Hybrid stars with hyperons: structure based on QCD sum rule coupling constants

This paper presents a comprehensive study of hybrid stars composed of hadrons, leptons, and quarks within a relativistic mean-field framework, utilizing QCD sum rule-derived coupling constants to construct equations of state for both hadronic and quark phases and subsequently predicting mass-radius relations, tidal deformabilities, and particle distributions for comparison with multimessenger astrophysical observations.

The Gist

Imagine the universe as a giant kitchen, and inside it, there are tiny, incredibly heavy "kitchen appliances" called neutron stars. These are the leftover cores of massive stars that have exploded. They are so dense that a single teaspoon of their material would weigh as much as a mountain on Earth.

For decades, scientists have been trying to figure out exactly what's happening inside these stellar appliances. The big question is: What is the "recipe" for the matter inside them?

The Mystery of the "Hyperon Puzzle"

Think of a neutron star like a crowded dance floor. At first, it's just filled with dancers called nucleons (protons and neutrons). But as the crowd gets denser and the music gets louder (pressure increases), new dancers try to join the party. These new dancers are called hyperons.

Here's the problem: In previous recipes, when these hyperons joined the dance, the floor got "squishy." The star couldn't hold its shape against gravity, and it would collapse. But we know from looking at the sky that some of these stars are huge and heavy (over twice the mass of our Sun). They should have collapsed if the recipe included hyperons, but they didn't. This contradiction is called the "Hyperon Puzzle."

The New Recipe: Listening to the "Ghost" of Quarks

The authors of this paper decided to rewrite the recipe. Instead of guessing the rules of how these particles interact (like guessing the ingredients of a cake without tasting it), they used a method called QCD Sum Rules.

Think of QCD (Quantum Chromodynamics) as the fundamental "physics of the universe" that governs how the tiniest building blocks (quarks) stick together. QCD Sum Rules are like a special listening device. Instead of guessing, the scientists used this device to "listen" to the whispers of the quantum vacuum to determine exactly how strongly the particles should interact.

By using these "listened-to" rules, they built a new model for the neutron star's interior.

The Two Ways the Party Can Change

The paper explores two different ways the matter inside the star can change from "normal" dancers (hadrons) to "freed" dancers (quarks):

1. The Gibbs Construction (The Smooth Transition): Imagine the dance floor slowly changing from a wooden floor to a rubber floor. The dancers gradually shift their style. There is a "mixed phase" where both types of dancers are on the floor at the same time, blending together smoothly.
2. The Maxwell Construction (The Sharp Cliff): Imagine the dance floor suddenly turning into a different material instantly. One moment you are on wood, the next you are on rubber. There is a sharp, hard line between the two, with no mixing in between.

What They Found

Using their new "listened-to" rules, the scientists ran simulations to see if these stars could survive.

• The Smooth Transition Wins: When they used the Gibbs (smooth) method, the stars held up perfectly! Even with the "squishy" hyperons joining the party, the star remained stable and heavy enough to match what we see in the sky (over 2 solar masses). The smooth mixing of matter acted like a shock absorber, preventing the star from collapsing.
• The Sharp Cliff Struggles: When they used the Maxwell (sharp) method, the stars were less stable. Only the "stiffest" version of this recipe could support a heavy star. If the recipe was slightly softer, the star would collapse under its own weight.

The "Squeeze Test" (Tidal Deformability)

The paper also checked how these stars would react if squeezed by a neighbor (like during a gravitational wave event). They calculated a number called tidal deformability.

• Their result was around 800.
• This is right at the edge of what was observed in a famous cosmic collision (GW170817). It suggests their star is "stiff" (hard to squeeze), which is good for keeping the star from collapsing, but it's a tightrope walk regarding the observational limits.

The Bottom Line

The paper claims that by using a method that connects the star's behavior directly to the fundamental laws of quarks (QCD Sum Rules), they found a way to solve the Hyperon Puzzle.

They showed that if the transition between normal matter and quark matter is smooth (Gibbs), the star can be heavy and stable, even with hyperons inside. This proves that the "squishiness" of hyperons doesn't have to be the end of the line for massive neutron stars; it just depends on how the matter inside them is allowed to mix.

In short: The universe's heaviest stars can exist, provided the "dance floor" inside them changes gradually rather than abruptly.

Technical Summary

Technical Summary: Hybrid Stars with Hyperons Based on QCD Sum Rule Coupling Constants

Problem Statement
The internal composition of neutron stars at supra-nuclear densities remains a significant open issue in nuclear astrophysics, particularly regarding the "hyperon puzzle." While the emergence of hyperons (such as $\Lambda$ and $\Xi^-$) in the dense cores of neutron stars is energetically favored, their inclusion in standard equations of state (EOS) typically softens the EOS, leading to maximum stellar masses below the $\sim 2 M_\odot$ threshold observed in massive pulsars (e.g., PSR J0740+6620, PSR J1614–2230). Furthermore, the coupling constants governing baryon-meson interactions in relativistic mean-field (RMF) models are often treated as purely phenomenological parameters, lacking a direct connection to the underlying Quantum Chromodynamics (QCD) dynamics. This work addresses the need for a microscopic foundation for these couplings and investigates whether a QCD-constrained framework can resolve the hyperon puzzle while remaining consistent with multimessenger astrophysical observations, including gravitational wave constraints on tidal deformability.

