Whether a bare strange-quark-matter surface fragments into a crystal crust of charged strangelets immersed in an electron gas is decided by a critical surface tension: fragmentation occurs only if the interface tension σ\sigma of cold quark matter against vacuum is below σcrit\sigma_{\mathrm{crit}}, of order a few MeV/fm2^2, a model-independent functional of the quark-matter equation of state. Neither σ\sigma nor the equation-of-state inputs entering σcrit\sigma_{\mathrm{crit}} are known from first-principles QCD at the relevant densities: model estimates still span orders of magnitude, and the problem of computing σ\sigma (and the quark-matter charge density nQn_Q and charge susceptibility χQ\chi_Q) with controlled accuracy, deciding σ\sigma versus σcrit\sigma_{\mathrm{crit}}, and deriving the resulting crust structure and observable signatures remains open.

Background

Strange quark matter (SQM) is hypothetical deconfined matter containing roughly equal numbers of up, down, and strange quarks, proposed by Bodmer, Witten, and Farhi and Jaffe to be possibly more tightly bound than nuclear matter (the 'strange matter hypothesis'). If the hypothesis is true, compact stars made of SQM ('strange stars') could exist, and small lumps of SQM called 'strangelets' could be stable or long-lived. Whether the surface of a strange star is a bare, essentially sharp quark-matter surface carrying a huge electric field (as in the classic Alcock–Farhi–Olinto picture), or instead fragments into a crystalline crust of charged strangelets in a neutralizing electron gas (as proposed by Jaikumar, Reddy, and Steiner, Phys. Rev. Lett. 96, 041101 (2006)), changes its thermal emission, cooling, oscillation, and accretion phenomenology. Alford, Rajagopal, Reddy, and Steiner (Phys. Rev. D 73, 114016 (2006)) settled the model-independent part of this question. Expanding the quark-matter pressure to second order in the electron chemical potential μe\mu_e (the energy cost of adding an electron), the total pressure of bulk quark matter at charge neutrality is pneutral(μ,ms)≃p0(μ,ms)−nQ2/(2χQ)p_{\mathrm{neutral}}(\mu,m_s) \simeq p_0(\mu,m_s) - n_Q^2/(2\chi_Q), where msm_s is the strange-quark mass, nQn_Q is the electric charge density (in units of the positron charge) that bulk quark matter carries at zero electron chemical potential, and χQ\chi_Q is the corresponding electric charge susceptibility, which measures the response of that charge density to μe\mu_e and controls Debye screening — the screening of electric fields over a length λD=1/4παχQ\lambda_D = 1/\sqrt{4\pi\alpha\chi_Q}. Working at zero external pressure and including electrostatics and Debye screening, they constructed charged spherical strangelets and found a critical surface tension

σcrit=(1−f) x∗ nQ212πα χQ3/2,\sigma_{\mathrm{crit}} = \frac{(1-f)\,x^*\, n_Q^2}{12\sqrt{\pi\alpha}\,\chi_Q^{3/2}},

with dimensionless constants x∗≃1.6x^*\simeq 1.6 (the optimal strangelet radius in units of the Debye length) and f≃0.505f\simeq 0.505 (fixed by the electrostatic charge profile), below which bulk quark matter is unstable to fission into strangelets of radius R∗≃x∗λDR^*\simeq x^*\lambda_D and strange-star surfaces fragment into a strangelet crust. For reasonable bag-model-type equations of state, σcrit\sigma_{\mathrm{crit}} is of order a few MeV/fm2^2. The unknown input is σ\sigma itself: the quark-matter/vacuum interface tension at cold, phenomenologically relevant densities (quark chemical potential of order 300–500 MeV at zero external pressure). Bag-model estimates in the tradition of Berger and Jaffe give a few to ten MeV/fm2^2 depending on msm_s and model assumptions, but the real interface has finite thickness, QCD interactions, and possibly color superconductivity (pairing of quarks near the Fermi surface, e.g. two-flavor 2SC or three-flavor CFL phases), which tend to raise σ\sigma. First-principles lattice QCD cannot currently address this regime because the finite chemical potential introduces a sign problem; existing lattice determinations of QCD surface tensions apply only to the hot, finite-temperature quark–gluon plasma relevant to the early universe. Later model work — the Nambu–Jona-Lasinio model (an effective chiral quark model with four-fermion interactions, 'NJL'), quasiparticle models with density-dependent masses, and geometrical/density-functional approaches — yields values from a few to tens of MeV/fm2^2 and beyond, with and without pairing, magnetic fields, finite temperature, and trapped neutrinos, so the sign of σ−σcrit\sigma-\sigma_{\mathrm{crit}} has never been decided. If σ<σcrit\sigma<\sigma_{\mathrm{crit}}, Alford and Eby showed within a generic equation-of-state parametrization that the crust prefers spherical strangelets throughout (rods, slabs, and voids being disfavored at small σ\sigma) and can be up to hundreds of meters thick; if σ>σcrit\sigma>\sigma_{\mathrm{crit}}, strange stars retain fluid, bare surfaces. Observable consequences differ sharply between the two cases: cooling and thermal relaxation, oscillation and magnetar activity, consistency of X-ray burst phenomenology of accreting compact stars, possible fast-radio-burst crust-collapse scenarios, and a predicted peak in the cosmic-ray strangelet mass-to-charge distribution at baryon numbers of several hundred to a few thousand, targeted by searches such as AMS-02 on the International Space Station (no strangelet has been detected to date).

Problem Statement

Determine the surface tension σ\sigma of the interface between cold strange quark matter and vacuum at zero external pressure (quark chemical potential set by the zero-pressure condition, roughly 300–500 MeV), from first-principles QCD or a controlled effective treatment that includes finite interface thickness, QCD interactions, and possible color-superconducting pairing, together with the equation-of-state functions nQ(μ,ms)n_Q(\mu,m_s) and χQ(μ,ms)\chi_Q(\mu,m_s) (the quark-matter charge density at zero electron chemical potential and the charge susceptibility entering the criterion, with the strange-quark mass treated self-consistently) at the same densities. Use these to decide the model-independent fragmentation criterion of Alford et al. (2006): whether σ<σcrit=(1−f)x∗nQ2/(12πα χQ3/2)\sigma<\sigma_{\mathrm{crit}}=(1-f)x^* n_Q^2/(12\sqrt{\pi\alpha}\,\chi_Q^{3/2}), in which case bulk strangelets and strange-star surfaces fragment into a crystalline crust of charged strangelets in an electron gas, or σ>σcrit\sigma>\sigma_{\mathrm{crit}}, in which case strange stars have bare fluid surfaces and large strangelets are stable against fission. Where fragmentation holds, determine the resulting crust structure — including deeper layers at nonzero external pressure, the preferred droplet geometries (spheres, rods, slabs, voids) and the role of curvature energy, which is negative for voids — and derive its observable consequences (thermal emission and cooling, oscillation modes, consistency with X-ray burst phenomenology of accreting compact stars, and the predicted peak in the cosmic-ray strangelet baryon-number-to-charge distribution with its detection reach). The fragmentation criterion is fixed by σ\sigma, nQ(μ,ms)n_Q(\mu,m_s), and χQ(μ,ms)\chi_Q(\mu,m_s), and is decided by the sign of σ−σcrit\sigma-\sigma_{\mathrm{crit}}; the crust-structure and observable-consequence items hold only where fragmentation holds.

References

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