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arxiv: 2606.31389 · v1 · pith:BYYVYQ7Rnew · submitted 2026-06-30 · ⚛️ nucl-th · astro-ph.HE

Cooling of Hybrid Stars with a 2SC+<dd> Phase

Tsuneo Noda , Akira Dohi , Nobutoshi Yasutake , Huan Chen , Toshiki Maruyama , Toshitaka Tatsumi This is my paper

Pith reviewed 2026-07-01 02:53 UTC · model grok-4.3

classification ⚛️ nucl-th astro-ph.HE
keywords hybrid starsquark superfluidityneutron star coolingcolor superconductivity2SC phase3P2 superfluiditybeta decay suppression
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The pith

Hybrid stars with the 2SC+dd phase cool more slowly than those with the 2SC phase and resemble the CFL phase in their thermal evolution.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines how a newly proposed color-superconductive state called the 2SC+dd phase affects the cooling of hybrid stars, which contain both hadronic and quark matter. This phase is proposed to connect smoothly to low-density baryon superfluidity, inheriting the neutron 3P2 superfluidity to unpaired d-quarks on the high-density side. As a result, the suppression of quark beta decay makes these stars hotter during cooling compared to standard 2SC quark matter, bringing their behavior closer to color-flavor-locked stars. A sympathetic reader would care because this could allow distinguishing the phase through observations of cool pulsars such as Vela and 3C58.

Core claim

Hybrid stars incorporating the 2SC+dd phase become hotter than those with the traditional 2SC phase during their thermal evolution and approach the cooling characteristics of the CFL phase. The 3P2 superfluidity plays a key role in the cooling curves specifically for the 2SC+dd phase by suppressing quark beta decay, unlike in the 2SC case. This is the first study of the thermal evolution under this scenario proposed by Fujimoto, Fukushima and Weise.

What carries the argument

The 2SC+dd phase, which smoothly connects low-density baryon superfluidity to high-density quark matter, enabling the inheritance of neutron 3P2 superfluidity by unpaired d-quarks and thereby suppressing beta decay processes.

If this is right

  • Neutron stars with 2SC+dd phase exhibit higher temperatures at given ages compared to 2SC phase models.
  • The 3P2 superfluidity significantly influences cooling only when the 2SC+dd phase is present.
  • Low-temperature observations of pulsars like Vela, 3C58, Vela Jr. could distinguish this phase if the scenario holds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the 2SC+dd phase is realized, it may require revisiting assumptions about quark pairing in dense matter models.
  • Future simulations could test how this phase affects other observables such as neutrino emission rates in hybrid stars.

Load-bearing premise

The 2SC+dd phase exists in hybrid stars and allows the neutron 3P2 superfluidity to be inherited by unpaired d-quarks from the low-density side.

What would settle it

A measurement showing that observed cooling curves of pulsars like Vela match predictions for the 2SC phase rather than the hotter curves for 2SC+dd would indicate the phase does not occur as proposed.

read the original abstract

Recently, Fujimoto, Fukushima & Weise (2019) have proposed a new colour-superconductive state, 2SC+$<dd>$ phase, which can be smoothly connected to the low-density baryon superfluidity in contrast to the 2SC phase. In this scenario, the neutron ${}^3P_2$ superfluidity on the low-density side of the phase transition is inherited by unpaired $d$-quarks in the 2SC phase on the high-density side. Since this could be realized in hybrid stars (neutron stars containing hadronic and quark matter), the 2SC+$<dd>$ phase may change the properties of neutron stars compared to the traditional 2SC phase. In this work, we study the thermal evolution of hybrid stars with the 2SC+$<dd>$ phase for the first time. We find that NSs with the 2SC+$<dd>$ phase become hotter than those with the 2SC phase, and are close to the CFL phase. The ${}^{3}P_2$ superfluidity plays an important role in cooling curves with not the 2SC but 2SC+$<dd>$ phases due to the suppression of quark $\beta$ decay. We therefore point out that, if the scenario of 2SC+$<dd>$ phase is true, it could be specified through low-temperature observations such as Vela, 3C58, Vela Jr., and Vela-like pulsar.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript studies the thermal evolution of hybrid stars containing the 2SC+<dd> color-superconducting phase proposed by Fujimoto, Fukushima & Weise (2019). In this phase the neutron 3P2 superfluidity is inherited by unpaired d-quarks, suppressing quark β-decay processes. The central result is that hybrid stars with the 2SC+<dd> phase cool more slowly than those with the conventional 2SC phase and produce cooling curves close to those of the CFL phase; the 3P2 gap is shown to be decisive only in the 2SC+<dd> case. The authors suggest that low-temperature observations of pulsars such as Vela could distinguish the phase.

