Hybrid Star Properties with NJL and MFTQCD Model: A Bayesian Approach

This study employs a Bayesian approach to generate hybrid star equations of state combining hadronic and quark matter models, demonstrating that vector and multiquark interactions are essential for supporting massive neutron stars consistent with NICER and pQCD constraints while predicting a positive trace anomaly in their cores.

The Gist

Imagine a neutron star as the ultimate cosmic pressure cooker. It's a dead star that has collapsed so tightly that a single teaspoon of its material would weigh as much as Mount Everest. Inside this pressure cooker, the rules of physics get weird. We know the outer layers are made of atoms (protons and neutrons), but what happens in the very center, where the pressure is so intense it might crush atoms apart?

This paper is like a team of cosmic detectives trying to solve the mystery of what's inside these stars. They are asking: Does the core turn into a soup of free-floating quarks (the building blocks of protons and neutrons), or does it stay as crushed-up atoms?

Here is the story of their investigation, broken down into simple concepts:

1. The Two Suspects: "The Hadron" and "The Quark"

Think of the matter inside a neutron star as having two possible identities:

• The Hadron Phase: This is the "normal" state where protons and neutrons are still distinct, like individual people in a crowded room holding hands.
• The Quark Phase: This is the "deconfined" state where the pressure is so high that the "hands" are broken, and the people (quarks) are running free in a chaotic, super-dense soup.

The scientists wanted to see if a neutron star could be a Hybrid Star—a mix of both. Maybe the outer layers are "people holding hands" (Hadrons), but the deep core is a "quark soup."

2. The Detective Tool: The Bayesian "Gamble"

Instead of guessing one specific answer, the team used a method called Bayesian Inference. Imagine you are trying to guess the weight of a mystery box.

• Old way: You guess one number.
• This paper's way: You generate thousands of possible weights based on what you know, then check them against real-world clues. If a guess doesn't fit the clues, you throw it away. If it fits, you keep it.

They ran a massive computer simulation (a "Markov-Chain Monte Carlo") to generate 8,000 different scenarios for how matter behaves. They then filtered these scenarios to see which ones matched real observations from space.

3. The Clues from Space

To filter their thousands of guesses, they used two main types of clues:

• The "NICER" Clues: NASA's NICER telescope acts like a cosmic scale and ruler. It measures the mass and size of specific neutron stars (like PSR J0030+0451). If a scientist's model predicts a star that is too heavy or too small, that model is wrong.
• The "pQCD" Clues: This is a bit like a physics speed limit. At extremely high densities (far beyond what we can test in a lab), our best math (Perturbative QCD) tells us how matter should behave. The scientists checked if their "quark soup" models obeyed these universal speed limits.

4. The Two Models of the "Quark Soup"

To describe the quark phase, they used two different "recipes" (mathematical models):

• The NJL Model: Think of this as a recipe where the quarks interact with each other in complex ways, like a dance where they sometimes pair up and sometimes form groups of four or eight.
• The MFTQCD Model: This is a different recipe, more like a simplified version of the famous "MIT Bag Model," where the quarks are trapped in a bag of energy.

5. The Big Findings

After running the simulations and filtering out the impossible scenarios, here is what they discovered:

• Hybrid Stars are Possible: The data supports the idea that neutron stars can have a core made of free quarks. They aren't just crushed atoms all the way through.
• The "Speed Limit" Matters: When they applied the "pQCD" speed limit (the physics rule about how fast sound can travel in the star), it forced the models to be more realistic. It lowered the maximum possible weight of these stars slightly, but they could still be very heavy (about 2 to 2.3 times the mass of our Sun).
• The "Stiffness" of the Core: To support a star that heavy, the core needs to be "stiff" (resistant to being squished). The NJL model needed a specific "glue" (a vector interaction) to keep the star from collapsing. Without it, the star would be too squishy and would fall in on itself.
• Size Matters: The MFTQCD model was better at explaining a specific, very small and light neutron star (HESS J1731-347). It suggested that some stars could be surprisingly small (under 12 km wide) if they have a quark core.
• No "Perfect" Soup: Even in the center of the heaviest stars, the quark matter wasn't "perfect" or "conformal" (a theoretical state where particles don't interact much). It was still a very sticky, interacting soup.

The Takeaway

This paper is a success story of combining theory (mathematical models of quarks) with observation (telescope data).

They didn't just say, "Quarks are there." Instead, they said, "If we assume quarks are there, and we use these specific rules of physics, we can create a universe of possible stars that perfectly matches what we see in the sky."

It's like saying, "We don't know exactly what's in the box, but if we assume it's made of quarks, we can build a box that fits perfectly under the doorframe of the universe." The existence of these Hybrid Stars is now a very strong possibility in the world of astrophysics.

Technical Summary

1. Problem Statement and Context

The internal composition of neutron star (NS) cores remains a major open question in nuclear astrophysics. While perturbative QCD (pQCD) predicts deconfined quark matter at extremely high densities, the intermediate density regime (relevant for NS cores, $\sim 1\text{--}10\rho_0$) is difficult to study due to the sign problem in Lattice QCD.

• The Challenge: Determining whether neutron stars contain a "quark core" (hybrid stars) and identifying the specific Equation of State (EOS) that describes the transition from hadronic to quark matter.
• The Gap: Previous studies often relied on "agnostic" (model-independent) EOS descriptions. This paper aims to bridge the gap by using microscopic, physically justified models for both the hadronic and quark phases to test the existence of hybrid stars against observational data.

2. Methodology

The authors employ a Bayesian inference framework using Markov-Chain Monte Carlo (MCMC) sampling (specifically the nested sampling algorithm via PyMultinest) to generate ensembles of hybrid EOSs that satisfy observational constraints.

