Isentropic hybrid stars in the Nambu-Jona-Lasinio model: effects of neutrino trapping

This paper investigates the thermodynamics and structural properties of isentropic hybrid stars using a Nambu-Jona-Lasinio model for quark matter and a covariant density functional for hadronic matter, revealing that neutrino trapping significantly alters particle composition, delays the onset of deconfinement to higher densities, and leads to hybrid stars with larger radii and slightly higher maximum masses compared to their cold, neutrino-free counterparts.

The Gist

Imagine the universe's most extreme weightlifting competition: a Neutron Star. These are the dead cores of massive stars, crushed so tightly that a single teaspoon of their material would weigh a billion tons on Earth. They are essentially giant atomic nuclei, made of protons and neutrons packed together.

But what happens when two of these stars crash into each other, or when a new one is born in a supernova explosion? The environment becomes a chaotic, super-hot, high-pressure soup. This is where the story of the paper comes in.

Here is the breakdown of the research, translated into everyday language with some creative metaphors.

1. The Setting: A Cosmic Pressure Cooker

Normally, neutron stars are cold and calm. But during a collision or a birth, they are like a pressure cooker on maximum heat.

• The Heat: The temperature is millions of degrees.
• The "Trapped" Neutrinos: Neutrinos are ghost-like particles that usually zip right through matter. But in this super-dense soup, they get stuck, like flies in a jar. They can't escape immediately, so they get trapped inside, adding extra pressure and changing the recipe of the star.

2. The Ingredients: From Bricks to Quarks

The scientists are trying to figure out what happens to the "stuff" inside the star under these extreme conditions.

• Hadronic Matter (The Bricks): At lower pressures, the matter is made of protons and neutrons. Think of these as sturdy bricks building a wall.
• Quark Matter (The Soup): If you squeeze hard enough, those bricks might break apart into their smaller components: quarks. This is like smashing the bricks until they turn into a thick, chaotic soup.
• The Hybrid Star: The researchers are studying a "Hybrid Star," which is a mix of both. It has a core of quark soup surrounded by a crust of neutron bricks.

3. The Big Question: When do the bricks break?

The main goal of the paper is to answer: At what point does the "brick wall" turn into "quark soup"?

The scientists used a complex mathematical recipe (called the Nambu-Jona-Lasinio model) to simulate this. They found two major surprises:

Surprise A: The "Trapped Ghosts" Delay the Breakup

Usually, you might think that adding more stuff (like trapped neutrinos) would make the bricks break apart sooner. But the opposite happened!

• The Analogy: Imagine a crowded dance floor (the star). If you add a bunch of people who refuse to leave the dance floor (trapped neutrinos), the crowd gets so packed and organized that it becomes harder to break the formation.
• The Result: The presence of trapped neutrinos makes the "brick wall" (hadronic matter) more stable. It takes more squeezing (higher density) to break the bricks into quark soup than it would if the neutrinos could just float away.

Surprise B: The "Melting Zone" is a Slope, Not a Cliff

In older theories, scientists thought the transition from bricks to soup happened like a cliff: one moment you are solid, the next you are liquid.

• The New View: The paper shows that with trapped neutrinos, the transition is more like a ramp. There is a "mixed phase" where you have a chunk of bricks and a chunk of soup existing together.
• Why it matters: Because the neutrinos are trapped, the pressure doesn't stay flat during this mix; it keeps rising. It's like a ramp that gets steeper the more you mix the ingredients.

4. The Consequences: Bigger, Hotter, and Heavier

What does this mean for the star itself?

• The "Hot" Star is Fatter: A hot, neutrino-rich star is larger (has a bigger radius) than a cold, dead star of the same weight. It's like a hot air balloon; the heat and trapped particles make it puff up.
• The "Cooling" Collapse: As the star cools down over time and the neutrinos finally escape, the star will shrink. It's like a deflating balloon.
• The Internal Shake-up: As the star shrinks, the pressure inside changes. This could trigger a sudden shift where the core suddenly turns from bricks to soup, or vice versa. This is like a building suddenly rearranging its internal structure as it cools, which could cause a "starquake" or a burst of gravitational waves.

