Existence and observational identification of quark stars: determining the cold dense quark-matter equation of state (including the bag constant) from QCD, and distinguishing quark stars from neutron stars

Quantum chromodynamics (QCD) predicts that hadronic nuclear matter deconfines into quark matter at sufficiently high energy density, but whether cold compact stars contain deconfined quark matter — as quark-matter cores inside neutron stars or as entire self-bound quark (strange) stars — remains undecided. The source paper (Li, Luo, and Zong, Phys. Rev. D 82, 065017 (2010), on a Dyson–Schwinger-equation-based equation of state with a bag constant) formulates the issue precisely: the vacuum bag constant $B$ is a phenomenological input that cannot currently be computed from QCD; the stellar models omit chemical/thermodynamic equilibrium, electric charge neutrality, crust structure, and the hadron–quark phase transition; and it is unknown whether quark stars exist or how to distinguish them from neutron stars observationally. More than fifteen years of subsequent work — perturbative-QCD EOS calculations to next-to-next-to-next-to-leading ($\mathrm{N}^3\mathrm{LO}$) order, multimessenger constraints from GW170817, two-solar-mass pulsars and NICER radius measurements, and proposed gravitational-wave phase-transition signatures — has narrowed the allowed equation-of-state band but resolved none of these three linked open issues. The open core is: establish from QCD-based microphysics and observation whether cold dense compact-star matter is (or contains) deconfined quark matter, with the first-principles determination or tight constraint of the equation of state at a few times nuclear saturation density, including the vacuum-pressure (bag) contribution, as the prerequisite, and an unambiguous observational discriminator between self-bound quark stars, hybrid stars, and purely hadronic neutron stars as the target.

Background

Quantum chromodynamics (QCD), the theory of the strong interaction, confines quarks and gluons inside hadrons (protons, neutrons) at low density, but at sufficiently high energy density hadronic matter is expected to undergo deconfinement into quark matter — quarks and gluons no longer bound into hadrons. This transition is observed in ultrarelativistic heavy-ion collisions, but those collisions probe hot, nearly baryon-free matter. Compact stars probe the complementary, still unexplored regime: cold matter at several times nuclear saturation density (nuclear saturation density $n_0\approx0.16$ fm$^{-3}$, the density inside ordinary nuclei). Inside neutron stars — the densest directly observable stellar objects — the central density may reach several $n_0$, exactly where QCD predicts deconfinement could occur, and no terrestrial experiment or first-principles computation currently settles what phase of matter is realized there.

A quark star (or strange star) is a hypothetical compact star made entirely of deconfined quark matter. The relevant possibility, going back to Bodmer and Witten, is strange quark matter: a roughly equal mixture of up, down, and strange quarks that, if absolutely stable (lower energy per baryon than iron-56), would be the true ground state of baryonic matter; a star made of it would be self-bound (held together by the strong interaction rather than gravity alone), in contrast to neutron stars, which are gravitationally bound and would disperse without gravity. A hybrid star is an intermediate possibility: a neutron star with a deconfined quark-matter core surrounded by hadronic matter.

The properties of a compact star follow from its equation of state (EOS): the relation $P(\varepsilon)$ between pressure $P$ and energy density $\varepsilon$ of the matter, inserted into the Tolman–Oppenheimer–Volkoff (TOV) equations of general relativistic hydrostatic equilibrium, which yield the stellar mass–radius curve. The key microscopic input is thus the EOS of cold (zero-temperature) dense QCD matter at baryon chemical potentials of order 1 GeV. Here a fundamental obstruction stands: lattice QCD, the controlled first-principles numerical method for QCD, fails at nonzero baryon chemical potential because of the fermion sign problem (the path-integral weight is no longer positive), and this has persisted for decades. At low density (below roughly $2n_0$) chiral effective field theory — an effective description of interacting nucleons and pions — provides controlled constraints, and at very high density (above roughly $40n_0$) weak-coupling perturbative QCD does; the intermediate regime where any quark star or quark core would live is accessible to neither, so it is bridged by phenomenological models.

