light quarks carry a mass of a few tens of MeV that decreases slowly towards $m_u$.

The strange quark behaves differently. Its larger current mass keeps $M_s$ close to its vacuum value until $\mu\approx560$–$700\ \mathrm{MeV}$, depending on $\eta$, and even beyond that point it stays heavy: at $\mu=950\ \mathrm{MeV}$ we find $M_s\approx60\ \mathrm{MeV}$ for $\eta=0$ and between $120$ and $330\ \mathrm{MeV}$ for $1/4\le\eta\le3/2$. In neutral matter this delays the appearance of strangeness, as discussed in the next section.

Neutral quark matter in beta equilibrium

Matter in a neutron star is electrically neutral and in chemical equilibrium under weak interactions. For $u$, $d$, and $s$ quarks together with electrons and muons, $\beta$ equilibrium fixes

so that all chemical potentials are determined by the baryon chemical potential $\mu_B = \mu_u+2\mu_d$ and the electron chemical potential $\mu_e$. The latter is fixed by charge neutrality,

The leptons are treated as free Fermi gases. The total pressure is $P=P_q+P_\ell$, with $P_q$ from Eq. (14), and the energy density and baryon density follow from

where the sum runs over quarks and leptons. Because the rainbow kernel does not couple different flavors, the flavors interact only through Eqs. (15) and (16). Each quark flavor can therefore be solved on a grid of $\mu_f$ once, and neutral matter is then assembled by root finding in $\mu_e$ at every $\mu_B$.

The first thing to notice is where the pressure becomes positive. In the chirally broken phase the quarks behave as quasiparticles with the vacuum masses of Table 1, and a Fermi sea forms as soon as $\mu_u$ and $\mu_d$ exceed $M_u$. The pressure therefore turns positive at $\mu_B \simeq3M_u$, that is, at 1102 MeV without dressing and 1030 MeV with it, independently of $\eta$. The energy per baryon never falls below 1110 and 1037 MeV, respectively. Both values lie well above the 930 MeV of the most bound nuclei, so CI strange quark matter is not self-bound and the Bodmer–Witten hypothesis [31, 32] is not realized in this model. The result has a direct consequence for stellar structure. The low-density phase of the model is a dilute gas of constituent quarks, which has no counterpart in nature, and any stellar configuration whose outer layers are made of it should not be taken at face value. We come back to this point in § 4.

Table 1 Vacuum properties of the contact interaction with and without flavor dressing for $N_f = 3$. Masses in MeV, coupling in GeV$^{-2}$.

	$\alpha_{\mathrm{eff}}$	$M_u = M_d$	$M_s$	$(-\langle\bar{q}q\rangle)^{1/3}$
Without dressing	57.37	367.5	533.0	242
With dressing ($N_f = 3$)	54.67	343.2	518.5	240

Chiral symmetry is restored for the light quarks when $\mu_u$ and $\mu_d$ cross the value $\mu_\chi$ discussed in § 2.4. Strangeness appears later, once $\mu_d$ exceeds the in-medium strange mass. The corresponding baryon density depends strongly on the vector coupling. With flavor dressing, strange quarks appear at $n_B=4.0\,n_0$ for $\eta=1/4$, $3.2\,n_0$ for $\eta=1/2$, $2.3\,n_0$ for $\eta=1$, and $1.6\,n_0$ for $\eta=3/2$; without dressing the numbers differ by less than 10%. The trend has a simple origin. Vector repulsion raises the chemical potential needed to reach a given density, and the strange threshold, which is fixed in terms of $\mu_d$, is then crossed at a lower density. Once present, the strange quarks reduce the need for electrons. The electron chemical potential never exceeds 120 MeV, so muons appear only in a narrow range and in small numbers, with no visible effect on the EoS.

Figure 2 shows the resulting pressure as a function of the energy density. The vector interaction stiffens the EoS at high density: for $\eta\ge1/4$ the squared speed of sound exceeds the conformal value 1/3 and reaches about 0.5 at the highest densities considered, while staying well below the causal limit. The same interaction lowers the pressure at a given chemical potential, which will matter for the competition with hadronic matter in § 5.

