⚛️ nuclear theory

Cooling of Hybrid Stars with a 2SC+$$ Phase

This paper presents the first study of the thermal evolution of hybrid stars containing a newly proposed 2SC+$$ phase, finding that the inherited ${}^3P_2$ superfluidity suppresses quark $\beta$ decay to produce hotter stars than the traditional 2SC phase, a signature potentially detectable through low-temperature observations of specific pulsars.

Original authors: Tsuneo Noda, Akira Dohi, Nobutoshi Yasutake, Huan Chen, Toshiki Maruyama, Toshitaka Tatsumi

Published 2026-07-01

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CC BY 4.0

Original authors: Tsuneo Noda, Akira Dohi, Nobutoshi Yasutake, Huan Chen, Toshiki Maruyama, Toshitaka Tatsumi

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

Imagine a neutron star as the universe's ultimate pressure cooker. It is a dead star so heavy that a single teaspoon of its material would weigh a billion tons on Earth. Inside this cosmic pressure cooker, the rules of physics get weird. The atoms, which usually keep their protons and neutrons separate, get crushed so hard that they melt into a soup of free-floating particles called quarks.

This paper is like a detective story trying to figure out how hot or cold these cosmic pressure cookers get as they age. The authors are investigating a specific "flavor" of this quark soup and how it changes the star's cooling speed.

The Main Characters: The "Soup" and the "Ice Cubes"

To understand the paper, you need to know two things about the inside of a neutron star:

The Soup (Quark Matter): Deep inside the star, the matter turns into a fluid of quarks. Usually, scientists think this fluid behaves like a super-conductor (a material that conducts electricity with zero resistance). There are two main theories on how this happens:
- The "2SC" Phase: Imagine a dance floor where most dancers pair up, but a few specific dancers (the "blue" ones and the "strange" ones) are left standing alone, shivering in the cold. These unpaired dancers are very active and can easily escape the star, carrying away heat. This makes the star cool down very fast.
- The "CFL" Phase: Imagine a dance floor where everyone pairs up perfectly. No one is left standing alone. Because everyone is locked in a tight embrace, it's very hard for them to escape and carry away heat. This makes the star cool down very slowly.
The Ice Cubes (Superfluidity): Before the star gets crushed into quark soup, the outer layers are made of neutrons. These neutrons can also form "ice cubes" (a state called superfluidity). Think of this as a thick, sticky gel that slows down the movement of particles. If this gel is strong, it acts like a blanket, keeping the star warm. If it's weak, the star loses heat quickly.

The New Discovery: The "Hybrid" Dance Floor

For a long time, scientists thought the transition from the "neutron ice" to the "quark soup" was a messy, abrupt jump. But a new theory (proposed by Fujimoto, Fukushima, & Weise) suggests a smoother transition.

They propose a new state called 2SC+⟨dd⟩.

The Analogy: Imagine the dance floor again. In the old "2SC" theory, the unpaired dancers were just standing there, shivering and letting heat escape. In this new "2SC+⟨dd⟩" theory, those unpaired dancers suddenly decide to hold hands with each other (forming a special pair called a "d-d condensate").
The Result: Even though they are still in the "2SC" zone, they are now holding hands. This stops them from running away with the heat as easily. It makes the star behave more like the slow-cooling "CFL" phase, even though it's technically in the 2SC zone.

The Investigation: How Fast Do They Cool?

The authors of this paper built a computer model to simulate how these stars cool down over millions of years. They tested three scenarios:

Standard 2SC: The unpaired dancers run away fast (Star cools too fast).
CFL: Everyone is paired up (Star cools slowly).
2SC+⟨dd⟩: The unpaired dancers hold hands (Star cools slowly, similar to CFL).

They compared their computer models against real observations of actual neutron stars, such as the Vela pulsar and the remnant of a supernova called 3C58. These real stars are surprisingly warm for their age.

