Viability of self-bound colour-flavour-locked quark stars for HESS J1731-347, GW190814, and GW170817 under rapid rotation, flavour-dependent parameters, and perturbative-QCD consistency

The central compact object in the supernova remnant HESS J1731-347 (M approximately 0.77 solar masses, R approximately 10.4 km) and the approximately 2.6-solar-mass secondary of the gravitational-wave event GW190814 bracket the observed compact-star mass range and motivate self-bound colour-flavour-locked (CFL) quark stars as a common explanation. A 2023 analysis of static CFL stars found a viable parameter window but with three unresolved strains: the analysis was restricted to non-rotating stars; it assumed flavour- and density-independent pairing gap and bag constant; and it required a perturbative-QCD quartic coefficient a_4 <= 0.594, below the canonical expectation of about 0.65, with the GW170817 tidal bounds forcing gaps above 200 MeV into a region where the positive trace-anomaly bound is violated. The question is whether the CFL quark-star interpretation of these three events survives when these simplifying assumptions are lifted.

Background

At supranuclear density, cold quark matter is expected to become a colour superconductor: quarks near the Fermi surface pair, in analogy with BCS superconductivity, opening a gap $\Delta$ in their excitation spectrum. In three-flavour (up, down, strange) matter at high density the favoured pattern is the colour-flavour-locked (CFL) phase, in which pairing locks colour and flavour symmetries and enforces exact electric charge neutrality without electrons; at intermediate densities two-flavour (2SC) pairing, in which only up and down quarks pair, competes with it. If such paired strange quark matter were absolutely stable — the true ground state of strong-interaction matter, a hypothesis dating to the 1980s — compact stars made of it would be self-bound: bound by confining forces rather than gravity, with mass and radius set by the equation of state (EoS) down to arbitrarily low mass, unlike gravitationally bound neutron stars.

A standard phenomenological description is an MIT bag-model thermodynamic potential for CFL matter, $\Omega_{\mathrm{CFL}} = -(3a_4/4\pi^2)\mu^4 + (3(m_s^2 - 4\Delta^2)/4\pi^2)\mu^2 + B_{\mathrm{eff}}$, where $\mu$ is the quark chemical potential, $m_s$ the strange-quark mass, $\Delta$ the pairing gap, $B_{\mathrm{eff}}$ the effective bag constant encoding non-perturbative vacuum pressure, and $a_4$ a quartic coefficient summarizing leading perturbative-QCD (pQCD) corrections of order $\alpha_s$; in bag-style parametrizations $a_4 \approx 0.65$ is the canonical reference value from fits to the perturbative EoS (Fraga, Pisarski and Schaffner-Bielich, 2001; Alford, Braby, Paris and Reddy, 2005). It is conventional to repackage the $\mu^2$ coefficient as $w = (m_s^2 - 4\Delta^2)/\sqrt{a_4}$. The stellar structure follows by integrating the Tolman-Oppenheimer-Volkoff (TOV) equations of general relativity; the maximum mass of the non-rotating sequence is denoted $M_{\mathrm{TOV}}$.

Two observational anchors frame the question. First, spectral modeling of the central compact object XMMU J173203.3-344518 in HESS J1731-347 (Doroshenko et al., 2022) gave $M = 0.77^{+0.20}_{-0.17}$ solar masses and $R = 10.4^{+0.86}_{-0.78}$ km — the lightest known compact object and a strong candidate for a self-bound star, since standard neutron-star formation channels rarely produce such low masses. Second, the GW190814 gravitational-wave event contains a secondary of about $2.5$-$2.6$ solar masses whose nature is unknown; requiring $M_{\mathrm{TOV}} \ge 2.6$ solar masses is a demanding constraint on any EoS. A third anchor is GW170817, whose tidal deformability measurement (the dimensionless deformability $\Lambda$ of the inspiralling stars, which constrains the EoS stiffness) must also be respected; throughout, the "GW170817 tidal bounds" denote the tidal-deformability constraint as imposed in the 2023 analysis of Oikonomou and Moustakidis described below.

Conformality provides theoretical bounds on the EoS. Fujimoto, Fukushima, McLerran and Praszałowicz (2022) proposed the normalized trace anomaly, built from $\langle\Theta\rangle = \mathcal{E} - 3P$ (the trace of the energy-momentum tensor, with $\mathcal{E}$ the energy density and $P$ the pressure), as a measure of how close cold dense matter is to conformal (scale-invariant) behavior, suggesting $\langle\Theta\rangle \ge 0$ at all densities; the speed of sound should then approach but not substantially exceed the conformal value $c_s^2 = 1/3$. Within the CFL bag EoS, the sign of $w$ controls these properties: for $w \ge 0$ both $\langle\Theta\rangle \ge 0$ and $c_s^2 \le 1/3$ hold, while $w < 0$ (i.e. $\Delta > m_s/2$ at the relevant chemical potential) violates the conformal sound-speed bound and can drive the trace anomaly negative.

