⚛️ nuclear theory

Coexistence of strange quark stars and neutron stars: metastability and nucleation in proto-neutron stars

This paper investigates the thermal nucleation of strange quark matter droplets in proto-neutron stars to demonstrate that a hadron-quark surface tension of at least 67 MeVfm $^{-2}$ is required to allow hadronic neutron stars to survive as metastable objects alongside stable strange quark stars in the two-families scenario.

Original authors: Mirco Guerrini, Giuseppe Pagliara, Luca Passarella, Alessandro Drago

Published 2026-10-02

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CC BY 4.0

Original authors: Mirco Guerrini, Giuseppe Pagliara, Luca Passarella, Alessandro Drago

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep in the cosmos, where gravity crushes matter to densities found nowhere else, two distinct types of stellar corpses may exist side by side. One is the familiar neutron star, a city-sized sphere of neutrons packed so tightly that a single teaspoon would weigh a billion tons. The other is a stranger, more exotic possibility: a strange quark star. In this scenario, the neutrons themselves have dissolved, breaking apart into their constituent building blocks—quarks—specifically a mix of up, down, and strange quarks. If this strange matter is truly stable, it would be the ultimate ground state of matter, more stable than the iron found in our own blood or the stars. This creates a fascinating cosmic puzzle: if strange matter is so stable, why haven't all neutron stars instantly transformed into these strange quark stars? Why do we still see ordinary neutron stars, and could both types actually coexist in the universe?

This question lies at the heart of a new study by physicists Mirco Guerrini, Giuseppe Pagliara, Luca Passarella, and Alessandro Drago. They set out to solve the mystery of how a neutron star could survive long enough to become a stable, cold object without immediately turning into a strange quark star. To understand their approach, one must first realize that no star is born cold. Every neutron star begins its life as a "proto-neutron star," a scorching, turbulent object born in the violent explosion of a dying massive star. During this hot, early phase, the conditions inside the star are far more extreme than they will ever be again. If the transformation to strange matter is going to happen, it is most likely to happen right then, while the star is still hot and dense. The researchers asked a simple but critical question: under what specific conditions does this transformation trigger, and is it possible for some stars to avoid it entirely?

The team focused on the mechanics of how this change begins. They imagined the process not as a sudden switch, but as the formation of a tiny seed, or a droplet, of strange matter appearing within the sea of normal matter. For this droplet to grow and take over the whole star, it must reach a certain critical size. If it is too small, surface tension—the force that tries to keep the droplet together—will crush it back into normal matter. If it is large enough, the energy gained by becoming strange matter will outweigh the cost of its surface, and the droplet will expand uncontrollably, converting the entire star. The researchers calculated the likelihood of such a droplet forming spontaneously due to thermal fluctuations, essentially asking how often the heat inside the star might accidentally create a seed big enough to start the conversion.

A key innovation in their work was how they treated the internal structure of these tiny droplets. In the densest regions of a star, quarks are expected to pair up in a state known as color superconductivity, which lowers their energy and makes the strange matter even more stable. However, this pairing requires a certain amount of space to form. The researchers realized that the very first, smallest seeds of strange matter might be too small to support this pairing. They treated these tiny seeds as "unpaired" and only allowed them to become superconducting once they grew large enough. This subtle distinction turned out to be crucial, as it made the initial seeds harder to form and slightly more difficult to grow, effectively raising the barrier against conversion.

By running these calculations against the known evolution of proto-neutron stars, the team discovered a narrow window of possibility where both types of stars could survive. They found that for a standard neutron star, weighing about 1.4 times the mass of our Sun, to survive its hot birth and cool down into a stable, long-lived object, the force holding the surface of the strange matter droplet together must be strong enough to resist the heat. Specifically, they determined that the surface tension between normal matter and strange matter must be at least 67 MeV per square femtometer. If the tension is lower than this, the barrier is too weak, and any star heavy enough to become a neutron star would almost certainly convert into a strange quark star during its hot youth.