The case $\eta=0$ deserves a separate remark. The light-quark transition is then of first order, and within the coexistence region no homogeneous neutral solution exists; neutral matter would have to be built as a mixed phase with two conserved charges. Since this limit gives the softest EoS of the set, with $c_s^2$ never above 1/3, and since vector repulsion is generated by the Fierz rearrangement of the kernel itself, we keep $\eta=0$ only as a reference for the EoS and do not compute stellar configurations for it.

Pure quark stars

Stellar structure and tidal deformability

For a static, spherically symmetric star the metric functions and the pressure follow from the Tolman–Oppenheimer–Volkoff equations [42, 43], which in units $G=c=1$ read

The radius $R$ is the point where the pressure vanishes, and $M=m(R)$. The response of the star to a static quadrupolar tidal field is measured by the dimensionless tidal deformability $\Lambda=(2/3)k_{2}C^{-5}$, where $C=M/R$ is the compactness and $k_{2}$ the second Love number [44]. We obtain $k_{2}$ from the function $y(r)$, which satisfies

with $y(0)=2$, $e^{\lambda}=(1-2m/r)^{-1}$, and

The Love number then follows from the standard expression in terms of $C$ and $y_{R}=y(R)$ [44]. Wherever the energy density jumps by $\Delta\varepsilon$ at fixed pressure, as happens at a Maxwell phase transition or at the surface of a self-bound object, $1/c_{s}^{2}$ contains a delta function and Eq. (19) must be supplemented by the matching condition $y\to y-4\pi r^{3}\Delta\varepsilon/m$ at that radius [45, 46]. We integrate Eqs. (18) and (19) together and apply this condition explicitly at every discontinuity. The numerical scheme reproduces the known results for a bag-model EoS and for an $n=1$ polytrope in the Newtonian limit, as shown in Appendix § 8.

Mass–radius relation and tidal deformability

The left panel of Figure 3 shows the mass–radius relation of stars made entirely of neutral CI quark matter, with flavor dressing, for $1/4\leq\eta\leq3/2$. Every sequence displays two families of configurations. At low central densities the star is a large object, with radii between 20 and 60 km, whose outer layers consist of the dilute gas of constituent quarks found in § 3. At high central densities the star contracts into a compact configuration with a chirally restored core. For most values of $\eta$ the two families are separated by an unstable segment on which $dM/d\varepsilon_{c}<0$, so that the sequence has two local maxima of the mass. As $\eta$ grows, the maximum of the extended family rises faster than that of the compact one, from $1.19$ to $2.62\,M_{\odot}$ against $1.49$ to $2.09\,M_{\odot}$ with dressing, and for $\eta\geq1$ the largest masses are reached on the extended family. Table 2 lists the relevant numbers. The compact branch never reaches two solar masses with a radius compatible with the NICER measurement of PSR J0740+6620. For $\eta=1/4$ with dressing it produces a $1.4\,M_{\odot}$ star with $R_{1.4}=13.5$ km and $\Lambda_{1.4}=$ 162, both within the ranges allowed by NICER and GW170817, but the maximum mass of the same sequence is only 1.49 $M_\odot$. Increasing $\eta$ raises the maximum mass and simultaneously moves the 1.4 $M_\odot$ configuration to the extended family, where $R_{1.4}$ exceeds 30 km and $\Lambda_{1.4}$ reaches values between $10^4$ and $10^6$. Masses above 2 $M_\odot$ appear only for $\eta\ge1$ with dressing and $\eta\ge3/2$ without it, always on the extended family, with radii of 24 to 27 km and central densities of about $2n_0$. Figure 4 makes

	$\eta$	$M_{\max}^{c}$	$R^{c}$	$M_{\max}^{e}$	$R^{e}$	$R_{1.4}$	$\Lambda_{1.4}$
With dressing	$1/4$	$1.49$	$12.0$	$1.19$	$34.1$	$13.5$	$162$
	$1/2$	$1.66$	$14.3$	$1.47$	$29.1$	–	–
	$3/4$	$1.86$	$17.1$	$1.78$	$26.2$	–	–
	$1$	$2.09$	$20.4$	$2.09$	$25.7$	–	–
	$3/2$	–	–	$2.62$	$26.6$	–	–
Without dressing	$1/4$	$1.31$	$10.6$	$0.99$	$29.9$	–	–
	$1/2$	$1.47$	$13.0$	$1.28$	$25.3$	$15.2$	$308$
	$3/4$	$1.67$	$16.3$	$1.62$	$22.7$	–	–
	$1$	–	–	$1.93$	$22.4$	–	–
	$3/2$	–	–	$2.45$	$24.2$	–	–

the same point in terms of $\Lambda(M)$: the sequences that pass through the GW170817 interval stop well short of $2M_{\odot}$.

