Multi-Quark Clustering in Neutron-Star Matter from Color-Spin Molecular Dynamics

Abstract: We study the equation of state of neutron-star matter with color-spin molecular dynamics. The calculation includes the internal color and spin degrees of freedom and their time evolution. The matter composition, including strangeness under beta equilibrium, is determined by energy minimization. We find two main trends. First, within the present CSMD framework and under the adopted clustering criterion along the stable neutron-star branch, isolated quark-like configurations do not appear; instead, color-magnetic interactions favor the self-consistent formation of multi-quark clusters. Within the same criterion, the cluster-size distribution is concentrated at quark numbers that are multiples of three, corresponding to integer baryon numbers. Second, the interaction between strange and light quarks has a strong impact on neutron-star radii. This suggests that future radius measurements may help constrain flavor-sector interactions, including those involving strangeness.

• The paper demonstrates that incorporating color-magnetic interactions via CSMD leads to multi-quark clustering, which maintains color confinement in neutron-star interiors.
• The study uses a self-consistent treatment of strangeness and many-body forces to produce EOSs that satisfy heavy neutron star mass and radius constraints.
• The model reveals that flavor-dependent repulsion, particularly in the qK* channel, critically modulates the onset of strangeness and impacts tidal deformability and stellar radii.

Multi-Quark Clustering in Neutron-Star Matter: A CSMD Approach

Introduction and Motivation

This work investigates the structure and properties of dense baryonic matter relevant to neutron star interiors via molecular dynamics incorporating internal color and spin degrees of freedom—termed Color-Spin Molecular Dynamics (CSMD). Addressing persistent open questions in the neutron star EOS, particularly the so-called “hyperon puzzle,” the paper emphasizes the role of quark-level interactions and clustering in the emergence of hadronic and multi-quark configurations at supranuclear densities.

The study advances previous MD approaches by including self-consistent treatment of strangeness, time-evolution of color and spin, and explicit modeling of spin-dependent (color-magnetic) interactions. Fundamental constraints from heavy pulsar mass measurements ($M_\mathrm{max} \gtrsim 2M_\odot$), tidal deformability, and NS radius observations are used as benchmarks.

Theoretical Framework: Color-Spin Molecular Dynamics

Variational Ansatz and Internal Degrees of Freedom

The system is modeled with a variational ansatz in which each quark is a Gaussian wave packet characterized by time-dependent center coordinates, momenta, and internal color-spin-flavor state variables. This approach moves beyond simple two-body interactions by encoding many-body effects through effective nonlinear terms in the interaction Hamiltonian and by tracking the time evolution of internal color and spin orientations. Strangeness is incorporated through self-consistent minimization under charge neutrality and $\beta$-equilibrium constraints.

Interaction Terms

The Hamiltonian includes:

• Relativistic kinetic energy (with constituent masses fitted to baryon spectra),
• A confining potential and one-gluon-exchange (OGE) term for color interactions,
• Color-magnetic (hyperfine) interactions explicitly dependent on spin and color, which are essential for realistic baryon mass splittings,
• Effective quark-meson couplings representing baryonic force channels via $\sigma$, $\omega$, $\rho$, $\phi$, and, crucially, the novel inclusion of a $K^*$-like vector channel for strange-light ($qs$) quark interactions,
• An effective Pauli potential enforcing Fermi statistics at the mean-field level.

Nonlinear effective factors $\epsilon_i$ encode many-body correlations in the quark-meson couplings, which are required to achieve sufficiently stiff EOSs in the presence of strangeness.

Calibration and Parameter Sampling

Quark masses, various coupling constants, and effective ranges are set to reproduce empirical baryon masses. The uncertain $qK^*$ coupling is systematically varied over five values ($\beta$0) to probe its impact on neutron star structure.

Numerical Results

Equation of State and Saturation Properties

The CSMD reproduces empirical saturation properties when strangeness is neglected, subject to some excess in symmetry energy slope ($\beta$1), though consistent with recent speculations on the possible range of $\beta$2. Including strangeness, the onset density for $\beta$3 quarks increases with the strength of the repulsive $\beta$4 coupling—models with $\beta$5 give onset densities at or above $\beta$6, consistent with phenomenological expectations from hyperonic matter.