The MIT bag model is the classic such model: quarks are treated as free inside a region of perturbative vacuum, while confinement is represented by a pressure $B$ (the bag constant) that the vacuum exerts on the quark phase — equivalently, the energy-density difference between the perturbative and true QCD vacua. $B$ is not calculable reliably from first principles and is fitted phenomenologically; different acceptable values span a range (in the source paper, $B=(65.5\ \mathrm{MeV})^4$ to $(108\ \mathrm{MeV})^4$; other models use up to $(164\ \mathrm{MeV})^4$), and stellar masses and radii are sensitive to it. Modern refinements include model EOSs built from Dyson–Schwinger equations (a continuum nonperturbative framework that resums quark–gluon interactions; the source paper belongs to this program) and density-dependent bag constants. The source paper additionally simplified its stellar models by omitting chemical and thermodynamic equilibrium, electric charge neutrality, any hadron–quark phase transition, and crust structure, and by treating the three quark flavors symmetrically (neglecting a distinct finite strange-quark mass) — all of which affect the mass–radius relation and observable predictions.

Observationally, several channels now constrain the EOS: two-solar-mass pulsars (measured by radio timing, using the Shapiro delay — the extra propagation time that radio pulses suffer in the gravitational field of the pulsar's companion star) demand a sufficiently stiff EOS; the gravitational-wave event GW170817 constrained the tidal deformability (the deformability $\Lambda$ of a star in an external tidal field, measurable from the gravitational-wave signal of a neutron-star merger), bounding the radius of a $1.4M_\odot$ star; and pulse-profile modeling with NICER (the Neutron star Interior Composition Explorer, an X-ray timing telescope mounted on the International Space Station) measures masses and radii jointly. Reviews of the field (e.g. Baym et al. 2018) and state-of-the-art Bayesian analyses (Annala et al. 2018, 2020) combine these with the microscopic limits, but the composition question — hadronic, hybrid, or self-bound quark matter — remains degenerate: many observables, including mass–radius curves, are similar across these families. Proposed discriminators include a shift of the dominant postmerger gravitational-wave frequency away from its empirical relation with tidal deformability, which simulations indicate occurs only if a strong first-order phase transition (a discontinuous transition with latent heat) forms an extended quark-matter core in the merger remnant, and asteroseismological, cooling, and glitch signatures.

Problem Statement

The problem, in the source's own formulation and the field's current form of it, is threefold and hierarchical — (a) and (b) are the principal inputs to the overarching objective (c):

(a) Determine, or tightly constrain, from QCD the equation of state of cold (zero-temperature) quark matter at baryon chemical potentials corresponding to a few times nuclear saturation density — including the vacuum-pressure contribution, the energy-density offset that appears as the bag constant $B$ in MIT-bag and Dyson–Schwinger models, where it is treated as a phenomenological input. Equivalently stated in the source's terms: compute the bag constant $B$ from first principles, or constrain it from microphysical arguments tightly enough that the stellar mass–radius relation becomes insensitive to its precise value.

(b) Use such an EOS, together with chemical and thermodynamic equilibrium, electric charge neutrality, the finite strange-quark mass, a possible hadron–quark phase transition, and crust/stratified structure, to compute realistic stellar models — mass–radius curves, tidal deformabilities, and stability limits — for self-bound quark (strange) stars and for hybrid stars with quark-matter cores.

(c) The overarching objective: establish whether deconfined quark matter actually occurs in compact stars — as quark-matter cores or as entire self-bound quark stars — and provide an unambiguous way to distinguish quark stars from neutron stars observationally; or, alternatively, rigorously exclude the possibility (for example by demonstrating from QCD that strange quark matter is not absolutely stable, or by showing that no self-bound or hybrid configuration is compatible with all observational constraints).

