Quarkyonic EOS for Dense QCD Matter

• Quarkyonic EOS is a theoretical framework that unifies quark and hadronic degrees of freedom by featuring a stratified Fermi shell structure.
• It employs QCD confinement and chiral symmetry restoration mechanisms to capture the transition between nuclear matter and deconfined quark matter.
• The model provides practical insights into neutron star masses, tidal deformability, and solutions to astrophysical puzzles like the hyperon problem.

The Quarkyonic Equation of State (EOS) describes dense strongly interacting matter by capturing a regime where aspects of both quark and hadronic degrees of freedom coexist, governed by QCD confinement and chiral symmetry. In this framework, quarkyonic matter interpolates between conventional nuclear (hadronic) matter at low densities and deconfined quark matter at asymptotically high densities, with a distinctive thermodynamic structure rooted in both microphysical modeling and phenomenological extensions. The quarkyonic EOS plays a pivotal role in the theoretical modeling of neutron stars, core-collapse supernovae, heavy-ion collisions, and in resolving astrophysical puzzles such as the hyperon problem. Its defining feature is a duality in which quarks populate the deep Fermi sea while nucleons (and possibly hyperons or resonances) reside in a shell-like region near the Fermi surface, naturally enforcing confining effects and embedding the fundamental QCD quantum statistics.

1. Defining Structure and Thermodynamic Formulation

The central physical mechanism underpinning the quarkyonic EOS is the realization of QCD symmetry and confinement at high baryon density. In such matter, the interior of the Fermi sea is filled with nearly-free quarks, while a thin shell of momentum space near the Fermi surface contains baryons, stabilized by nonperturbative confining interactions. This stratified structure can be formalized by partitioning the total baryon density as follows (Kumar et al., 2023, Gao et al., 22 Oct 2024):

$n = n_n + n_p + \frac{n_u + n_d}{3}$

$$n = \frac{g_s}{6\pi^2} \left[ (k_{F_n}^3 - k_{0_n}^3) + (k_{F_p}^3 - k_{0_p}^3) + \frac{ (k_{F_u}^3 + k_{F_d}^3) }{3 } \right]$$

Here, $k_{F_{(n,p)}}$ are the nucleon Fermi momenta, $k_{F_{(u,d)}}$ are the quark Fermi momenta, and $k_{0_{(n,p)}}$ represents a lower integration limit, determined by the QCD scale, marking the onset of quark degrees of freedom. The transition density $n_t$ and the QCD confinement scale $\Lambda$ (or $\Lambda_{\rm cs}$) are essential parameters, fixing, respectively, the baryon density where quarks "drip out" and the momentum cutoff separating quark and baryon domains (Kumar et al., 2023, Dey et al., 4 Jan 2024).

The total energy density and pressure are then written as: $$\varepsilon = \varepsilon_B(n; k\geq k_0) + \varepsilon_{\text{quarks}}$$

$P = P_B(n; k\geq k_0) + P_{\text{quarks}}$

where baryonic ($\varepsilon_B, P_B$) and quark ($\varepsilon_{\text{quarks}}, P_{\text{quarks}}$) contributions are computed using, respectively, a relativistic mean-field (RMF) or effective field-theory approach for baryons and a degenerate Fermi gas for quarks (often including interactions or color superconductivity).

2. Mechanisms of Stiffening and Shell Formation

A salient feature of the quarkyonic EOS is its "stiffening" at high density, required by neutron star mass observations. When the Fermi momenta of nucleons exceed a defined threshold, the energetically-favored configuration is to fill low-momentum states with quarks, while nucleonic (baryon) states are restricted to a shell near the Fermi surface (Gao et al., 22 Oct 2024, Kumar et al., 2023). This rearrangement increases the pressure due to the large degeneracy of color degrees of freedom for quarks and the efficient filling of available phase space.

This structure is encapsulated by the matching condition (Duarte et al., 2020, Gao et al., 22 Oct 2024): $k_F^{(B)} \approx N_c \, k_F^{(Q)}$ with $N_c$ the number of colors. The shell width, $\Delta$, decreases with increasing density and is controlled by the QCD scale ($\Delta \sim \Lambda_{QCD}^3 / (k_F^2)$), ensuring the baryonic shell contains a numerically small but thermodynamically significant fraction of the total particles.

The hard–soft evolution of the EOS—where the EOS stiffens at high densities but remains appropriately soft at intermediate densities to avoid overpredicting radii—is a direct manifestation of this shell–core duality (Duarte et al., 2020, Folias et al., 23 Aug 2024).

3. Comparison with Conventional and Hybrid Models

Traditional bag models parameterize the quark–gluon plasma (QGP) EOS as (Begun et al., 2010): $$\varepsilon(T) = \sigma_{SB} T^4 + B \quad ; \quad P(T) = \frac{\sigma_{SB}}{3} T^4 - B$$ where $B$ is a positive bag constant. However, this form fails to reproduce the suppressed Stefan–Boltzmann limit and proper approach to the conformal limit as seen in lattice QCD simulations (Begun et al., 2010). The quarkyonic EOS introduces a negative bag constant ($B < 0$) and a linear-in-temperature correction to the pressure, parameterized by an integration constant $A$: $p(T) = (\sigma/3) T^4 - B - A T$ This negative bag constant ensures that at low temperature limits, normalized quantities (e.g., $\varepsilon/T^4$) approach their limits from below, matching lattice QCD data (Begun et al., 2010).

