Effects of short-range correlations at high densities on neutron stars with and without DM content: role of the repulsive self-interaction

This study demonstrates that incorporating short-range correlations into relativistic hadronic models softens the equation of state and reduces neutron star maximum mass when only quadratic vector self-interactions are present, but stiffens the equation of state and increases maximum mass when fourth-order terms are included, thereby partially compensating for the mass reduction caused by dark matter content while remaining consistent with current astrophysical constraints.

The Gist

Imagine a neutron star as the ultimate cosmic pressure cooker. It's a dead star so dense that a single teaspoon of its material would weigh as much as a mountain. Inside this pressure cooker, protons and neutrons are packed tighter than sardines in a can. For decades, physicists have tried to write the "recipe" (called the Equation of State) for how this matter behaves under such extreme pressure.

This paper investigates two main ingredients in that recipe: Short-Range Correlations (SRC) and Dark Matter (DM).

Here is the breakdown of their findings using simple analogies.

1. The "Crowded Dance Floor" (Short-Range Correlations)

In the old models, physicists imagined the neutrons and protons inside a star were like people standing in a perfectly organized grid, all moving at the same speed limit. They didn't interact much with their immediate neighbors.

However, experiments show that particles actually do something wild: they occasionally bump into each other and form tight, high-speed pairs. This is Short-Range Correlation (SRC).

• The Analogy: Imagine a crowded dance floor. The old model assumed everyone was just swaying gently in place. The new model realizes that occasionally, two dancers grab each other and spin wildly, creating a "high-momentum tail" (a few people moving much faster than the rest).

2. The "Spring" vs. The "Stiff Rod" (The Vector Self-Interaction)

The paper tests two different ways these particles push back against gravity. Think of the force holding the star up as a spring.

• Model A (The Quadratic Spring): This is a standard spring. If you push it, it resists.
• Model B (The Quadratic + Quartic Spring): This is a spring with a special "stiffener" added to it. It gets much harder to compress as you push harder.

3. The Big Surprise: How SRC Changes the Rules

The authors found that adding the "wild dancers" (SRC) changes the star's behavior, but only if you know which type of spring you are using.

• Scenario 1: The Standard Spring (Model A)

  • What happens: When you add the wild dancers (SRC) to this model, the star actually becomes softer and easier to crush.
  • The Result: The star can't hold as much weight. The maximum mass of the star decreases. It's like adding a few rowdy dancers to a fragile dance floor; the floor sags more easily.

• Scenario 2: The Stiffened Spring (Model B)

  • What happens: When you add the wild dancers to the model with the "stiffener" (the fourth-order term), something magical happens. The star becomes stiffer and stronger.
  • The Result: The star can hold more weight. The maximum mass increases. It's like adding those rowdy dancers to a reinforced steel floor; their energy actually helps lock the structure together, making it harder to collapse.

Why does this matter? We have observed neutron stars that are incredibly heavy (about 2 times the mass of our Sun). If we only used the "Standard Spring" model, adding SRC would make these heavy stars impossible to exist. But with the "Stiffened Spring" model, SRC actually helps explain how these massive stars survive.

4. The Ghost in the Machine (Dark Matter)

Now, the authors ask: "What if there is invisible Dark Matter hiding inside the star?"

• The Analogy: Imagine the neutron star is a house made of bricks (visible matter). Dark Matter is like a swarm of invisible ghosts living in the walls.
• The Effect: Usually, adding ghosts (Dark Matter) makes the house weaker. The ghosts don't push back against gravity like the bricks do, so the house collapses at a lower weight. The maximum mass of the star goes down.

5. The Heroic Rescue

Here is the paper's most exciting conclusion:
When they combined the Stiffened Spring model (Model B) with Dark Matter, the "wild dancers" (SRC) came to the rescue!

• The Dark Matter tried to make the star collapse (lowering the max mass).
• But the SRC effects in the Stiffened Spring model made the star stronger.
• The Outcome: The strengthening effect of SRC partially canceled out the weakening effect of the Dark Matter.

