When a self-bound strange quark star cools through the critical temperature of the color-spin-locked (CSL) phase — a spin-one color-superconducting state of dense quark matter that behaves as a Type-I electromagnetic superconductor — a complete Meissner effect is expected, and the phase transition can expel a small fraction of the star's magnetic flux from a thin layer near the quark-core surface on short timescales, with an available magnetic energy of up to about 104410^{44} erg. The source analysis (Song, Phys. Rev. D 111, 063018 (2025)) established only an energy and timescale budget and conjectured that this expelled flux could act as an inducement mechanism powering typical fast radio bursts (FRBs); the actual coupling of the expelled flux to the stellar surface, possible nuclear crust, and magnetosphere — reconnection with external field lines, particle acceleration, pair creation, and the conversion efficiency of expelled magnetic energy into radiation — was explicitly left uncomputed, as was the real-time general-relativistic cooling and CSL-front evolution that sets the expulsion geometry. This record formulates that open problem: construct a self-consistent quantitative account linking the cooling-driven CSL-front propagation, the Type-I intermediate-state flux expulsion, and the magnetospheric response, in order to determine the conversion efficiency of the expelled magnetic energy into observable radiation and the resulting luminosity, spectrum, duration, and event rate — or to demonstrate quantitatively that no observable emission can result.

Background

At supranuclear density, quark matter may undergo color superconductivity: pairing of quarks in color-antisymmetric channels, analogous to BCS superconductivity but in the color gauge sector. The best-studied pattern is the color-flavor-locked (CFL) phase — a pairing pattern that locks the color and flavor indices of the paired quarks — at asymptotically high density; at the moderate densities relevant to compact-star cores, single-flavor and spin-one pairings compete, including the color-spin-locked (CSL) phase, in which all quarks pair so that color and spin degrees of freedom are locked. The CSL condensate has small gaps, of order 10 keV to 1 MeV, with correspondingly low critical temperatures of order 1 MeV or below, so a CSL core would form only in the late stages of compact-star cooling. The equation-of-state frameworks used for such matter are the MIT bag model (quarks confined in a bag with perturbative QCD corrections) and the Nambu–Jona-Lasinio (NJL) model, an effective chiral model with diquark interactions, which is solved with model parameters (coupling ratios, cutoff) that carry significant uncertainty. A strange quark star is a self-bound compact star made of approximately bulk-neutral strange quark matter; whether such stars exist — the Bodmer–Witten hypothesis that strange quark matter is absolutely stable — is itself observationally unestablished, and the source paper finds that the NJL model does not support absolutely stable quark matter, while the MIT bag model with QCD corrections can reproduce stable massive strange-quark-star configurations. A Type-I electromagnetic (EM) superconductor expels magnetic fields completely (the Meissner effect) and, in an applied field below the critical field BcB_{\rm c}, forms an intermediate state of alternating normal and superconducting domains; this contrasts with Type-II superconductors, which admit quantized flux tubes (vortices). The CSL phase is predicted to be a Type-I EM superconductor with BcB_{\rm c} of order 101610^{16} G, so a cooling quark star whose interior field is below criticality develops an intermediate-state domain structure as the CSL front advances. Fast radio bursts (FRBs) are millisecond-duration radio flashes with isotropic-equivalent radiated energies of 103510^{35}–104210^{42} erg, increasingly attributed to magnetars — neutron stars with 101410^{14}–101510^{15} G fields — whose emission is generally modeled via crust fracture launching Alfvén waves (magnetohydrodynamic waves of the magnetized plasma) that drive charge-starved magnetospheric discharges in an electron–positron pair plasma, producing coherent radiation, or via magnetospheric reconnection. Soft gamma-ray repeaters (SGRs) and their giant flares are energetic magnetar bursts of 104310^{43}–104610^{46} erg. The source paper (Song, Phys. Rev. D 111, 063018 (2025), arXiv:2409.15811) analyzed the CSL scenario in this setting: it computed NJL and bag-model stellar configurations hosting CSL quark matter, treated the flux expulsion from the Type-I intermediate state using classical superconductivity estimates (domain periodicity from wall-energy and penetration-depth parameters, expulsion timescale τexp\tau_{\rm exp} set by the quark-matter conductivity), and compared the expelled magnetic energy EB,expE_{B,\rm exp} and timescales with observed high-energy transient phenomenology. Its conclusions are budgetary: in special cases (millisecond-duration expulsion from a layer of thickness ∼10−5\sim 10^{-5} km with B′∼1016B^{\prime}\sim 10^{16} G), EB,expE_{B,\rm exp} can reach about 104410^{44} erg, exceeding typical FRB energies, so the process could act as an FRB inducement mechanism, but it cannot alone power gamma-ray bursts or giant flares; for hybrid stars with a nuclear crust the expulsion timescale is stretched to 101010^{10}–101110^{11} s, suppressing any prompt signal. Two adjacent model lineages exist but do not cover this case: Meissner/vortex expulsion from CFL Type-II superconducting quark cores with reconnection heating was modeled for SGR reheating by Ouyed et al. (A&A 420, 1025 (2004)) and Niebergal et al. (Phys. Rev. D 81, 043005 (2010)) — a different superconducting type and a rotationally driven, not cooling-driven, expulsion; and the magnetar FRB machinery of crust fracture, Alfvén-wave propagation, and coherent pair-plasma emission (e.g., Yang & Zhang, ApJ 919, 89 (2021)) has not been driven by the CSL Type-I expulsion boundary condition. The CSL phase itself remains a live hypothesis in dense-matter research: NJL-model studies place it in the phase diagram at compact-star densities (Aguilera, Astrophys. Space Sci. 308, 443 (2007)), and a recent one-gluon-exchange plus renormalization-group analysis (Fujimoto, arXiv:2508.19222) finds the CSL pairing channel favored when the quark Fermi momenta are strongly mismatched.