5. Why Should We Care?

We live in an era of Multimessenger Astronomy. This means we don't just "see" the universe; we "hear" it (gravitational waves) and "feel" it (neutrinos).

• When two neutron stars collide, they send out ripples in space-time (gravitational waves).
• The shape of these ripples depends on how "squishy" or "stiff" the star is.
• If we don't understand how trapped neutrinos change the "stiffness" of the star, we might misinterpret the signals from these collisions.

The Bottom Line

This paper is like a recipe book for the universe's most extreme kitchens. It tells us that if you cook a neutron star with trapped neutrinos, the ingredients behave differently than we thought. The "bricks" hold together longer, the star puffs up bigger, and the transition to "quark soup" happens more gradually.

By understanding these details, we can better interpret the signals from the most violent events in the cosmos, helping us decode the secret life of the densest matter in the universe.

Technical Summary

1. Problem Statement

The paper addresses the thermodynamic and structural properties of dense matter in extreme astrophysical environments, specifically proto-neutron stars (PNS) and the remnants of binary neutron star (BNS) mergers. In these transient phases, matter is characterized by:

• High temperatures ($T \sim 10\text{--}50$ MeV).
• High lepton fractions due to neutrino trapping (neutrinos cannot escape on dynamical timescales).
• Potential phase transitions from hadronic matter to deconfined quark matter.

Existing studies often focus on cold, neutrino-free neutron stars or assume isothermal conditions. There is a lack of consistent microphysical models that simultaneously treat finite entropy, trapped neutrinos (fixed lepton fraction), and the hadron-quark phase transition using a unified framework that includes color superconductivity. This work aims to fill that gap to better interpret future multimessenger observations (gravitational waves, neutrinos).

2. Methodology

The authors construct a hybrid equation of state (EoS) by combining distinct models for the hadronic and quark phases, linked via a thermodynamically consistent phase transition.

A. Hadronic Phase (Low Density)

• Model: Covariant Density Functional Theory (CDFT).
• Specifics: Uses the DDLS(50) parametrization (based on Ref. [13, 14]), which incorporates the full $JP=1/2^+$ baryon octet.
• Parameters: Symmetry energy slope $L_{sym} = 50$ MeV and skewness $Q_{sat} = 400$ MeV.
• Implementation: Utilizes finite-temperature EoS tables from the CompOSE database, extended to include trapped neutrinos by enforcing $\beta$-equilibrium with a fixed neutrino chemical potential ($\mu_{\nu_e}$).

B. Quark Phase (High Density)

• Model: Extended Nambu–Jona-Lasinio (NJL) model.
• Features:
  • Includes repulsive vector interactions ($G_V$).
  • Includes the 't Hooft determinant interaction ($K$) to break axial $U_A(1)$ symmetry.
  • Incorporates Two-Flavor Color Superconductivity (2SC) pairing ($G_D$) between up and down quarks.
• Parameters: Standard NJL parameters are used, with a bag constant $B^*$ adjusted to set the deconfinement transition density. The vector coupling is set to $\eta_V = G_V/G_S = 1$.

C. Phase Transition Construction

• Approach: Gibbs construction (mixed-phase prescription) rather than the Maxwell construction.
• Rationale: The system possesses two globally conserved charges: baryon number ($B$) and electron lepton number ($L_e$).
• Conditions:
  • Local electric charge neutrality is enforced in each phase.
  • Global conservation of $B$ and $L_e$ is maintained across the mixed phase.
  • Thermal, chemical, and mechanical equilibrium ($T, \mu_B, \mu_{Le}, P$) are satisfied between phases.
• Trajectories: The study follows isentropic trajectories (fixed entropy per baryon, $S = 1, 2$) at fixed lepton fractions ($Y_{Le} \approx 0.4$ for PNS, $0.1$ for merger remnants).

D. Stellar Structure

• Macroscopic properties are derived by integrating the Tolman–Oppenheimer–Volkoff (TOV) equations.
• Low-density matching is performed with the HS(DD2) model for consistency in hot, dense matter simulations.