In this setting, Oikonomou and Moustakidis (Phys. Rev. D 108, 063010 (2023)) constrained flavour-independent CFL and 2SC bag models by the HESS J1731-347 mass-radius posterior, $M_{\mathrm{TOV}} \ge 2.6$ solar masses, and absolute-stability conditions, finding CFL favoured over 2SC and a permitted parameter window in $(\Delta, B_{\mathrm{eff}}^{1/4})$ — but only if $a_4 \le 0.594$ (for $M_{\mathrm{TOV}} = 2.6$ solar masses) to keep $w \ge 0$, a value below the canonical pQCD expectation of about 0.65; and with the additional GW170817 tidal bounds forcing very high gaps $\Delta > 200$ MeV that lie only in the $w < 0$ region, where the positive trace-anomaly bound is violated at densities of a few times nuclear saturation density $n_0 \approx 0.16$ fm$^{-3}$. Stellar rotation is treated by computing equilibrium sequences of rotating stars; uniform rotation can support supramassive stars above $M_{\mathrm{TOV}}$ up to the mass-shedding (Kepler) limit where the equator rotates at orbital frequency. The 2023 analysis used only non-rotating configurations and flagged rapid rotation, flavour-dependent parameters, and the status of the required $a_4$ as its principal open issues.

Problem Statement

Can self-bound colour-flavour-locked (CFL) quark stars simultaneously explain the HESS J1731-347 central compact object (through its measured mass-radius posterior), the approximately 2.6-solar-mass secondary of GW190814, and the tidal-deformability limits of GW170817, once the three simplifying assumptions of the 2023 analysis of Oikonomou and Moustakidis are lifted? Specifically, within the CFL bag-model equation of state with perturbative-QCD corrections (parameters $\Delta$, $B_{\mathrm{eff}}$, $a_4$, $m_s$) or a more general density- and flavour-dependent framework:

1. Rotation: the 2023 analysis used only non-rotating (TOV) sequences. Determine how uniform rapid rotation - equilibrium sequences from the slow-rotation limit to the mass-shedding (Kepler) limit, including supramassive configurations gravitationally supported only by rotation - modifies the permitted $(\Delta, B_{\mathrm{eff}}^{1/4}, a_4)$ parameter space for the HESS J1731-347 object and the GW190814 secondary, and whether it relaxes the tension between the $M_{\mathrm{TOV}} \ge 2.6$-solar-mass requirement and the $w \ge 0$ conditions (non-negative trace anomaly $\langle\Theta\rangle = \mathcal{E} - 3P \ge 0$ and conformal sound-speed bound $c_s^2 \le 1/3$).

2. Flavour/density dependence: the 2023 analysis assumed the same gap $\Delta$ and bag constant $B_{\mathrm{eff}}$ for all flavours, independent of density. Reassess the relative stability of CFL versus 2SC quark matter and the true-ground-state question under density- and/or flavour-dependent parameters (or within a field-theoretic treatment such as NJL-type models), under the same observational constraints as the static 2023 baseline (HESS posterior, $M_{\mathrm{TOV}} \ge 2.6$ solar masses, GW170817 tidal bounds), independently of any rotation-relaxed constraint set considered in item 1.

3. pQCD consistency of $a_4$: the $w \ge 0$ window requires $a_4 \le 0.594$ for $M_{\mathrm{TOV}} = 2.6$ solar masses, below the canonical bag-model perturbative expectation of about 0.65. Determine whether higher-order perturbative or non-perturbative QCD contributions shift the expected effective quartic coefficient enough to accommodate this bound, or whether the simultaneous description of GW170817, GW190814, and HESS J1731-347 necessarily requires very high gaps ($\Delta > 200$ MeV) in the $w < 0$ region, where the positive trace-anomaly bound is violated at densities of a few times nuclear saturation.

These three deliverables jointly support one objective: establishing whether the self-bound CFL quark-star hypothesis for these three observational landmarks survives the removal of its own limiting assumptions. A complete answer states, for each extension, the modified stability window or viability verdict, and then answers the joint question - does the scenario survive, and, if so, at what parameter cost (in particular, whether accepting $\Delta > 200$ MeV with $w < 0$ is unavoidable); or, if it fails, which constraint set excludes it.

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