Methodology
The authors construct a comprehensive model of hybrid stars (composed of hadrons, leptons, and quarks) within a relativistic mean-field (RMF) framework. The methodology proceeds through several key stages:

1. Hadronic Sector: The hadronic phase is modeled using the $\sigma-\omega-\rho$ model. A distinctive feature of this approach is that the baryon-meson coupling constants ($g_{BB\sigma}, g_{BB\omega}, g_{BB\rho}$) are not fitted to nuclear saturation properties but are derived directly from QCD sum rules (QCDSR). The study focuses on nucleons ($n, p$) and the hyperons $\Lambda$ and $\Xi^-$, excluding $\Sigma$ hyperons due to experimental evidence of their repulsive potentials and the lack of direct QCDSR calculations for their couplings.
2. Quark Sector: The deconfined quark phase is described using two distinct models: the MIT bag model (with bag constants $B = 60, 80, 100$ MeV fm$^{-3}$) and the Nambu–Jona-Lasinio (NJL) model (using RKH, HK, and LKW parameter sets).
3. Phase Transition: The transition from hadronic to quark matter is analyzed using two limiting constructions:
   • Maxwell Construction: Assumes local electric charge neutrality in each phase, resulting in a sharp, first-order phase transition with a constant-pressure plateau.
   • Gibbs Construction: Assumes global electric charge neutrality, allowing for a mixed phase where hadronic and quark matter coexist over a finite density interval.
4. Stellar Structure: The Tolman-Oppenheimer-Volkoff (TOV) equations are solved numerically to determine the mass-radius ($M-R$) relations. The study also calculates the dimensionless tidal deformability ($\Lambda$) and the tidal Love number ($K_2$) for a $1.4 M_\odot$ star to compare with gravitational wave data (e.g., GW170817).

Key Contributions and Results

• Microscopic Consistency: The QCDSR-derived coupling constants successfully reproduce empirical nuclear matter properties at saturation density (saturation density $\rho_0 \approx 0.153$ fm$^{-3}$, binding energy $E/A \approx -16.2$ MeV, symmetry energy $a_{sym} \approx 28$ MeV) and hypernuclear single-particle potentials ($U_\Lambda \approx -30$ MeV, $U_{\Xi^-} \approx -10$ MeV) without tuning the couplings to these specific observables. This establishes a direct link between quark-gluon dynamics and macroscopic stellar properties.
• Resolution of the Hyperon Puzzle: The study demonstrates that stable hybrid stars with maximum masses exceeding $2 M_\odot$ can be achieved even in the presence of hyperons.
  • Gibbs Construction: All Gibbs configurations (both MIT bag and NJL models) yield maximum masses above the $2 M_\odot$ threshold. The extended mixed phase provides sufficient pressure support to counteract the softening effect of hyperons. The maximum mass for the Gibbs + MIT ($B=60$) case is $2.18 M_\odot$.
  • Maxwell Construction: The results are more restrictive. Only the Maxwell + MIT case with $B=60$ MeV fm$^{-3}$ supports $M_{max} > 2 M_\odot$ ($2.03 M_\odot$). For higher bag constants ($B=80, 100$), the maximum mass drops below $2 M_\odot$ due to the softening of the EOS and the abrupt density jump at the phase boundary. Notably, the Maxwell construction could not be realized for the NJL model within the tested parameter sets, as the quark pressure remained below the hadronic pressure.
• Internal Structure and Composition:
  • In the Gibbs scenario, the transition is smooth, with quark volume fractions increasing continuously. Hyperons ($\Lambda, \Xi^-$) appear at intermediate densities, and quark degrees of freedom coexist with hadrons over a finite radial interval.
  • In the Maxwell scenario, the transition is discontinuous, creating a sharp interface between a pure hadronic envelope and a pure quark core.
• Tidal Deformability: For a $1.4 M_\odot$ star, the central density in all models lies below the phase transition onset. Consequently, the tidal deformability is determined solely by the hadronic EOS. The calculated value is $\Lambda_{1.4} \approx 800$. This value sits at the upper bound of the constraints inferred from GW170817 ($\Lambda_{1.4} \le 800$ at 90% confidence) but is consistent with theoretical predictions for relatively stiff hadronic equations of state.

Significance
The paper claims that its primary significance lies in demonstrating that a unified, microscopic framework anchored in QCD sum rules can simultaneously satisfy low-energy nuclear constraints, reproduce the existence of massive neutron stars ($>2 M_\odot$), and remain consistent with current multimessenger observations. By employing QCDSR-derived couplings, the study reduces the arbitrariness of phenomenological models and offers a viable path to alleviating the hyperon puzzle. The results suggest that the choice of phase transition mechanism (Gibbs vs. Maxwell) and the specific quark matter model are critical factors in determining the stability and maximum mass of hybrid stars. The findings provide quantitative predictions for the mass-radius relation and tidal deformability that can be tested against future astrophysical observations.