A. Theoretical Models

The study constructs hybrid EOSs by combining a hadronic phase with a quark phase, connected via a Maxwell construction (first-order phase transition).

1. Hadronic Phase:

   • Based on the Relativistic Mean Field (RMF) model with non-linear meson terms.
   • Two extreme EOS sets from previous work [25] are used to bound the parameter space:
     • BMPF 220: A "soft" EOS.
     • BMPF 260: A "stiff" EOS.
   • Hyperons are excluded in this specific study to isolate the effects of the quark phase.

2. Quark Phase (Two Models):

   • Nambu-Jona-Lasinio (NJL) Model: An $SU(3)$ chiral symmetric model including 4-quark and 8-quark interaction terms. It incorporates vector interactions and a bag constant ($B$) to control deconfinement.
   • Mean Field Theory of QCD (MFTQCD): Derived from the QCD Lagrangian by decomposing gluon fields into soft (condensates) and hard (mean field) components. It resembles a MIT bag model but includes a vector term derived from hard gluon contributions.

B. Constraints and Priors

The Bayesian analysis imposes the following constraints to filter the generated EOS ensembles:

• NICER X-ray Observations: Mass and radius measurements for PSR J0030+0451 and PSR J0740+6620.
• Phase Transition Density: Constrained to occur between $0.15$ and $0.40 \, \text{fm}^{-3}$ (above nuclear saturation density).
• pQCD Constraints: For half of the generated sets, the EOS is required to satisfy perturbative QCD calculations at high density ($\rho_B = 7\rho_0$), ensuring causality and thermodynamic consistency.
• Priors: Uniform distributions are applied to model parameters (coupling constants, bag constant $B$, etc.).

3. Key Contributions

• Microscopic Hybrid EOS Generation: Unlike agnostic approaches, this work generates EOSs based on specific microscopic interactions (NJL with multi-quark terms and MFTQCD), allowing for a direct interpretation of the underlying physics.
• Impact of pQCD Constraints: The study rigorously quantifies how imposing high-density pQCD constraints affects the maximum mass and speed of sound in hybrid stars, particularly for the NJL model.
• Analysis of Conformality Indicators: The paper evaluates proposed indicators for deconfined matter, specifically the renormalized trace anomaly ($\Delta = 1/3 - P/\epsilon$) and the conformality measure ($d_c = \sqrt{\Delta^2 + \Delta'^2}$), within the context of these microscopic models.

4. Key Results

A. Compatibility with Observations

• Hybrid stars are fully compatible with current observational data (NICER, GW170817).
• Maximum Mass ($M_{max}$): Even with pQCD constraints, the models support stars with $M_{max} \approx 2.1 - 2.3 \, M_\odot$.
  • The pQCD constraint reduces $M_{max}$ more significantly for the NJL model (due to its 8-quark vector term) than for MFTQCD.
• Radius:
  • MFTQCD can reproduce smaller radii ($< 12$ km) for medium-mass stars, potentially compatible with the compact object HESS J1731-347.
  • NJL (especially with the stiff hadronic EOS) predicts larger radii ($\sim 13$ km) for massive stars.

B. Phase Transition Characteristics

• Transition Density:
  • MFTQCD: Allows transitions at lower densities (near saturation density, $\sim 0.15\text{--}0.25 \, \text{fm}^{-3}$).
  • NJL: Transitions occur at higher densities ($\gtrsim 0.25 \, \text{fm}^{-3}$), typically above $2\rho_0$.
• Speed of Sound ($c_s^2$):
  • In the NJL model, the 8-quark interaction term ($\xi_{\omega\omega}$) is crucial for reaching $2M_\odot$ but can lead to superluminal speeds of sound if not constrained by pQCD.
  • The pQCD constraint forces $\xi_{\omega\omega}$ to lower values, reducing $M_{max}$ but ensuring causality.
  • MFTQCD predicts a smoother, weaker increase in $c_s^2$ after the phase transition compared to the sharp dip and rise seen in NJL.

C. Conformality and Trace Anomaly

• Trace Anomaly ($\Delta$): The study finds sets where the renormalized matter trace anomaly is always positive, challenging the idea that $\Delta$ must be negative in the deconfined phase.
• Conformality Limit ($d_c$): The conformal limit ($d_c < 0.2$) is not attained at the center of the most massive stars in these models.
  • The NJL model's 8-quark vector interactions can cause $d_c$ to rise above 0.2 even in the deconfined phase, indicating that the matter remains strongly interacting rather than conformal.
  • This suggests that the presence of a quark core does not necessarily imply the system has reached the conformal limit.

5. Significance and Conclusion

This work demonstrates that hybrid stars with quark cores are a viable solution consistent with current astrophysical data, provided the quark matter is described by realistic microscopic models with strong vector interactions.

• Vector Interactions: They are essential for supporting $2M_\odot$ stars with large quark cores.
• Model Dependence: The choice of quark model (NJL vs. MFTQCD) significantly impacts predicted radii and transition densities. MFTQCD favors earlier transitions and smaller radii, while NJL favors later transitions and larger radii.
• Conformality: The results challenge the hypothesis that the conformal limit is easily reached in NS cores; instead, quark matter in these models remains strongly interacting.
• Future Directions: The authors plan to extend this Bayesian approach to the hadronic phase to fully map the EOS parameter space and compare the probability of hybrid stars versus purely hadronic stars.

In summary, the paper provides a robust, model-based confirmation that neutron stars can harbor deconfined quark matter without violating observational constraints, while highlighting the critical role of multi-quark interactions and pQCD constraints in shaping the properties of these extreme objects.