The Findings

Here is what the paper discovered, using simple terms:

The "Too Cold" Problem: If the star is in the standard "2SC" phase (where unpaired dancers run away), it cools down so fast that it should be freezing cold by now. But the real stars we see are much warmer than that. The standard 2SC theory doesn't fit the data.
The "Just Right" Solution: The new 2SC+⟨dd⟩ phase acts like a thermal blanket. Because the unpaired particles hold hands, they can't carry away heat as easily. This keeps the star warmer, matching the temperatures we actually observe in stars like Vela and 3C58.
The Importance of the "Ice Cubes": The paper also found that the "ice cubes" (the neutron superfluidity) play a huge role. If the ice cubes are too weak, the star cools too fast. If they are just right (a specific temperature range), they help keep the star warm enough to match our observations.

The Conclusion

The paper argues that if this new "2SC+⟨dd⟩" theory is true, it solves a major mystery: Why are some old neutron stars still warm?

Old Theory: Stars should be freezing cold because unpaired particles escape easily.
New Theory: The particles hold hands (the 2SC+⟨dd⟩ phase), trapping the heat inside.

The authors suggest that by looking at the temperatures of specific cold neutron stars (like Vela, 3C58, and Vela Jr.), astronomers might be able to prove that this "holding hands" state exists inside them. It's a way of seeing the invisible quantum dance happening deep inside the most extreme objects in the universe.

In short: The paper proposes that neutron stars have a secret "hand-holding" mechanism among their particles that keeps them warmer than we thought, and this mechanism fits the real-world data much better than the old theories.

Technical Summary: Cooling of Hybrid Stars with a 2SC+⟨dd⟩ Phase

Problem Statement
Neutron stars (NSs) contain matter at supra-nuclear densities where a transition from hadronic matter to deconfined quark matter is expected. A major challenge in understanding NS cores is establishing the continuity between the neutron $^3P_2$ superfluidity in the hadronic phase and the color-superconducting (CSC) phases in the quark phase. Traditional models often assume a first-order phase transition to a pure two-flavor color superconducting (2SC) phase. However, the 2SC phase leaves unpaired $d$ -quarks that allow for rapid neutrino emission via quark $\beta$ -decay, leading to cooling rates that are often too fast to match observations of cold neutron stars (e.g., Vela, 3C58). Conversely, the color-flavor-locked (CFL) phase suppresses these emissions effectively but is theoretically expected to appear only at extremely high densities, potentially beyond the reach of typical neutron star cores. Recently, Fujimoto, Fukushima, and Weise (2019) proposed a "2SC+⟨dd⟩" phase, where unpaired $d$ -quarks form a condensate ( $\langle dd \rangle$ ), allowing for a smooth connection to the low-density neutron $^3P_2$ superfluidity. The thermal evolution of hybrid stars containing this specific phase had not been previously investigated.

Methodology
The authors constructed a hybrid equation of state (EOS) incorporating a first-order phase transition between hadronic and quark matter, utilizing the Gibbs construction to model the mixed phase with finite-size effects (droplet, rod, slab, bubble geometries).

Hadronic Phase: Modeled using the Brueckner-Hartree-Fock method with realistic two-body (Bonn-B) and phenomenological three-body (Urbana UIX) potentials. Hyperons were excluded, justified by the assumption that quark matter onset occurs at similar or lower densities.
Quark Phase: Modeled using the Dyson-Schwinger approach with a Gaussian-type effective interaction. The model includes up, down, and strange quarks with current masses $m_u=m_d=0$ and $m_s=115$ MeV.
Cooling Processes: The thermal evolution was calculated by solving the TOV equation and thermal transport equations. Neutrino emissivity included:
- Hadronic: Modified Urca, bremsstrahlung, nucleon Direct Urca (DU), and Pair Breaking and Formation (PBF) processes.
- Quark: Quark $\beta$ -decay (dominant in unpaired 2SC), electron-electron scattering (dominant in CFL), and PBF processes.
Superfluidity Models: The study introduced four models (A–D) for the critical temperature ( $T_c$ ) of the $^3P_2$ superfluidity for both neutrons and unpaired $d$ -quarks in the 2SC+⟨dd⟩ phase. The $T_c$ for $d$ -quarks was assumed equal to that of neutrons to ensure continuity.
Observational Constraints: The cooling curves were compared against observational data for Vela, 3C58, Vela Jr., and a Vela-like pulsar, as well as constraints from massive pulsars (PSR J0740+6620, PSR J1614−2230) and gravitational wave events (GW 170817).

