Quarkyonic Quark-Meson Coupling Model for Nuclear and Neutron Matter

Technical Summary: Quarkyonic Quark-Meson Coupling Model for Nuclear and Neutron Matter

Problem Statement
The paper addresses the "hyperon puzzle" in neutron star physics, where conventional nuclear calculations predict a softening of the Equation of State (EoS) at high densities due to the appearance of new degrees of freedom (e.g., hyperons), contradicting observations of massive neutron stars ($>2M_\odot$) which require a stiff EoS. While the concept of quarkyonic matter—proposed as a phase of soft deconfinement where quark wavefunctions overlap but confinement persists—offers a potential solution, existing models like the ideal dual quarkyonic (IdylliQ) model neglect nuclear interactions. Conversely, standard quark-meson coupling (QMC) models incorporate interactions but often lack a unified framework for the crossover region between hadronic and quark matter. Furthermore, naive quarkyonic models exhibit singular behaviors (discontinuities in chemical potential/pressure and divergences in sound velocity) at the quark saturation density ($\rho_{sat}$), rendering them thermodynamically inconsistent.

Methodology
The authors construct a novel framework, the Quarkyonic Quark-Meson Coupling (QQMC) model, by unifying the dual quarkyonic picture with the QMC model. The methodology proceeds in three stages:

1. Relativistic Gaussian Wavefunction: Unlike the IdylliQ model which uses a Yukawa-type wavefunction, this study adopts a relativistic, Gaussian quark wavefunction derived from a scalar-vector harmonic oscillator (HO) confinement potential. This allows for analytical calculations of nucleon properties, including center-of-mass corrections and spin correlations, to reproduce the nucleon mass and charge radius.
2. Ideal Fermi Gas (GQ Model): The authors first apply this wavefunction to an ideal Fermi gas (the Gaussian Quarkyonic or GQ model) to study Pauli blocking at the quark level without nuclear interactions. They identify that the naive GQ model results in discontinuous thermodynamic quantities at $\rho_{sat}$. To remedy this, they introduce an infrared regulator ($\alpha_\nu$) to smear the sharp Fermi surface in the nucleon momentum distribution, ensuring continuity and smoothness of physical quantities.
3. Unification with QMC (QQMC Model): The GQ model is then combined with the QMC model, which describes nuclear matter via mean fields of $\sigma$, $\omega$, and $\rho$ mesons. In this unified framework, both quark and nucleon degrees of freedom couple to common mean scalar and vector fields. The model assumes duality holds (nuclear density and energy density can be expressed via either nucleon or quark distributions) and calculates physical quantities using the nucleon degrees of freedom with an in-medium effective mass ($M^*_N$). The quark saturation density in matter ($\rho^*_{sat}$) is determined self-consistently, accounting for the swelling of the nucleon due to the attractive scalar field.

Key Contributions

• Novel Model Construction: The paper presents the QQMC model, the first to simultaneously incorporate quark-level Pauli blocking (characteristic of quarkyonic matter) and medium modifications of nucleon structure (scalar polarizability) within a relativistic mean-field framework.
• Resolution of Singularities: By introducing an infrared regulator to the boundary conditions of the momentum distributions, the authors successfully remove the unphysical divergences in sound velocity and discontinuities in chemical potential and pressure at the quark saturation density.
• Relativistic Corrections: The study highlights the importance of using a relativistic wavefunction. The lower component of the Dirac spinor enhances the high-momentum tail, resulting in a higher quark saturation density compared to non-relativistic calculations.
• Parameter Sensitivity: The model demonstrates a strong dependence of $\rho_{sat}$ and other physical quantities on the proton root-mean-square radius ($r_p$).

Results

• Quark Saturation Density: In the ideal GQ model, $\rho_{sat}$ is found to be $1.63\rho_0$ (for $r_p=0.8$ fm), significantly higher than non-relativistic predictions. When nuclear interactions are included in the QQMC model, the nucleon swelling reduces $\rho^*_{sat}$ to approximately $1.46\rho_0$, yet it remains above the nuclear saturation density $\rho_0$, consistent with low-energy nuclear experiments.
• Sound Velocity ($v_s$): The QQMC model successfully reproduces the behavior of the sound velocity inferred from neutron star observations. Specifically, for $r_p \approx 0.6 - 0.8$ fm, the model predicts a rapid rise in $v_s^2$ around $2-4\rho_0$, reaching a maximum consistent with Bayesian inference and neural network analyses of neutron star data, followed by a gradual decrease or saturation. This contrasts with the monotonic increase seen in standard QMC models.
• Pressure Constraints: The pressure calculated for symmetric nuclear matter (SNM) and pure neutron matter (PNM) falls within the constraints deduced from high-energy heavy-ion collision (HIC) flow data and kaon production data.
• Role of Interactions: The inclusion of nuclear interactions is shown to be critical for quantitatively describing the EoS. While the quarkyonic phase enhances pressure and sound velocity, the meson exchanges (particularly $\omega$ and $\rho$) provide the necessary repulsion to match experimental constraints.

Significance and Claims
The authors claim that the QQMC model provides a unified framework valid from low densities to the crossover region where the transition from baryonic to quark matter begins. By integrating quark-level Pauli blocking with medium-induced nucleon modifications, the model offers a realistic description of dense nuclear matter that resolves the hyperon puzzle without violating observational constraints on neutron star masses and radii. The paper emphasizes that the choice of nucleon size ($r_p \approx 0.6-0.8$ fm) is crucial for the model's success in matching astrophysical data.

The authors remain modest regarding future work, noting that the boundary conditions used to determine momentum distributions are currently borrowed from the IdylliQ model and require further rigorous justification. Additionally, the regularization of singularities is treated as an effective parametrization, with the ultimate resolution expected to come from fundamental QCD dynamics. The model is also noted as a potential candidate for extension to finite temperature and hyperonic matter.