An answer settles the question only if the existence/nonexistence conclusion is model-independent in the sense that it does not hinge on one fitted EOS family: a positive identification must survive a quantitative account of alternative hadronic explanations, and an exclusion must hold across the EOS families allowed by chiral effective field theory at low density, perturbative QCD at high density, and all accepted astronomical constraints (two-solar-mass pulsars, NICER radii, GW170817 tidal deformabilities). No restriction to any particular EOS model, bag-model variant, stellar mass range, or observation channel is imposed; these are the delivery routes, not the problem.

References

1. Hua Li, Xin-Lian Luo, Hong-Shi Zong, 'Bag model and quark star', Physical Review D 82, 065017 (2010). DOI: 10.1103/PhysRevD.82.065017; arXiv:1008.5019; https://doi.org/10.1103/PhysRevD.82.065017
2. Hua Li, Xin-Lian Luo, Yu Jiang, Hong-Shi Zong, 'Model Study of a Quark Star', Physical Review D 83, 025012 (2011). DOI: 10.1103/PhysRevD.83.025012; arXiv:1101.1744; https://doi.org/10.1103/PhysRevD.83.025012
3. Gordon Baym, Tetsuo Hatsuda, Toru Kojo, Philip D. Powell, Yifan Song, Tatsuyuki Takatsuka, 'From hadrons to quarks in neutron stars: a review', Reports on Progress in Physics 81, 056902 (2018). DOI: 10.1088/1361-6633/aaae14; arXiv:1707.04966; https://doi.org/10.1088/1361-6633/aaae14
4. Eemeli Annala, Tyler Gorda, Aleksi Kurkela, Aleksi Vuorinen, 'Gravitational-wave constraints on the neutron-star-matter Equation of State', Physical Review Letters 120, 172703 (2018). DOI: 10.1103/PhysRevLett.120.172703; arXiv:1711.02644; https://doi.org/10.1103/PhysRevLett.120.172703
5. Eemeli Annala, Tyler Gorda, Aleksi Kurkela, Joonas Nättilä, Aleksi Vuorinen, 'Evidence for quark-matter cores in massive neutron stars', Nature Physics 16, 907–910 (2020). DOI: 10.1038/s41567-020-0914-9; arXiv:1903.09121; https://doi.org/10.1038/s41567-020-0914-9
6. Andreas Bauswein, Niels-Uwe F. Bastian, David B. Blaschke, Katerina Chatziioannou, James A. Clark, Tobias Fischer, Micaela Oertel, 'Identifying a first-order phase transition in neutron star mergers through gravitational waves', Physical Review Letters 122, 061102 (2019). DOI: 10.1103/PhysRevLett.122.061102; arXiv:1809.01116; https://doi.org/10.1103/PhysRevLett.122.061102
7. Tyler Gorda, Aleksi Kurkela, Risto Paatelainen, Saga Säppi, Aleksi Vuorinen, 'Soft Interactions in Cold Quark Matter', Physical Review Letters 127, 162003 (2021). DOI: 10.1103/PhysRevLett.127.162003; arXiv:2103.05658; https://doi.org/10.1103/PhysRevLett.127.162003
8. Salil Joshi, Sovan Sau, Soma Sanyal, 'Quark cores in extensions of the MIT Bag model', Journal of High Energy Astrophysics 30, 16–23 (2021). DOI: 10.1016/j.jheap.2021.03.001; arXiv:2002.07647; https://doi.org/10.1016/j.jheap.2021.03.001
9. J. Sedaghat, G. H. Bordbar, S. M. Zebarjad, 'Constraint on the equation of state of strange quark star: Perturbative QCD along with a density-dependent bag constant', Annals of Physics 472, 169864 (2025). DOI: 10.1016/j.aop.2024.169864; arXiv:2411.12048; https://doi.org/10.1016/j.aop.2024.169864