None of the sequences satisfies the mass and tidal constraints together, and the configurations that come closest owe their mass to an envelope of constituent quarks at densities where nuclear matter is known to be hadronic. We regard this as evidence against pure CI quark stars rather than as a tension to be resolved by further tuning. The natural next step is to replace the unphysical low-density phase by a realistic hadronic EoS and to ask where, if anywhere, deconfinement takes place inside the star.

Hadronic matter and the deconfinement transition

Hadronic equation of state

For the hadronic phase we use a relativistic mean-field model of nucleons interacting through the exchange of $\sigma$, $\omega$, and $\rho$ mesons, with cubic and quartic self-interactions of the scalar field [38]. The energy density of nuclear matter with proton and neutron densities $n_p$ and $n_n$ reads

where $n=n_p+n_n$, $m^*$ is the effective nucleon mass, $\varepsilon_{\mathrm{FG}}$ is the energy density of a free Fermi gas, $c_i=(g_i/m_i)^2$, and $U(x)=\frac{b}{3}m_Nx^3+\frac{c}{4}x^4$. Rather than taking the couplings from the literature, we fixed them by requiring the GM1 saturation properties [38]: $n_0=0.153\,\mathrm{fm}^{-3}$, binding energy $16.3\,\mathrm{MeV}$, incompressibility $K=300\,\mathrm{MeV}$, $m^*/m_N=0.70$, and symmetry energy $a_{\mathrm{sym}}=32.5\,\mathrm{MeV}$. The fit returns $c_\sigma=11.785\,\mathrm{fm}^2$, $c_\omega=7.148\,\mathrm{fm}^2$, $c_\rho=4.410\,\mathrm{fm}^2$, $b=0.002948$, and $c=-0.001071$, in agreement with the published GM1 parameter set. Nucleons, electrons, and muons are kept in $\beta$ equilibrium with zero net charge. Below $0.57n_0$ the core EoS is joined to the SLy crust [47].

Hadronic GM1 stars reach $M_{\max} = 2.35\,M_\odot$ at $R = 11.9$ km, with a central density of $5.67n_0$, and a $1.4\,M_\odot$ star has $R_{1.4} = 13.7$ km and $\Lambda_{1.4} \approx810$. The model is thus comfortably compatible with the $2M_\odot$ constraint but, like other stiff mean-field models, exceeds the GW170817 bound on $\Lambda_{1.4}$. We keep GM1 because it provides the off-equilibrium pressure $P_H(\mu_B,\mu_e)$ needed for a Gibbs construction, which tabulated EoSs do not. As a check that our conclusions do not hinge on this choice, we repeated the Maxwell analysis with the tabulated SLy4 and DD2 EoSs [47, 48] distributed with LALSuite [49]; our integrator reproduces their maximum masses, radii, and tidal deformabilities (Appendix § 8).

Maxwell and Gibbs constructions

Hadronic and quark matter share two conserved charges, baryon number and electric charge. If the surface tension between the two phases is large, each phase stays locally neutral, and the transition occurs at the baryon chemical potential where the neutral pressures coincide, $P_H(\mu_B) = P_Q(\mu_B)$. This Maxwell construction produces a jump in the energy density at constant pressure. If the surface tension is small, the phases can carry opposite charges and coexist in a mixed phase that is neutral only globally [39]. The Gibbs conditions then read

where $\chi$ is the volume fraction of quark matter and $q_H$, $q_Q$ are the charge densities of the two phases including the common lepton background. The energy density and baryon density of the mixed phase are the corresponding volume averages. Gibbs matter has no density jump, and the mixed phase begins at a lower pressure than the Maxwell transition. The true behavior lies between the two limits and depends on a surface tension that is poorly known, so we present both.