It's as if the ghosts tried to weaken the house, but the rowdy dancers started doing push-ups and reinforcing the beams, keeping the house standing taller than it would have otherwise.

Summary for the General Audience

1. Neutron stars are extreme labs where matter is squished to its limit.
2. Short-Range Correlations are like particles forming high-speed pairs, which changes how the star pushes back against gravity.
3. The Twist: Whether these pairs make the star weaker or stronger depends on the specific physics model used. In the most realistic model (which includes a strong repulsive force), these pairs make the star stronger.
4. Dark Matter usually makes neutron stars weaker and smaller.
5. The Conclusion: If neutron stars contain Dark Matter, the "wild pairs" of particles might be the reason they are still able to support such massive weights, keeping them from collapsing into black holes.

This work helps us understand why the heaviest neutron stars we see in the universe exist, and it suggests that if Dark Matter is hiding inside them, it's not destroying them—it's being held in check by the complex dance of the particles inside.

Technical Summary

1. Problem Statement

The internal structure of neutron stars (NSs) at supranuclear densities is governed by the Equation of State (EoS) of dense matter. Standard Relativistic Mean-Field (RMF) models typically approximate nucleons as an independent Fermi gas with a step-function momentum distribution. This approximation neglects Short-Range Correlations (SRC), which are experimentally observed to create a high-momentum tail (HMT) in the nucleon distribution.

Furthermore, the potential presence of Dark Matter (DM) within neutron stars introduces additional complexity. While DM is often modeled as a non-interacting fluid, its accumulation can significantly alter stellar structure. A critical open question is how the inclusion of SRC affects the EoS at high densities, particularly in the context of different vector meson self-interaction terms (quadratic vs. quartic), and how these effects interact with a DM component to satisfy current astrophysical constraints (e.g., $2M_\odot$ pulsars, NICER data, and gravitational wave events).

2. Methodology

A. Hadronic Sector (Relativistic RMF with SRC)

The authors utilize a Lagrangian density for hadronic matter including scalar ($\sigma$), vector ($\omega$), and isovector-vector ($\rho$) mesons. They investigate two specific versions of the vector self-interaction:

1. Quadratic Model ($C=0$): Vector self-interactions up to second order ($\omega^2$).
2. Quartic Model ($C \neq 0$): Includes a fourth-order term ($\omega^4$) to regulate high-density stiffness.

Implementation of SRC:
Instead of a step-function momentum distribution, the nucleon momentum distribution $n(k)$ is modified to include a high-momentum tail ($k > k_F$) proportional to $k^{-4}$. The distribution is defined by:

• A depletion factor $\Delta$ for $k < k_F$.
• A coefficient $C$ and a cutoff factor $\phi$ for the HMT region.
  These parameters are constrained by normalization conditions and empirical trends regarding isospin asymmetry.

The authors derive exact analytical expressions for thermodynamic quantities (energy density, pressure, chemical potentials) incorporating the full density dependence of the SRC parameters, rather than treating them as constants.

B. Dark Matter Sector (Two-Fluid Formalism)

The study adopts a two-fluid formalism where the visible (hadronic) and dark sectors interact only via gravity.

• DM Model: Fermionic DM is modeled as a relativistic Fermi gas with a repulsive vector self-interaction (mediated by a dark vector field $V_\mu$).
• Parameters: Dark fermion mass $m_\chi = 1.9$ GeV and coupling ratio $C_V = 3.26$ fm.
• Equations: The generalized Tolman-Oppenheimer-Volkoff (TOV) equations are solved for coupled fluids, determining mass-radius relations for various Dark Matter fractions ($F_{dm} = M_{dm}/M_{total}$).

C. Constraints and Calibration

• Nuclear Empirical Parameters (NEPs): Models are calibrated to reproduce saturation density ($\rho_{sat}$), binding energy ($E_{sat}$), symmetry energy ($E_{sym}$), and effective mass ($m^*$), using the NL3* parametrization as a baseline.
• Astrophysical Constraints: Results are tested against:
  • NICER and XMM-Newton data for pulsars PSR J0030+0451 and PSR J0740+6620.
  • Gravitational wave event GW190425.

