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

Abstract: Recently, Fujimoto, Fukushima & Weise (2019) have proposed a new colour-superconductive state, 2SC+$&lt;dd&gt;$ phase, which can be smoothly connected to the low-density baryon superfluidity in contrast to the 2SC phase. In this scenario, the neutron ${}<sup>3P_2$ 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+$&lt;dd&gt;$ 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+$&lt;dd&gt;$ phase for the first time. We find that NSs with the 2SC+$&lt;dd&gt;$ phase become hotter than those with the 2SC phase, and are close to the CFL phase. The ${}<sup>{3}P_2$ superfluidity plays an important role in cooling curves with not the 2SC but 2SC+$&lt;dd&gt;$ phases due to the suppression of quark $β$ decay. We therefore point out that, if the scenario of 2SC+$&lt;dd&gt;$ phase is true, it could be specified through low-temperature observations such as Vela, 3C58, Vela Jr., and Vela-like pulsar.

• The paper introduces a novel 2SC+<dd> phase that bridges baryonic superfluidity and quark pairing in hybrid stars.
• It employs a hybrid EOS combining BHF-based hadronic and Dyson-Schwinger quark models to study neutrino emission and pairing gap effects.
• The results show that the 2SC+<dd> phase moderates rapid cooling, aligning theoretical models with observed cold neutron star data.

Cooling Phenomenology of Hybrid Stars with a 2SC+$\langle dd \rangle$ Phase

Introduction: Theoretical Context and Motivation

Understanding the thermal evolution of neutron stars (NSs), especially in the regime where hadronic matter transitions to deconfined quark matter, requires detailed modeling of phases such as color superconductivity (CSC). Two canonical CSC phases are established in the QCD phase diagram: the color-flavor-locked (CFL) phase, in which all quarks are paired and gapped, and the two-flavor color superconducting (2SC) phase, where only some quarks (typically up and down) are paired, and others (such as strange quarks and a subset of colors) remain gapless [1999NuPhB.537..443A, 2008RvMP...80.1455A]. The nature of the transition from baryonic superfluidity (notably, the neutron ${}^3P_2$ phase) to color superconductivity, and its imprint on NS cooling, remains an open question.

This paper addresses a scenario proposed by Fujimoto et al. [2020PhRvD.101i4009F], in which a novel 2SC+$\langle dd \rangle$ phase mediates quark-hadron continuity. Here, diquark condensates of paired down quarks emerge alongside the usual 2SC pairing, providing a symmetry bridge between baryonic ${}^3P_2$ superfluidity and quark pairing, with concomitant consequences for the thermal evolution of hybrid stars (HSs).

EOS Construction and Phase Structure

A hybrid equation of state (EOS), built from a BHF-based hadronic sector (Bonn-B two-body + Urbana UIX three-body forces) and a Dyson-Schwinger-model quark sector, is adopted. Particle fractions, pressure, and energy density are computed under beta equilibrium and charge-neutrality constraints, with first-order phase transition modeling and finite-size effects included via the Wigner-Seitz cell method. Hyperons are neglected; their threshold densities closely align with the onset of the quark phase in the adopted EOS parameterization.

Key EOS features include:

• Direct Urca (DU) threshold at $$\rho_{\rm DU}=1.46\times10^{15}\,\mathrm{g\,cm^{-3}}$$.
• Quark-hadron mixed phase begins at $$\rho_t\simeq 1.77\times10^{15}\,\mathrm{g\,cm^{-3}}$$.
• Maximum neutron star mass of $2.13\,M_\odot$ with a radius of 13 km for $1.4\,M_\odot$, within multi-messenger and X-ray observational bounds.

Microphysics of Cooling: Neutrino Emission and Pairing Gaps

Thermal evolution is governed by standard and enhanced neutrino emission processes:

• Modified Urca and nucleon bremsstrahlung ("slow cooling") with emissivities $\propto T^8$.
• Neutrino emission from Cooper pair breaking and formation (PBF), efficient near $T\sim 0.2$-${}^3P_2$0 for both ${}^3P_2$1 and ${}^3P_2$2 SF.
• Direct Urca (DU) and analogous quark beta decay processes open above density and composition thresholds, with emissivities up to ${}^3P_2$3 for ${}^3P_2$4 K.

