The color-spin-locked (CSL) phase is a spin-one color-superconducting state of dense quark matter in which same-flavor quarks pair with spin and color locked, making the star-sized sample an electromagnetic type-I superconductor. Song (Phys. Rev. D 111, 063018 (2025)) proposed that during cooling of a massive strange quark star the transition to the CSL phase could expel a small fraction of the internal magnetic field from a thin surface layer on millisecond timescales, and that this could act as an inducement mechanism for fast radio bursts (FRBs), though not as the sole energy source for gamma-ray bursts or magnetar giant flares. The proposal rests on unverified legs: the CSL phase has not been shown to be the ground state against competing spin-one and spin-zero pairings across the relevant density–magnetic-field region and model classes; the real-time general-relativistic formation of CSL domains and expulsion of flux during stellar cooling (including the nuclear-crust barrier and realistic conductivities) has not been modelled; and the conversion of expelled magnetic energy into coherent radio emission, via reconnection with the magnetosphere and crust, has not been computed. The problem is to establish, quantitatively delimit, or exclude this mechanism as a predictive FRB powering channel by resolving these three jointly required links.

Background

At supranuclear density, quarks may pair into a color-superconducting ground state, the dense-matter analogue of ordinary superconductivity in which Cooper pairs carry color charge (the charge of the strong interaction) rather than electric charge (the pairing gaps are the energy gains from condensation). Because the attractive single-flavor channel is spin-one, if two-flavor (2SC) or three-flavor (CFL) spin-zero pairing is blocked — for example by unequal Fermi momenta or neutrality constraints — the favored fallback can be a spin-one state. In the color-spin-locked (CSL) phase a single flavor (effectively up quarks) pairs so that spin and color directions are locked; the condensate breaks the residual electromagnetic gauge symmetry, so bulk CSL quark matter behaves as an electromagnetic superconductor. Schmitt, Wang, and Rischke (Phys. Rev. Lett. 91, 242301 (2003)) showed that CSL quark matter exhibits the electromagnetic Meissner effect — the expulsion of magnetic fields from the interior of a superconductor — as a type-I superconductor (magnetic fields are completely expelled below a thermodynamic critical field BcB_c of order 101610^{16} G, rather than admitted as quantized vortices as in type-II), with a coherence length (the spatial extent of a Cooper pair) far larger than the magnetic penetration depth (the depth over which an applied field is screened). Two model classes are used for stellar applications: the Nambu–Jona-Lasinio (NJL) model, an effective chiral quark model with contact interactions, ultraviolet cutoff Λ\Lambda, and coupling constants (GSG_S, diquark GDG_D, 't Hooft term HH, compression KK) fixed elsewhere; and the MIT bag model, which confines quarks in a bag with perturbative QCD corrections. The 2005 NJL phase-diagram studies (Rüster et al., Phys. Rev. D 72, 034004; Aguilera et al., Phys. Rev. D 72, 034008) mapped windows where CSL and related spin-one phases (polar, A, planar) compete, finding CSL favored only in narrow regions, with gaps of order MeV or less. Fast radio bursts (FRBs) are millisecond, extragalactic radio flashes with isotropic energies up to ∼2×1042\sim 2\times 10^{42} erg; the prevailing models tie them to magnetars (strongly magnetized neutron stars), e.g. magnetospheric releases (Yang & Zhang, Astrophys. J. 919, 89 (2021)), while quark-matter alternatives (e.g. Meissner-expulsion and vortex-dynamics models for soft gamma repeaters — neutron stars emitting recurrent low-energy gamma-ray flares — Ouyed et al., Astron. Astrophys. 420, 1025 (2004); flux dynamics in color-superconducting cores, Iida & Baym, Phys. Rev. D 66, 014015 (2002)) have stayed speculative. Song (Phys. Rev. D 111, 063018 (2025)) asked whether compact stars could 'hideout' CSL quark matter and whether the CSL transition could power high-energy electromagnetic emission. Key findings: (i) within the NJL model, absolutely stable quark matter and conventional pure quark stars are not supported, and hybrid stars (neutron stars with a deconfined-quark core surrounded by nuclear matter) with CSL cores are unstable above 2M⊙2M_\odot; the CSL hideout survives only as massive strange quark stars — stars made of absolutely stable strange quark matter in the Bodmer–Witten sense, i.e. the hypothesis that three-flavor strange quark matter is more strongly bound than nuclear matter and stable even at zero pressure — in the MIT bag model with QCD corrections; (ii) the CSL condensation during cooling is a type-I transition, so most internal magnetic field is frozen inside the quark core, and only in special configurations can a thin layer expel a fraction of magnetic energy: when cooling is much faster than expulsion, an intermediate-state layer of thickness a∼10−7a\sim 10^{-7} km at B∼Bc/2B\sim B_c/2 carries less than 104210^{42} erg, whereas if expulsion proceeds fast enough for the flux to reach the surface, a layer of thickness d∼10−5d\sim 10^{-5} km at internal field B′∼1016B'\sim 10^{16} G matches millisecond burst durations with an upper limit EB,exp≲1044E_{B,\mathrm{exp}}\lesssim 10^{44} erg; (iii) energetics and timescales allow this to act as an inducement mechanism for typical FRBs, but not as the sole energy source for gamma-ray bursts or magnetar giant flares; (iv) in hybrid stars the nuclear crust stretches the expulsion timescale to 101010^{10}–101110^{11} s, enclosing the process inside the star. The author explicitly flags what is missing: no explicit calculation of the gapless CFL (gCFL) phase — the CFL variant in which neutrality stresses leave some quark quasiparticles ungapped — in the same model, no comparison against the polar, A, and planar spin-one phases, no general-relativistic cooling/transport dynamics of the CSL front (described in the paper as very complex and not discussed in this work), and no computation of how expelled flux couples to the magnetosphere and crust to produce coherent radio emission; the Discussion further notes the dependence on the hypothesis that conventional strange quark stars actually exist and on the effects of the nuclear crust.