This finding places a strict lower limit on the properties of the universe. It does not prove that strange quark stars exist, nor does it prove that they don't. Instead, it defines the conditions required for the "two-families" scenario to be true. If the surface tension is high enough, the universe can host a mix of ordinary neutron stars and strange quark stars, with the former surviving because they were born in conditions where the conversion was too difficult to start. If the tension is lower, the universe would likely be filled only with strange quark stars, and the ordinary neutron stars we observe would be impossible. The study also calculated that if a conversion does occur, it would release a massive burst of energy, potentially visible as a delayed flash of light or gravitational waves, offering a way for astronomers to spot these events if they happen.

The researchers were careful to note that their results depend on the specific models they used for the behavior of matter at these extreme densities. They simulated the process using a reference timescale of one millisecond, which represents the rapid pace of events inside a collapsing star, and found that their conclusion about the minimum surface tension holds up even if they change the details of their simulation slightly. The value of 67 MeV per square femtometer is a hard boundary derived from their models; below it, the coexistence of the two families of stars is ruled out. Above it, the door remains open. This work provides a concrete, testable constraint for future theories of matter, turning a theoretical debate about the nature of the universe's densest objects into a question of measurable physical properties. It suggests that the survival of the neutron stars we see today is not guaranteed by chance, but by a specific, robust physical force that keeps them from dissolving into something even more fundamental.

Technical Summary: Coexistence of Strange Quark Stars and Neutron Stars

Problem Statement
The paper addresses the "two-families scenario" in compact star astrophysics, which posits that if strange quark matter (SQM) is absolutely stable (the Bodmer–Witten hypothesis), both hadronic neutron stars (NSs) and strange quark stars (QSs) could coexist. A critical theoretical challenge for this scenario is explaining the metastability of NSs. If SQM is the true ground state, NSs should spontaneously convert into QSs. However, observational evidence (e.g., glitches, quasi-periodic oscillations) suggests the existence of ordinary NSs with nuclear crusts. The paper investigates the conditions under which a hadronic proto-neutron star (PNS) can survive its hot, lepton-rich birth phase without nucleating a critical SQM droplet, thereby remaining a metastable NS upon cooling.

Methodology
The authors employ a microphysical treatment of thermal nucleation within the framework of a first-order phase transition. Key methodological components include:

Equations of State (EOS):
- Hadronic Phase: Modeled using the relativistic mean-field model SFHo-H $\Delta$ , extended to include hyperons and $\Delta$ -isobars.
- Quark Phase: Modeled using the $\alpha$ Bag model (MIT bag model with perturbative QCD corrections). Two phases are considered: unpaired quark matter and the color-flavor-locked (CFL) superconducting phase.
Nucleation Formalism:
- The nucleation rate is calculated based on the formation of a critical SQM droplet. The energy barrier $W^*$ is computed as the sum of bulk, finite-size (surface), and Coulomb terms.
- Flavor Fluctuations: Unlike previous geometric criteria, the authors allow the flavor composition of the droplet to fluctuate locally to minimize the energy barrier, enforcing conservation of baryon number, electric charge, and strangeness only globally or via strong interaction timescales.
- Finite-Size Pairing: A novel treatment accounts for the suppression of color superconductivity in small droplets. Since the diquark coherence length $\xi \sim 1/\Delta$ (where $\Delta$ is the pairing gap) is comparable to the size of early sub-critical droplets, the authors impose a minimal radius $R_\Delta$ . Droplets with $R < R_\Delta$ remain in the unpaired phase, while those with $R > R_\Delta$ can transition to the CFL phase.
PNS Evolution:
- The study focuses on two representative thermodynamic snapshots of a hadronic PNS: the moment of core-collapse supernova explosion ( $t_0$ ) and the epoch of maximum central temperature ( $t_{Tmax}$ , $\sim 10$ s later).
- Nucleation conditions are defined as the thermodynamic state where the nucleation time $\tau$ equals the dynamical timescale of the system ( $\tau_{dyn} \simeq 10^{-3}$ s).