3. Key Contributions

1. Analytical Derivation of SRC Effects: The paper provides exact analytical expressions for the chemical potentials and thermodynamic quantities in the presence of SRC, moving beyond previous approximations that neglected the density dependence of SRC parameters.
2. Duality of SRC Effects: The study demonstrates that the impact of SRC on the EoS stiffness is not universal; it depends critically on the order of the vector self-interaction term:
   • Quadratic Case ($C=0$): SRC softens the EoS.
   • Quartic Case ($C \neq 0$): SRC stiffens the EoS.
3. Compensation Mechanism: The authors show that in models with quartic self-interactions, the stiffening effect of SRC can partially compensate for the softening effect introduced by the presence of Dark Matter, allowing the star to support higher masses despite the DM content.
4. Stability Analysis: A rigorous stability analysis using the two-fluid formalism (checking eigenvalues of the particle number variation matrix) is performed to determine the maximum stable mass for admixed stars.

4. Key Results

A. High-Density Behavior (Pure Neutron Stars)

• $C=0$ Models: The inclusion of SRC increases the coefficient of the non-leading term in the pressure expansion ($\alpha_{SRC} > \alpha$), leading to a softer EoS. Consequently, the maximum mass ($M_{max}$) of the neutron star decreases compared to the standard RMF model.
  • Example: For the NL3* model, $M_{max}$ drops from 2.76 $M_\odot$ to 2.73 $M_\odot$.
• $C \neq 0$ Models: The inclusion of SRC decreases the relevant coefficient ($\beta_{SRC} < \beta$) due to the refitting of coupling constants (specifically $G_\rho^2$) required to maintain symmetry energy constraints. This leads to a stiffer EoS and an increase in maximum mass.
  • Example: For the TMA model, $M_{max}$ increases from 1.99 $M_\odot$ to 2.14 $M_\odot$.

B. Neutron Stars with Dark Matter

• Mass Reduction: In all scenarios, increasing the DM fraction ($F_{dm}$) leads to a linear decrease in the maximum stable mass. This is attributed to the formation of a DM core (since $m_\chi = 1.9$ GeV is heavy), which reduces the overall pressure support.
• Radius Behavior: The stellar radius is determined by the visible (hadronic) sector. The softer EoS (RMF without SRC) results in smaller radii, while the stiffer EoS (RMF-SRC with $C \neq 0$) yields larger radii.
• No Dark Halos: For the specific parameters used, the DM distribution remains confined within the hadronic surface ($R_{dm} < R_{vis}$); no "dark halo" configurations were found.
• Compensation Effect: In the $C \neq 0$ models, the mass increase induced by SRC offsets the mass loss due to DM. For instance, at $F_{dm} = 6\%$, the model with SRC maintains a higher mass than the model without SRC, effectively allowing the star to sustain a larger DM fraction while remaining consistent with the $2M_\odot$ constraint.

C. Consistency with Observations

All parametrizations (with and without SRC, with and without DM) were found to be consistent with:

• The mass and radius constraints from PSR J0030+0451 and PSR J0740+6620.
• The total mass of the binary system in GW190425.

5. Significance

This work resolves a critical ambiguity in nuclear astrophysics regarding the role of Short-Range Correlations. It establishes that SRC do not universally soften the EoS; their effect is contingent on the underlying vector self-interaction structure of the nuclear model.

• Theoretical Impact: It highlights the necessity of including higher-order vector terms ($\omega^4$) to correctly model high-density matter when SRC are present. Ignoring these terms could lead to incorrect predictions of neutron star maximum masses.
• Astrophysical Implications: The finding that SRC can stiffen the EoS in specific models suggests that neutron stars containing Dark Matter might be more massive than previously thought if the nuclear EoS includes quartic vector interactions. This provides a viable pathway for reconciling the existence of heavy pulsars with the potential accumulation of Dark Matter in their cores.
• Future Outlook: The results provide a robust framework for interpreting future data from X-ray missions (e.g., eXTP, STROBE-X) and gravitational wave detectors (e.g., Einstein Telescope), which will further constrain the dense-matter EoS and the nature of Dark Matter interactions.