Significance. If the 2SC+<dd> phase is realized, the work supplies a concrete, observationally testable signature that distinguishes it from the standard 2SC phase through the inherited superfluidity and the resulting suppression of quark Urca processes. The explicit linkage between the low-density baryon superfluidity and the high-density quark pairing is a clear modeling strength.

major comments (2)
  1. [Results] §3 (or equivalent results section): the quantitative temperature offset between 2SC+<dd> and 2SC curves is presented without a sensitivity study on the amplitude of the inherited 3P2 gap; because the suppression of β-decay is the load-bearing mechanism, the robustness of the offset to plausible variations in that gap must be shown.
  2. [Model] §2 (model description): the matching condition at the hadron-quark interface that transmits the 3P2 gap is stated only qualitatively; an explicit functional form or table of gap values across the transition density is required to make the cooling calculation reproducible.
minor comments (2)
  1. [Introduction] The notation 2SC+<dd> is introduced without a brief reminder of its microscopic definition in the introduction; a one-sentence restatement would aid readers unfamiliar with Fujimoto et al. (2019).
  2. [Figures] Figure captions for the cooling curves should state the stellar mass, the hadronic EOS, and the quark gap parameters used, rather than referring only to the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation for minor revision. We address each major comment below and will incorporate the requested improvements into the revised manuscript.

read point-by-point responses

  1. Referee: [Results] §3 (or equivalent results section): the quantitative temperature offset between 2SC+<dd> and 2SC curves is presented without a sensitivity study on the amplitude of the inherited 3P2 gap; because the suppression of β-decay is the load-bearing mechanism, the robustness of the offset to plausible variations in that gap must be shown.

    Authors: We agree that a sensitivity study on the inherited 3P2 gap amplitude is necessary to demonstrate the robustness of the temperature offset, given that gap suppression of quark β-decay is central to the result. In the revised manuscript we will add calculations varying the gap amplitude over a plausible range (consistent with the Fujimoto et al. 2019 framework) and include the corresponding cooling curves to quantify how the offset between the 2SC+<dd> and 2SC cases responds to these changes. revision: yes

  2. Referee: [Model] §2 (model description): the matching condition at the hadron-quark interface that transmits the 3P2 gap is stated only qualitatively; an explicit functional form or table of gap values across the transition density is required to make the cooling calculation reproducible.

    Authors: We acknowledge that the interface matching was described qualitatively in the original text. The 3P2 gap is transmitted from the hadronic 3P2 superfluidity to the unpaired d-quarks according to the 2SC+<dd> construction of Fujimoto, Fukushima & Weise (2019). To ensure reproducibility we will add an explicit functional form (or a table of gap values versus density) for the inherited gap across the hadron-quark transition density in the model section of the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results are conditional consequences of an external phase model

full rationale

The paper adopts the 2SC+<dd> phase from Fujimoto, Fukushima & Weise (2019) as an external premise and computes standard cooling curves for hybrid stars, finding hotter temperatures due to inherited 3P2 superfluidity suppressing quark beta decay. No quoted equations or steps reduce these outcomes to fitted inputs renamed as predictions, self-definitions, or load-bearing self-citations by the same authors. The central claims are explicitly conditional on the phase scenario being true and are presented as direct computational consequences rather than independent derivations that loop back to the paper's own inputs. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is supplied, so the ledger cannot be populated with concrete free parameters, axioms, or invented entities from the manuscript; the central claim rests on the existence of the 2SC+<dd> phase as defined in the cited 2019 work and on standard assumptions about neutrino emissivity in quark matter.

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discussion (0)

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