Hybrid and excluded-volume models, such as those incorporating a baryon core with an excluded volume correction and dynamically generated quark degrees of freedom, recover the essential hard–soft structure and flexible phase behavior required by multimessenger observations (Duarte et al., 2020). In three-flavor extensions, additional parameters (e.g., flavor-dependent excluded volumes) are introduced to accommodate hyperonic components and electromagnetic/weak equilibrium (Duarte et al., 2020).

4. Chiral Symmetry, Confinement, and QCD Duality

Unified EOS models for quarkyonic matter combine chiral symmetry restoration and confinement via mean-field or parity-doublet frameworks, often leveraging a chiral invariant mass term for both baryons and constituent quarks (Gao et al., 22 Oct 2024, Marczenko et al., 2020). The order parameter for chiral symmetry ($\sigma$) and the parity splitting govern the nucleon mass: $$m_\pm = \frac{1}{2} \left[ \sqrt{(g_1 + g_2)^2 \sigma^2 + 4 m_0^2} \mp (g_1 - g_2)\sigma \right]$$ as $\sigma \to 0$ at high densities, both parity states degenerate to the invariant mass $m_0$, signaling chiral restoration (Gao et al., 22 Oct 2024). The constituent quark mass is simultaneously tied to chiral symmetry through either an explicit relation to the nucleon mass, $M_Q = m_+ / 3$, or a more general interpolation $M_Q = m_+ / w(\sigma)$ (Gao et al., 22 Oct 2024).

Confinement is modeled dynamically with fields (e.g., an auxiliary scalar $b$, which enforces an infrared cutoff for quarks and an ultraviolet cutoff for nucleons) or through explicit momentum-space exclusions. This ensures quarks cannot freely access low-momentum states unless the density crosses the critical threshold for deconfinement (Marczenko et al., 2020).

5. Impact on Astrophysical Observables and Tidal Phenomena

Quarkyonic EOSs are critically assessed through their ability to reproduce neutron star observations such as high maximum mass ($\gtrsim 2 M_\odot$), relatively small radii ($\sim 11–14\,\mathrm{km}$ for a $1.4 M_\odot$ star), and GW/EM-messenger constraints on tidal deformability ($\Lambda$) (Kumar et al., 2023, Dey et al., 4 Jan 2024, Pattnaik et al., 8 Sep 2025). Key results include:

• Maximum masses up to $\sim 2.5–2.8\,M_\odot$ for reasonable transition densities ($n_t \sim 0.3–0.5\,\mathrm{fm}^{-3}$) and moderate confinement scales (Kumar et al., 2023, Pattnaik et al., 8 Sep 2025).
• Speed of sound $c_s^2$ exhibits a spike near the onset of the quarkyonic phase, then saturates at the conformal limit ($c_s^2 \to 1/3$) in the high-density regime (Kumar et al., 2023, Folias et al., 23 Aug 2024).
• Tidal deformabilities for canonical-mass stars ($M = 1.4 M_\odot$) consistent with GW170817 and GW190425 constraints (Kumar et al., 2023, Pattnaik et al., 8 Sep 2025).
• In binary merger simulations, the transition to the quarkyonic regime affects merger time, collapse threshold, post-merger frequencies, and gravitational wave power spectra, providing distinct observational markers (Kumar et al., 2023).

In dark-matter-admixed quarkyonic stars, the EOS can be further tuned by balancing the stiffening due to the quarkyonic phase against the softening introduced by degenerate dark matter components, modeled as additional Fermi gases with their own interaction parameters and couplings (Pattnaik et al., 8 Sep 2025, Dey et al., 4 Jan 2024).

6. Hyperons, Pauli Blocking, and the Hyperon Puzzle

Quarkyonic models incorporating three flavors provide a natural solution to the hyperon puzzle by leveraging the quark substructure of baryons (Fujimoto et al., 30 Oct 2024). As neutrons populate the low-momentum down-quark states, hyperons (which contain down quarks) cannot be formed at low energy due to Pauli blocking in momentum space. The hyperon chemical potential threshold is shifted: $\mu_B^\text{onset} = 2 M_Y - M_N$ delaying their appearance to higher densities ($\sim 5–6\,n_\text{sat}$), and when they do appear, their occupation is restricted to high-momentum states, yielding only a mild softening of the EOS. This maintains the required stiffness for explaining two-solar-mass neutron stars, which is problematic in conventional EOSs with early hyperon onset.

7. Extensions, Constraints, and Multi-Messenger Implications

Recent work generalizes 1D (cold, $\beta$-equilibrated) quarkyonic EOSs into full three-dimensional tables (density, temperature, electron fraction) via Fermi-liquid theory, incorporating thermal corrections, composition effects, and phase transitions by Maxwell construction with hadronic EOSs (e.g., DD2) (Zhu et al., 21 Jun 2025). This allows consistent application in gravitational (TOV) evolution and core-collapse supernova/hydrodynamic simulations, where the phase transition to quarkyonic matter can induce core bounce, drive shock expansion, or alter the relaxation of neutron stars. The hybrid EOSs constructed in this fashion maintain thermodynamic and dynamical consistency across a range of astrophysical scenarios.

The quarkyonic EOS framework thus provides a theoretically robust, phenomenologically versatile approach to modeling dense QCD matter, capturing the essential microscopic duality between baryons and quarks, and offering a compelling explanation for key features of compact star physics and QCD phase structure as probed by both laboratory and astrophysical observations.