Density-induced dark-baryon conversion in $\Delta-$admixed hypernuclear neutron stars

Technical Summary: Density-Induced Dark-Baryon Conversion in $\Delta$-Admixed Hypernuclear Neutron Stars

Problem Statement
The internal composition of neutron stars (NSs) at densities exceeding nuclear saturation ($n_0$) remains uncertain. While the existence of $\sim 2 M_\odot$ pulsars constrains the Equation of State (EOS) to be sufficiently stiff, the inclusion of non-nucleonic degrees of freedom—specifically hyperons ($Y$) and $\Delta(1232)$ resonances—typically softens the EOS, potentially violating these mass constraints (the "hyperon puzzle"). Concurrently, there is interest in whether dark matter (DM) can exist within NSs. Previous phenomenological models often assumed a fixed admixture of dark matter or relied on gravitational capture, which is generally insufficient to alter NS observables significantly. Furthermore, models proposing dark baryons ($\chi$) often treat their abundance as an input parameter rather than deriving it from the thermodynamic conditions of the star. This work addresses the gap in understanding how a density-induced conversion of neutrons into a neutral dark baryon $\chi$ interacts with the emergence of conventional non-nucleonic species (hyperons and $\Delta$ resonances) and how this competition affects the macroscopic properties of neutron stars.

Methodology
The authors investigate cold, charge-neutral, and $\beta$-equilibrated neutron star matter containing the baryon octet ($N, \Lambda, \Xi, \Sigma$), the complete $\Delta(1232)$ quartet, and a neutral spin-1/2 dark baryon $\chi$ carrying unit baryon number.

1. Hadronic Framework: The hadronic sector is modeled using a density-dependent covariant density-functional (CDF) theory with the DDME2 parameterization. This framework includes density-dependent meson-baryon couplings ($\sigma, \omega, \rho, \phi$) to describe strong interactions. The model incorporates the complete $\Delta$ quartet and hyperons, with coupling ratios for $\Delta$ baryons treated as free parameters (specifically $R_{\sigma\Delta}, R_{\omega\Delta}, R_{\rho\Delta}$).
2. Dark Sector Interaction: A scalar Higgs portal is included as a potential interaction channel between the dark and hadronic sectors. However, the primary mechanism for dark baryon production is an in-medium conversion channel ($n \leftrightarrow \chi$) motivated by neutron lifetime anomaly studies. This conversion is governed by a mixing parameter $g_c$.
3. Equilibrium Conditions: Unlike fixed-admixture models, the abundance of $\chi$ is determined self-consistently. The system is solved under the conditions of:
   • Total baryon number conservation ($n_B = n_H + n_\chi$).
   • Charge neutrality.
   • Chemical equilibrium, where the onset of $\chi$ occurs when the neutron chemical potential $\mu_n$ exceeds the in-medium dark baryon mass $m^*_\chi$, and equilibrium abundance is established when $\mu_n = \mu_\chi$.
4. Numerical Implementation: The coupled mean-field equations, charge neutrality, and chemical equilibrium constraints are solved simultaneously for total baryon densities. The resulting barotropic EOS is used to integrate the Tolman-Oppenheimer-Volkoff (TOV) equations to determine stellar structure (mass, radius, tidal deformability). The study focuses on benchmark dark baryon masses $m_\chi = 1250, 1300,$ and $1400$ MeV.

Key Contributions

• Self-Consistent Dark Baryon Abundance: The paper moves beyond fixed-mass-fraction assumptions by deriving the dark baryon population directly from chemical equilibrium and baryon conservation, treating $\chi$ as a density-generated species rather than a pre-existing admixture.
• Competition with Non-Nucleonic Species: The study explicitly analyzes the interplay between the appearance of hyperons/$\Delta$ resonances and the dark baryon. It demonstrates that the prior appearance of these conventional non-nucleonic degrees of freedom modifies the neutron chemical potential, thereby delaying the onset of the dark baryon and suppressing its abundance.
• Negligible Higgs Portal Effect: The authors quantitatively demonstrate that for the couplings considered, the scalar Higgs portal contribution to the mean field is numerically negligible. The observed effects are driven almost entirely by the conversion-induced rearrangement of the equilibrium composition, not by Higgs-mediated interactions.
• Mass Constraints on Dark Baryons: The work provides specific constraints on the viability of dark baryons based on the $2 M_\odot$ pulsar mass limit, showing a strong dependence on the dark baryon mass and the $\Delta$-meson couplings.