Color superconductivity suppresses quark beta decay in a gap-dependent manner:

• In 2SC, only one color of each flavor remains ungapped, leaving ${}^3P_2$5-quark beta decay active.
• In the CFL or 2SC+${}^3P_2$6 phases, quark beta decay is strongly suppressed across all colors.

The 2SC+${}^3P_2$7 phase preserves a sizable ${}^3P_2$8 pairing gap (up to ${}^3P_2$9 K), now inherited by $\langle dd \rangle$0-quarks in the quark sector, allowing continuity with neutron superfluidity. The cooling code incorporates these microphysical ingredients within a fully general relativistic, energy-conserving treatment.

Results: Cooling Curves, Observational Comparison, and CSC Discrimination

The model delivers a systematic analysis of surface temperature histories for $\langle dd \rangle$1, $\langle dd \rangle$2, and $\langle dd \rangle$3 stars using a sequence of pairing gap models (A-D). The $\langle dd \rangle$4 sequence highlights the essential differences between the 2SC, CFL, and 2SC+$\langle dd \rangle$5 phases:

• 2SC-only scenario: Rapid cooling persists due to unsuppressed quark beta decay in unpaired $\langle dd \rangle$6-quarks.
• CFL and 2SC+$\langle dd \rangle$7 scenarios: Cooling is moderated and compatible with the full range of observed "cold" neutron stars, due to efficient suppression of quark beta decay by robust pairing. The 2SC+$\langle dd \rangle$8 phase leads to temperature evolution nearly indistinguishable from the CFL case for surface observations.

The authors claim that, if the 2SC+$\langle dd \rangle$9 phase exists, thermal evolution models with even moderate ${}^3P_2$0 pairing (${}^3P_2$1-${}^3P_2$2 K) match recent surface temperature data for Vela, 3C58, Vela Jr., and Vela-like pulsars. In contrast, models assuming only 2SC phase produce excessive cooling incompatible with these observations.

Envelope composition (iron vs. helium) remains an external uncertainty, but does not qualitatively alter the CMSC discrimination.

Implications for the QCD Phase Diagram and Dense Matter Observables

This study provides strong evidence that the inclusion of the 2SC+${}^3P_2$3 phase, which realizes quark-hadron continuity via simultaneous pairing in baryonic and quark sectors, is favored by updated neutron star cooling data. The thermal signatures of this phase overlap with those for the CFL phase, though the latter is expected only at higher densities not typically probed in canonical-mass NSs.

The findings challenge the adequacy of standard 2SC modeling of dense matter and reinforce the theoretical case for continuous (as opposed to first-order) transition scenarios at the quark-hadron interface [2020PhRvD.101i4009F, 2021PhRvD.104f3036K]. Additionally, the necessity of non-negligible ${}^3P_2$4 pairing gaps for both baryonic and quark sectors is highlighted as an observational constraint from cooling data—strengthening the synergy between nuclear theory, QCD modeling, and astrophysical observation.

Should future radius and mass measurements become available for these thermally intriguing NSs, the scenario can be tested further, potentially discriminating between hybrid star models with different underlying QCD symmetry-breaking patterns.

Conclusion

The paper establishes that the 2SC+${}^3P_2$5 color superconducting phase yields cooling tracks for massive neutron stars that are consistent with recent observations of "cold" isolated pulsars. The model interpolates between hadronic and quark matter pairing regimes, resolving discrepancies inherent in simpler 2SC or purely hadronic scenarios. The results underscore the role of the 2SC+${}^3P_2$6 phase as a leading candidate for the dense core state of hybrid stars, with consequences for both the microphysics of dense QCD and the phenomenology of observable neutron star cooling (2606.31389).