Problem Statement

Determine whether the expulsion of magnetic flux during the cooling-driven phase transition to color-spin-locked (CSL) quark matter in a compact star can power observable electromagnetic emission, by constructing a self-consistent quantitative account of the full chain that the source paper left uncomputed: (a) the general-relativistic thermal evolution of the cooling quark star and the propagation of the CSL phase front (the true cooling timescale τcool\tau_{\rm cool} and the spatiotemporal structure of the Type-I intermediate state of normal and superconducting domains), which fixes the amount, geometry, and expulsion timescale τexp\tau_{\rm exp} of the magnetic flux leaving the quark core; (b) the coupling of this expelled flux to the stellar surface, to any nuclear crust, and to the magnetosphere, including reconnection with external magnetic field lines, the induced particle acceleration and pair creation, and the dynamical response of the ambient field; and (c) the conversion efficiency of the expelled magnetic energy EB,expE_{B,\rm exp} (up to roughly 104410^{44} erg in the special millisecond-duration cases identified in the source analysis) into radiated electromagnetic energy. These components jointly serve one objective: a predictive emission model — luminosity, spectrum, duration, and event rate — for the CSL flux-expulsion scenario, enabling a decision on whether it can act as the inducement (or sole) mechanism for fast radio bursts as conjectured, or whether, alternatively, a quantitative argument shows that no observable emission can result (for example because crustal or magnetospheric coupling quenches the process). The setting is the one established by the source paper: a self-bound strange quark star in the MIT bag model with QCD corrections hosting CSL quark matter, with the NJL-model hybrid-star case and its crust-suppressed expulsion as a controlled comparison.

References

  1. Xin-Ying Song, "Compact stars as hideouts for color-spin-locked quark matter: Implications for powering high-energy electromagnetic emissions", Phys. Rev. D 111, 063018 (2025), DOI: 10.1103/PhysRevD.111.063018, arXiv:2409.15811, https://arxiv.org/abs/2409.15811
  2. Xin-Ying Song, "Constraints from gamma-ray burst phenomenology on the hypothesis of quark stars as the central engine of gamma-ray bursts", Phys. Rev. D 111, 063040 (2025), DOI: 10.1103/PhysRevD.111.063040, arXiv:2501.03610, https://arxiv.org/abs/2501.03610
  3. Huai-Min Chen, Cheng-Jun Xia, Guang-Xiong Peng, "The properties of strange quark matter and evolution of strange quark stars", arXiv:2601.11106, https://arxiv.org/abs/2601.11106
  4. Mauro Mariani, Ignacio F. Ranea-Sandoval, "How Neutron Star Observations Point Towards Exotic Matter: Existing Explanations and a Prospective Proposal", Symmetry 18, 27 (2026), DOI: 10.3390/sym18010027, arXiv:2602.06170, https://arxiv.org/abs/2602.06170
  5. Yuki Fujimoto, "Classification of color superconductivity by one-gluon exchange helicity amplitudes and renormalization group equations", arXiv:2508.19222, https://arxiv.org/abs/2508.19222
  6. Xiao-Li Zhang, Yong-Feng Huang, Ze-Cheng Zou, "Recent progresses in strange quark stars", arXiv:2404.00363, https://arxiv.org/abs/2404.00363
  7. Yuan-Pei Yang, Bing Zhang, "Fast Radio Bursts and Their High-energy Counterparts from Magnetar Magnetospheres", Astrophys. J. 919, 89 (2021), DOI: 10.3847/1538-4357/ac14b5, arXiv:2104.01925, https://arxiv.org/abs/2104.01925
  8. R. Ouyed, Ø. Elgarøy, H. Dahle, P. Keränen, "Meissner effect and vortex dynamics in quark stars — A model for soft gamma-ray repeaters", Astron. Astrophys. 420, 1025 (2004), DOI: 10.1051/0004-6361:20034180, arXiv:astro-ph/0308166, https://arxiv.org/abs/astro-ph/0308166
  9. Brian Niebergal, Rachid Ouyed, Rodrigo Negreiros, Fridolin Weber, "Meissner effect and vortex expulsion in color-superconducting quark stars, and its role for reheating of magnetars", Phys. Rev. D 81, 043005 (2010), DOI: 10.1103/PhysRevD.81.043005, arXiv:0906.3043, https://arxiv.org/abs/0906.3043
  10. D. N. Aguilera, "Spin-one color superconductivity in compact stars? -- an analysis within NJL-type models", Astrophys. Space Sci. 308, 443 (2007), DOI: 10.1007/s10509-007-9359-1, arXiv:hep-ph/0608041, https://arxiv.org/abs/hep-ph/0608041