Deconfinement bag shift and its upper bound

The CI does not confine quarks at finite density, and its quark pressure carries no information about the energy cost of deconfinement. Following the idea of the vBag model [33], we allow for a constant shift of the quark pressure,

with $B_{\mathrm{dc}} \ge0$ favoring the quark phase. Applied without restriction, this shift has an unacceptable consequence. At $\mu_B \approx930$ MeV the quark pressure of Eq. (14) vanishes, so any positive $B_{\mathrm{dc}}$ would make quark matter more stable than nuclei. We therefore admit the deconfined phase only where it is physically meaningful, namely where the light quarks have restored chiral symmetry, $\mu_u \ge\mu_\chi$. This mirrors the assumption in vBag that deconfinement and chiral restoration occur together.

The restriction has a second consequence that fixes the size of the shift. At the baryon chemical potential $\mu_B^*$ where neutral quark matter first satisfies $\mu_u = \mu_\chi$, the shifted quark pressure must not exceed the hadronic one; otherwise the pressure of the favored phase would jump upwards at fixed $\mu_B$, which thermodynamics forbids. This yields

Table 3 lists $B_{\max}$ together with the hadronic density at $\mu_B^*$, which is the earliest possible onset of deconfinement for a given $\eta$. The bound grows with $\eta$, because vector repulsion lowers the quark pressure, but the earliest onset density grows as well and never falls below $3.4n_0$.

$\eta$	With dressing		Without dressing
	$B_{\max}$	$n_H^*/n_0$	$B_{\max}$	$n_H^*/n_0$
$1/4$	68.8	3.41	98.5	3.69
$1/2$	108.3	4.08	147.8	4.50
$3/4$	148.9	4.46	196.2	4.88
1	189.4	4.79	246.5	5.28

Hybrid stars

Where deconfinement sets in

Figure 5 shows the hadronic density at which quark matter first appears, as a function of $B_{\rm dc}/B_{\max}$, for the Maxwell construction and, at selected values, for the Gibbs construction. The dotted line marks the central density of the heaviest stable GM1 star, $5.67n_0$. A quark phase can exist inside a stable star only if its onset lies below this line.

Without any bag shift the Maxwell transition takes place at $5.1n_0$ for $\eta=1/4$ and at $6.3n_0$ for $\eta=1/2$ with flavor dressing. The first value lies inside the GM1 sequence, but only barely: the star whose center reaches the transition already has $2.335\,M_\odot$, almost the maximum mass. For $\eta\ge3/4$ the onset lies at $7.7n_0$ or above, when it occurs at all below $\mu_B=2.4$ GeV, and deconfinement never takes place in a stable star. The tabulated EoSs lead to the same picture. With SLy4 and DD2 and $B_{\rm dc}=0$, the Maxwell transition appears only for $\eta\le1/2$, at pressures between 360 and 820 MeV fm$^{-3}$, and in every case at or beyond the center of the maximum-mass configuration.

Raising $B_{\rm dc}$ lowers the onset monotonically. Near the upper bound of Table 3 it reaches $4.0n_0$ for $\eta=1/4$ and $4.9n_0$ for $\eta=1/2$ with dressing, and $4.2n_0$ and $5.0n_0$ without dressing. For $\eta=3/4$ even $B_{\rm dc}=0.98B_{\max}$ leaves the onset at $5.9n_0$ with dressing and $6.0n_0$ without it, above the central density of the heaviest hadronic star. For $\eta=1$ the onset stays above $7n_0$. The bound of Eq. (24) is what prevents a further reduction: a larger shift would pin the transition at $\mu_B^*$ with a discontinuous pressure.

The Gibbs construction moves the onset to lower densities, as expected. For $\eta=1/4$ with dressing and $B_{\rm dc}=0$ the mixed phase begins at $4.7n_0$ instead of $5.1n_0$ and extends up to about $10n_0$, well beyond the stellar center. The gain shrinks as $B_{\rm dc}$ increases. At $B_{\mathrm{dc}}=0.9B_{\max}$ no mixed phase forms at all, and the Gibbs result coincides with the Maxwell one. The reason is the chiral condition of § 5.3: in the mixed phase the electron chemical potential is set by the hadronic component and exceeds 200 MeV, which pushes $\mu_u=\mu_B/3-2\mu_e/3$ below $\mu_\chi$ and excludes the deconfined phase.