The presence of the color-magnetic interaction produces an increasingly attractive contribution at high density, facilitating energetically favorable cluster formation.

Causality, Sound Speeds, and Maximum Mass

All sampled models achieve $\beta$7 due to the incorporated effective many-body repulsion ($\beta$8). However, models with excessive $\beta$9 coupling ($\sigma$0) or without strangeness violate causality (superluminal sound speeds) before reaching peak compactness. Acceptable models fall in the range $\sigma$1.

Mass-Radius and Tidal Deformability Constraints

For sampled $\sigma$2 values, predicted radius and tidal deformability curves are consistent with current observational constraints—radii at $\sigma$3 fall within the range supported by X-ray and GW constraints for $\sigma$4. Models with minimal $\sigma$5 coupling fail to meet lower bounds from heavy pulsar mass-radius pairs, while overly stiff models exceed the upper bounds imposed by numerical relativity and GW data.

The optimal scenario within the chosen parameter grid is $\sigma$6, which delivers an acceptable strangeness onset, respects causality, and closely tracks current mass-radius and tidal deformability constraints.

Clustering: Multi-Quark Correlations and Deconfinement

A key result is the demonstration that the color-magnetic (hyperfine) interaction robustly suppresses isolated quark-like configurations on the stable branch, enforcing clustering of quarks predominantly into color-singlet aggregates of size $\sigma$7 ($\sigma$8), i.e., baryons and multi-baryonic/multi-quark clusters. This contrasts sharply with dynamics omitting color-magnetic effects, which produce a spectrum of cluster sizes and—at high density—isolated deconfined quark configurations.

Color and spin degrees of freedom are entangled within clusters, with cluster color neutrality achieved collectively; the resulting spin alignments within clusters are isotropic. Despite the effective treatment of internal degrees and the use of coarse clustering criteria, the model finds no evidence for deconfinement or isolated free quarks within the density range of stable neutron stars. Instead, matter at high density is best described as an assemblage of correlated multi-quark clusters, consistent with theoretical predictions of sexaquark and strangeon matter.

Theoretical and Astrophysical Implications

The findings illuminate several key facets of neutron-star matter:

• Resolution of the Hyperon Puzzle: Achieving $\sigma$9 stability with strangeness present requires explicit many-body repulsion, implemented here through nonlinearities in the interaction kernels.
• Flavor-Dependent Repulsion: The flavor structure of vector interactions, specifically the $\omega$0 channel, significantly impacts both the onset of strangeness in the core and the macroscopic radius, suggesting that precise future radius measurements may tightly constrain the strength of strange-light interactions at the quark level.
• Clustering and Non-Deconfinement: Color-magnetic interactions dynamically enforce clustering, indicating that even at central densities inside neutron stars, matter may remain confined at the quark level except perhaps in the most extreme cases—shedding light on the conditions for quark deconfinement and possible existence of stable multi-quark states.

Future Directions

Extensions to the approach are warranted:

• A systematic scan of coupling parameters and clustering thresholds,
• More rigorous treatment of spin and color correlations beyond mean-field/statistical ansätze,
• Incorporation of fully relativistic dynamics,
• Explicit modeling of finite-temperature and antiparticle effects,
• Application to observables sensitive to short-range correlations and baryonic substructure, such as nucleon pressure distributions and heavy-ion collisions.

Conclusion

This study establishes, via CSMD, that inclusion of color-magnetic and effective many-body forces is requisite for EOSs with strangeness capable of supporting observed heavy neutron stars. The flavor structure of repulsive interactions modulates both the star’s radius and the threshold for strange matter. Most notably, color-magnetic forces drive self-consistent clustering into color-singlet multi-quark states, precluding the emergence of isolated deconfined quarks in equilibrium neutron-star interiors under current observational constraints. The results thus provide a physically motivated quark-level mechanism for the observed macroscopic stability and composition of neutron stars and direct future observational, experimental, and theoretical avenues for probing the flavor structure and clustering of dense QCD matter.