Results

• EOS Softening and Thresholds: The appearance of $\chi$ softens the EOS by transferring baryon number from the strongly interacting neutron sector to the weakly interacting dark sector. However, the inclusion of hyperons and $\Delta$ resonances further softens the EOS but simultaneously delays the onset of $\chi$. For $m_\chi = 1250$ MeV, the dark baryon onset is delayed from $\sim 2.8n_0$ (in pure nucleonic matter) to $\sim 3.4n_0$ in the full $N+Y+\Delta+\chi$ composition.
• Maximum Mass Constraints:
  • For pure nucleonic matter with $\chi$, maximum masses are $2.00, 2.11,$ and $2.27 M_\odot$ for $m_\chi = 1250, 1300, 1400$ MeV, respectively.
  • When hyperons and $\Delta$ resonances are included, the maximum masses drop significantly to $1.806, 1.899,$ and $2.024 M_\odot$.
  • Consequently, the $m_\chi = 1250$ and $1300$ MeV benchmarks are incompatible with the $2 M_\odot$ observational constraint under the adopted couplings. Only the $m_\chi = 1400$ MeV case marginally satisfies the $2 M_\odot$ requirement.
  • A linear interpolation suggests the $2 M_\odot$ threshold is crossed near $m_\chi \approx 1.38$ GeV for the benchmark scalar coupling $R_{\sigma\Delta} = 1.2$.
• Radial Composition: For the heaviest benchmark ($m_\chi = 1400$ MeV), the dark baryon is absent in canonical ($1.4 M_\odot$) stars because their central densities remain below the conversion threshold. In the most massive stars, $\chi$ is confined to a compact inner core (a few kilometers), surrounded by a broader region of hyperons and $\Delta$ resonances.
• Tidal Deformability: The tidal deformability of canonical ($1.4 M_\odot$) stars is largely insensitive to the dark baryon mass in the heavy regime ($m_\chi \ge 1300$ MeV) because these stars do not reach the conversion threshold. Instead, tidal deformability is more sensitive to the $\Delta$-meson couplings.
• Sound Speed and Adiabatic Index: The onset of $\chi$ produces a pronounced local reduction in the squared sound speed ($c_s^2$) and the adiabatic index ($\Gamma$), signaling a softening of the EOS. Additional structures in these quantities correspond to the sequential appearance of hyperons and $\Delta$ resonances.

Significance and Claims
The paper claims that the viability of density-induced dark baryons in neutron stars is critically dependent on the presence of other non-nucleonic degrees of freedom. The "competition" between the dark sector and conventional hyperons/$\Delta$ resonances is the dominant factor shaping the stellar structure, rather than direct Higgs-mediated interactions.

The authors conclude that for the specific interaction scheme and chemical equilibrium limit adopted, the existence of massive pulsars ($\sim 2 M_\odot$) strongly disfavors lighter dark baryon benchmarks ($m_\chi < 1.38$ GeV). They emphasize that their results are conditional on the assumption of rapid chemical equilibrium ($n \leftrightarrow \chi$) and the specific hadronic interaction model (DDME2 with fixed $\Delta$ coupling ratios). The work suggests that future constraints on dark baryons must account for the complex rearrangement of the stellar composition caused by the simultaneous presence of hyperons and $\Delta$ resonances, rather than treating dark matter as an isolated additive component. The authors note that their findings do not support the free-neutron lifetime anomaly scenario (which requires $m_\chi < m_n$) but rather describe a heavier dark-sector baryon populated only in extreme densities.