Key Contributions

Finite-Size Suppression of Pairing: The paper introduces a scheme where the thermodynamics of the quark droplet depends on its radius relative to the diquark coherence length. This "unpaired-CFL" (unpCFL) scheme reveals that small critical seeds may remain unpaired, significantly altering the nucleation barrier compared to assuming immediate CFL pairing.
Thermodynamic Nucleation Criterion: Instead of using qualitative geometric proxies (e.g., inter-quark distance), the authors define nucleation conditions explicitly through the statistical expectation of a critical droplet forming within the dynamical timescale.
Universal Reduced Barrier: The analysis reveals that at the point of nucleation, the reduced energy barrier $W^*/T$ is nearly universal, with a value of approximately $167 \pm \text{few units}$ , largely independent of the specific EOS parameters or surface tension.

Results

Critical Surface Tension ( $\sigma_{crit}$ ): The existence of a metastable 1.4 $M_\odot$ NS requires that the hadron-quark surface tension $\sigma$ exceeds a critical threshold. If $\sigma < \sigma_{crit}$ , the nucleation barrier is too low, and the PNS converts to a QS.
Parameter Space Constraints: Scanning the SQM parameter space (bag constant $B$ , gap $\Delta_0$ , coupling $\alpha_s$ ), the authors find that for a canonical 1.4 $M_\odot$ NS to survive, the surface tension must satisfy:
$\sigma \gtrsim 67 \text{ MeV fm}^{-2}$
This lower bound is derived for the "softest" SQM parametrizations compatible with supporting massive pulsars ( $M_{max} > 2 M_\odot$ ).
Dependence on Observables: The bound tightens if heavier objects are identified as QSs. For instance, requiring the QS branch to support PSR J0952–0607 ( $2.18 M_\odot$ ) raises the bound to $\sigma \gtrsim 81 \text{ MeV fm}^{-2}$ , and for GW190814's secondary ( $2.6 M_\odot$ ), it rises to $\sigma \gtrsim 110 \text{ MeV fm}^{-2}$ .
Impact of Finite-Size Effects: Neglecting the finite-size suppression of pairing (assuming immediate CFL formation) would qualitatively misrepresent nucleation conditions, potentially underestimating the required surface tension by up to $\sim 21 \text{ MeV fm}^{-2}$ in specific regions of the parameter space.
Conversion Energy: If conversion occurs, the released binding energy is substantial ( $E_{conv} \sim 4\text{--}8 \times 10^{53}$ erg), potentially powering delayed neutrino bursts, gravitational waves, or gamma-ray bursts.

Significance and Claims
The paper claims to establish a necessary condition for the viability of the two-families scenario: the hadron-quark surface tension must be sufficiently large to prevent the prompt conversion of proto-neutron stars into strange quark stars.

The authors emphasize that $\sigma \gtrsim 67 \text{ MeV fm}^{-2}$ is a lower bound, not a preferred value. Values below this threshold would disfavor the coexistence of NSs and QSs by converting all massive PNSs into QSs.
The result is presented as a constraint on the microphysical properties of the QCD phase transition, derived from the requirement that ordinary neutron stars exist.
The study highlights that the "bottleneck" for NS survival is the epoch of maximum central temperature ( $t_{Tmax}$ ); if a PNS survives this stage without nucleation, it will cool into a stable NS.
The authors note that their results are conservative regarding the accessible droplet configurations; restricting the ensemble of reachable states (due to diffusion limits) would likely raise the barrier further, making nucleation even harder and potentially lowering the required $\sigma_{crit}$ .

The paper concludes that while the required surface tension is larger than some microscopic estimates, it lies within the broad range of theoretical predictions, keeping the two-families scenario viable provided the surface tension is not too small.

Technical Summary: Coexistence of Strange Quark Stars and Neutron Stars

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