Key Contributions

First Thermal Evolution Study of 2SC+⟨dd⟩: This work presents the first calculation of the thermal evolution of hybrid stars specifically incorporating the 2SC+⟨dd⟩ phase.
Mechanism of Suppression: The authors demonstrate that in the 2SC+⟨dd⟩ scenario, the $^3P_2$ superfluidity of the unpaired $d$ -quarks (inherited from the neutron $^3P_2$ state) suppresses the quark $\beta$ -decay process. This is a crucial distinction from the pure 2SC phase, where unpaired $d$ -quarks lead to rapid cooling.
Differentiation of CSC Phases: The study establishes that the cooling curves of stars with the 2SC+⟨dd⟩ phase are distinct from those with the pure 2SC phase and are instead similar to the CFL phase, despite the 2SC+⟨dd⟩ phase appearing at lower densities than the CFL phase.

Results

Cooling Behavior: Hybrid stars with the 2SC+⟨dd⟩ phase cool significantly slower than those with the pure 2SC phase. The 2SC+⟨dd⟩ cooling curves are comparable to the CFL phase, resulting in surface temperatures that are higher than the 2SC case.
Role of $^3P_2$ Superfluidity: The critical temperature of the $^3P_2$ $^{3} P_{2}$ superfluidity is a decisive factor.
- Model A (Weak $T_c$ ): Results in cooling that is too rapid for all observational data, regardless of the CSC phase.
- Model C (Intermediate $T_c \approx 5 \times 10^8$ K): Successfully explains the observational data for Vela, 3C58, Vela Jr., and the Vela-like pulsar when the 2SC+⟨dd⟩ (or CFL) phase is present. The pure 2SC phase fails to match these observations even with Model C.
- Model D (Strong $T_c \approx 10^9$ K): Can explain Vela but fails to explain the lower temperature limits of 3C58 and Vela Jr., suggesting that extremely strong $^3P_2$ superfluidity is unlikely.
Surface Composition: The study notes that surface composition (Fe vs. He envelopes) affects the cooling curves. A lighter He envelope results in higher surface temperatures, which could reconcile discrepancies between specific models and observations (e.g., Vela).
Mass Constraints: The models are consistent with the existence of $2M_\odot$ neutron stars and the radius constraints from GW 170817 and NICER.

Significance and Claims
The paper claims that if the 2SC+⟨dd⟩ scenario is realized in nature, it offers a viable explanation for the cooling behavior of cold neutron stars that is consistent with both the requirement for quark-hadron continuity and observational data. The authors argue that the 2SC+⟨dd⟩ phase resolves the tension between the density requirements for the appearance of CSC phases and the need to suppress rapid neutrino cooling. Specifically, the paper posits that:

The 2SC+⟨dd⟩ phase allows for the smooth inheritance of neutron $^3P_2$ superfluidity to the quark sector, suppressing quark $\beta$ -decay.
Observational data from Vela, 3C58, Vela Jr., and Vela-like pulsars can be used to distinguish between the pure 2SC phase and the 2SC+⟨dd⟩ phase.
A $^3P_2$ critical temperature in the range of $5 \times 10^8 \text{ K} \lesssim \max(T_c) < 10^9 \text{ K}$ is favored by the data, a result that coincidentally aligns with constraints derived from the cooling rate of Cassiopeia A.

The authors conclude that while their study relies on specific assumptions (first-order phase transition and continuous $T_c$ linking), it provides the first basis for investigating the 2SC+⟨dd⟩ state through thermal evolution, suggesting that future mass measurements of cold neutron stars could further constrain the internal composition and superfluidity of NS cores.

The Main Characters: The "Soup" and the "Ice Cubes"

The New Discovery: The "Hybrid" Dance Floor

The Investigation: How Fast Do They Cool?

The Findings

The Conclusion

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