Problem Statement

Determine whether the formation of color-spin-locked (CSL) quark matter — the spin-one color-superconducting phase of dense quark matter, in which single-flavor Cooper pairs lock spin and color directions so that the bulk behaves as an electromagnetic type-I superconductor expelling magnetic fields below a critical field BcB_c of order 101610^{16} G — in the core of a cooling compact star can quantitatively power fast-radio-burst-class electromagnetic emission, or exclude or delimit the mechanism, by establishing, as jointly required links of one viability argument: (a) the ground-state question: whether CSL is the true minimum of the free energy against the competing spin-one phases (polar, A, planar) and spin-zero phases (two-flavor 2SC and three-flavor CFL pairing, and the gapless CFL variant gCFL, in which neutrality stresses leave some quark quasiparticles ungapped), across the relevant chemical-potential–magnetic-field region — in the source's setting, the interiors of strange quark stars at T=0T=0 with NJL chemical potentials μ≈368\mu\approx 368–457457 MeV (light-quark onset through the CSL window) and internal magnetic fields up to the critical field Bc∼1016B_c\sim 10^{16} G — with the systematic model dependence mapped rather than fixed: for the NJL realization, the coupling ratio GD/GS∈{3/4,1}G_D/G_S\in\{3/4,1\} and couplings H=(3/8)GSH=(3/8)G_S, KΛ5=12.36K\Lambda^5=12.36, GSΛ2=1.835G_S\Lambda^2=1.835 (quark masses mu,d=5.5m_{u,d}=5.5 MeV, ms=140.7m_s=140.7 MeV), and the ultraviolet cutoff Λ=602.3\Lambda=602.3 MeV and cutoff scheme; for the MIT-bag-model strange-quark-star realization, the perturbative coupling αc∈{0.3,0.6}\alpha_c\in\{0.3,0.6\}, bag constant B1/4≈117B^{1/4}\approx 117–145145 MeV, strange-quark mass Ms≈110M_s\approx 110–300300 MeV, CFL gap ΔCFL≈10\Delta_{\mathrm{CFL}}\approx 10–≳50\gtrsim 50 MeV, and the CSL gap ΔCSL=1\Delta_{\mathrm{CSL}}=1 MeV that the source assumes rather than derives (with coherence length ξ∼200\xi\sim 200 fm following from ξ∼1/Δ\xi\sim 1/\Delta, and penetration depth λ∼1\lambda\sim 1–1010 fm and Bc∼1016B_c\sim 10^{16} G taken from Schmitt–Wang–Rischke); the source omits the polar, A, and planar phases paper-wide, borrows the gCFL results from an NJL-based study, and reduces the bag-model CSL-versus-CFL comparison to the approximate criterion ΔCFL≳Ms2/(4μ)\Delta_{\mathrm{CFL}}\gtrsim M_s^2/(4\mu), so the ground-state competition remains open within both model classes; (b) the dynamical question: the general-relativistic magneto-thermal evolution of a cooling star through the CSL transition — the true cooling timescale τcool\tau_{\mathrm{cool}} versus the flux-expulsion timescale τexp\tau_{\mathrm{exp}}, the spatiotemporal structure of the CSL phase-transition front, flux freezing versus expulsion in the type-I Meissner state, the barrier role of the nuclear crust, and realistic electrical conductivities (impurity scattering, strong-interaction and finite-temperature transport) governing how much magnetic energy, EB,expE_{B,\mathrm{exp}}, actually leaves the quark core on what timescale; and (c) the emission question: the conversion of the expelled magnetic flux into observable radiation — reconnection with the stellar magnetosphere and crust, particle acceleration, and the efficiency with which EB,expE_{B,\mathrm{exp}} becomes radiated energy — sufficient to yield a predictive signature (energy, duration, repetition, spectra) that can be confronted with FRB observations and with the counterargument that crust-enclosed expulsion suppresses the signal entirely. An answer may take the form of a validated quantitative viability model or a parameter-space exclusion (e.g. demonstrating that flux expulsion or ground-state favorability fails across the source-defined parameter ranges enumerated in leg (a)). Consistent with the source's own exclusion logic, a robust negative result for any single link — established across these ranges, e.g. CSL is never the free-energy minimum in the stellar-relevant window, or expulsion is always crust-enclosed or energetically insufficient for FRB-class output — suffices to exclude the mechanism; a positive viability verdict requires all three links jointly, since the source's FRB-inducement claim is conditional on the strange-quark-star host existing, on the expelled flux actually reaching the surface, and on the energetics/duration match.

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, https://doi.org/10.1103/PhysRevD.111.063018, arXiv:2409.15811, https://arxiv.org/abs/2409.15811 (the arXiv v1/v2 preprint carries an earlier title, 'Quark Stars as Hideouts For Color-spin-locked Quark Matter...')
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