3. Key Contributions

1. Unified Framework: Development of a thermodynamically consistent EoS for hybrid matter that simultaneously accounts for finite temperature, finite entropy, and neutrino trapping within a 2SC NJL framework.
2. Gibbs vs. Maxwell: Demonstration of how the presence of two conserved charges (baryon and lepton numbers) necessitates a Gibbs construction, leading to a density-dependent pressure in the mixed phase, unlike the constant pressure in a standard Maxwell construction.
3. Neutrino Trapping Effects: Quantification of how trapped neutrinos alter the particle composition and shift the phase transition boundaries compared to neutrino-free scenarios.
4. Isentropic Stellar Sequences: Calculation of static stellar configurations for hot, lepton-rich matter, providing a baseline for understanding the evolution of merger remnants and PNSs.

4. Key Results

Thermodynamics and Phase Structure

• Shift in Onset: Neutrino trapping significantly delays the onset of deconfinement to higher baryon densities. The fixed lepton fraction increases the proton fraction in the hadronic phase, stabilizing it against the formation of quark matter.
• Mixed Phase Behavior:
  • The pressure increases smoothly across the mixed phase (no plateau), determined by the volume fraction of quark matter.
  • Isentropic Curves: In the mixed phase, the ordering of isentropic curves can be partially reversed. Increasing entropy can lower the pressure in the mixed region because quark matter has higher entropy per baryon; increasing $S$ shifts the coexistence conditions to favor the quark phase at lower densities, compressing the mixed region.
• Temperature Drop: Along isentropic trajectories, the temperature decreases as the system enters the mixed phase. This is a thermodynamic consequence of the system gaining new degrees of freedom (quarks) at fixed entropy, requiring a temperature reduction to compensate.

Composition

• Hadronic Phase: Neutrino trapping leads to a significantly higher electron and proton fraction compared to the neutrino-free case.
• Quark Phase: The fixed lepton fraction prevents the strong suppression of electrons seen in neutrino-free matter. Consequently, the system requires a higher abundance of positively charged $u$-quarks to maintain charge neutrality, altering the charge balance of the quark matter.

Stellar Properties (Mass-Radius Relations)

• Radius Expansion: Hot, neutrino-rich configurations have larger radii than their cold counterparts. For a $1.4 M_\odot$ star, the radius can increase by $\sim 1$ km ($S=1$) to several km ($S=2$).
• Maximum Mass: The maximum mass of hybrid stars is slightly higher in the hot, neutrino-trapped regime compared to cold configurations.
• Evolutionary Path: As the star cools and deleptonizes (neutrinos escape), the radius contracts at approximately constant baryonic mass. This contraction can trigger changes in the internal phase structure, potentially causing transitions between different stellar branches (e.g., "twin" stars).

5. Significance

• Multimessenger Astronomy: The results provide critical inputs for interpreting gravitational wave signals from BNS mergers and future neutrino detections from supernovae. The larger radii and modified EoS in the hot, trapped regime affect the tidal deformability and post-merger oscillation frequencies.
• Phase Transition Dynamics: The study highlights that the "softening" or "stiffening" of the EoS during the phase transition is highly sensitive to lepton content and entropy. Ignoring neutrino trapping leads to incorrect predictions for the onset of quark matter.
• Stability and Evolution: The findings suggest that the evolution from a hot, lepton-rich PNS to a cold neutron star involves a significant structural contraction. This dynamic evolution could trigger phase transitions that alter the star's stability or lead to the formation of "twin" stars (stars with similar masses but different radii).
• Model Consistency: By employing a Gibbs construction with 2SC pairing, the paper offers a more realistic description of the QCD phase diagram in astrophysical environments than previous isothermal or Maxwell-construction-based studies.

In conclusion, the paper establishes that neutrino trapping and thermal effects are not minor corrections but fundamental drivers of the structure and evolution of hybrid stars, significantly delaying deconfinement and expanding stellar radii in transient astrophysical events.