Stellar sequences

The right panel of Figure 3 and Table 4 show the stellar sequences for the cases in which the transition lies inside the GM1 sequence. In all of them the quark core forms in stars between $2.21$ and $2.35\,M_\odot$, and the stable hybrid branch is short. The largest mass gain after the onset is $0.022\,M_\odot$, obtained with a Maxwell transition for $\eta=1/4$, flavor dressing, and $B_{\mathrm{dc}}=0.98B_{\max}$. The other cases gain less than $0.01\,M_\odot$, and several of them become unstable as soon as the core forms. We find no third family of compact stars in any of the cases studied.

	$\eta$	$B_{\mathrm{dc}}/B_{\max}$	$n_{\mathrm{on}}/n_0$	$\Delta\varepsilon/\varepsilon_H$ or $n_{\mathrm{end}}/n_0$	$M_{\mathrm{on}}$ ($M_\odot$)	$M_{\max}$ ($M_\odot$)
Maxwell	1/4	0	5.12	0.94	2.335	2.335
Maxwell	1/4	0.90	4.26	0.26	2.257	2.260
Maxwell	1/4	0.98	3.98	0.12	2.209	2.230
Maxwell	1/2	0.98	4.90	0.22	2.323	2.323
Gibbs	1/4	0	4.73	10.2	2.311	2.315
Gibbs	1/4	0.50	4.53	8.8	2.294	2.298
Gibbs	1/2	0.90	4.84	8.6	2.321	2.325
Maxwell, no dressing	1/4	0.98	4.23	0.15	2.253	2.262

The size of the density jump explains why the branches are short. For a Maxwell transition, small quark cores are stable when $\Delta\varepsilon/\varepsilon_H<1/2+3P_t/(2\varepsilon_H)$, where $P_t$ is the transition pressure [50, 51]. All the cases of Table 4 satisfy this condition, so the hybrid branch is connected to the hadronic one. The limitation comes from elsewhere. Because the onset lies so close to the central density of the heaviest hadronic star, there is almost no room left before the sequence turns over, and the soft quark EoS above the transition cannot support much additional mass.

Comparison with observations

The hybrid sequences inherit the maximum masses of GM1, between 2.23 and 2.35 $M_\odot$, and the radii of their heaviest stars, between 11.9 and 12.7 km, fall inside the NICER interval for PSR J0740+6620. The presence of a quark core changes the radius of the heaviest stars by at most 0.12 km relative to a purely hadronic star of the same mass, far below current observational uncertainties.

Stars of intermediate mass are unaffected. A 1.4 $M_\odot$ star has no quark core for any parameter set, so $R_{1.4}$ and $\Lambda_{1.4}$ are those of GM1, as Figure 4 shows. The value $\Lambda_{1.4} \approx810$ exceeds the GW170817 bound, but this is a property of the hadronic model and cannot be remedied by the quark phase. A softer hadronic EoS lowers $\Lambda_{1.4}$, as SLy4 does with $\Lambda_{1.4} = 297$, but it does not help the quark phase. With SLy4 and $B_{\mathrm{dc}} = 0$ the Maxwell transition moves up to $P_t \approx740\ \mathrm{MeV}\ \mathrm{fm}^{-3}$ for $\eta= 1/4$ and coincides with the maximum-mass configuration, so the tidal constraint can be met only by stars that contain no quark matter.

Conclusions

We have studied cold, neutral, $\beta$-equilibrated strange quark matter with a contact interaction whose parameters are fixed entirely in vacuum, and followed it into two kinds of compact stars. The three flavors enter on the same footing, the vector repulsion generated by the Fierz rearrangement is treated as a free fraction $\eta$ of the scalar coupling, and the flavor dressing of the coupling is either switched on or off to bracket its effect.

Regarding compact stars, those of pure quarks fail on two counts. The model does not produce self-bound matter, so these objects carry an envelope of constituent quarks that nature does not provide; and even setting that aside, no value of $\eta$ yields both a 2$M_\odot$ star and a tidal deformability compatible with GW170817. The only sequences that reach 2 $M_\odot$ do so on the extended branch, with radii near 25 km.

Hybrid stars tell a sharper story. The vector strength that pure quark stars need in order to become massive, $\eta\gtrsim1$, is also the one that pushes deconfinement beyond the center of the heaviest hadronic star. For $\eta\lesssim1/2$ the quark phase does appear, but never below about $4n_0$, and the resulting cores live only in stars of $2.2$ to $2.35\,M_\odot$ over a mass window of at most $0.02\,M_\odot$. This holds for Maxwell and Gibbs constructions, with and without flavor dressing, and for every admissible value of the deconfinement bag shift. The upper bound on that shift, which follows from the continuity of the pressure once deconfinement is tied to chiral restoration, is what makes the statement robust: the one free parameter that could move the transition is limited by thermodynamics rather than by choice.

These findings have a direct observational reading. Within this class of models, the mass, radius, and tidal deformability of stars below about $2.2\,M_\odot$ constrain only the hadronic EoS. Evidence for deconfined matter would have to come from the heaviest pulsars, and even there the imprint on the radius is about a tenth of a kilometer. A softer hadronic EoS that satisfies the GW170817 bound does not change this picture in the case we tested: with SLy4 the transition moves to higher pressure and coincides with the maximum-mass star.

Several limitations should be kept in mind. We worked at zero temperature and in the mean-field approximation, we did not include hyperons in the hadronic phase, and we did not consider diquark pairing in quark matter. Color superconductivity in particular adds a pairing energy that raises the quark pressure at intermediate densities and could lower the onset below the bound found here, in the same direction as a positive $B_{\rm dc}$ but without the constraint of Eq. (24); in NJL-type models, large pairing gaps are known to stabilize quark cores in hybrid stars [37]. Including the diquark channel of the contact interaction, which is fixed by the same Fierz rearrangement and by vacuum diquark masses, is therefore the natural extension of this work, and it is the one that could overturn the conclusion that deconfinement is confined to the heaviest neutron stars.

Acknowledgements

Acknowledgements. A. Ahmad and N. Akhter acknowledge the hospitality of the Institute of Physics, Gomal University. A. Raya acknowledges support from SECI-HTI (México) and CIC-UMSNH (México) under grants CBF-2026-465 and 18371, respectively.

Numerical checks

The per-flavor gap equations, Eqs. (11) and (12), are solved on a grid of quark chemical potentials with a spacing of 1 MeV up to $\mu_f = 950$ MeV. At each point all solutions are located by bracketing in $M_f$, and the one with the lowest $\Omega_f$ is retained. The functions $C$ and $D$ are evaluated in closed form and agree with direct quadrature to machine precision. Thermodynamic consistency, tested by comparing $P_q$ from Eq. (14) with the integral of $n_f$ over $\mu_f$, holds to a relative accuracy of $10^{-5}$.

The TOV and tidal equations are integrated with an adaptive eighth-order Runge–Kutta scheme, with relative tolerances between $10^{-7}$ and $10^{-8}$. Density discontinuities are located as events in the pressure, where the matching condition for $y$ is applied. We tested the integrator against four cases with known results. For a bag model with $\varepsilon= 3P + 4B$ and $B = 57\,\mathrm{MeV}\,\mathrm{fm}^{-3}$, including the surface discontinuity, we find $M_{\max} = 2.015\,M_\odot$ at $R = 11.0\,\mathrm{km}$. For an $n=1$ polytrope the Love number approaches the Newtonian value $k_2 = (15-\pi^2)/(2\pi^2) \simeq0.260$ as the compactness decreases, reaching $k_{2} = 0.256$ at $C = 2 \times10^{-3}$. For the SLy4 EoS [47] we obtain $M_{\max} = 2.049\,M_{\odot}$, $R_{1.4} = 11.71$ km, and $\Lambda_{1.4} = 297$. For DD2 [48] with the unified crust of the LALSuite tables [49] we obtain $M_{\max} = 2.418\,M_{\odot}$, $R_{1.4} = 13.20$ km, and $\Lambda_{1.4} = 689$. These values agree with those reported in the literature for the same EoSs.

For the hadronic model, the couplings obtained from the GM1 saturation properties coincide with the published parameter set to the quoted digits, and the saturation properties are recovered to better than one part in $10^{8}$. In the Gibbs construction the hadronic field equations are solved at fixed $(\mu_{B}, \mu_{e})$ from several starting points, and only converged solutions enter the search for the coexistence condition. This precaution proved necessary: non-converged solutions can produce spurious sign changes of the charge density and, with them, false roots of the neutrality condition.