Microscopic determination of the QCD-scale surface tension of the nuclear-CFL interface: sharp interface versus mixed phase in compact stars

The minimal transition from electrically neutral nuclear matter to the color-flavor-locked (CFL) phase of dense quark matter in a compact star is first order and can be realized in space either as a single sharp interface or as a broad charge-separated mixed phase. Alford, Rajagopal, Reddy and Wilczek (Phys. Rev. D 64, 074017 (2001)) constructed the sharp interface, including its calculable charged boundary layers (proton pileup on the nuclear side, a kaon condensate on the CFL side, boundary-layer tension of about 420 MeV/fm$^2$), and showed that the mixed phase is favored only if the QCD-scale micro-boundary surface tension $\sigma_{\rm QCD}$ is below roughly 40 MeV/fm$^2$, whereas dimensional analysis suggests $\sigma_{\rm QCD}\sim 300$ MeV/fm$^2$. No microscopic calculation of $\sigma_{\rm QCD}$ exists, so the choice between the sharp interface and the mixed phase has remained undecided. This problem asks for a controlled microscopic computation of $\sigma_{\rm QCD}(\mu,\mu_e)$, including the internal structure of the $\sim 1$ fm micro-boundary, and the resulting determination of whether the minimal nuclear-CFL transition is a sharp interface or a mixed phase.

Background

At nuclear saturation density, the matter of a compact star is hadronic: in weak (beta) equilibrium, electrically neutral bulk matter consists mostly of neutrons with a small admixture of protons and electrons. At asymptotically high baryon chemical potential, three-flavor QCD instead favors the color-flavor locked (CFL) phase, in which $u$, $d$ and $s$ quarks of all three colors form Cooper pairs through a single diquark condensate that locks color and flavor rotations; the CFL phase is a color superconductor and a superfluid, breaks chiral symmetry, contains equal numbers of $u$, $d$ and $s$ quarks, and is electrically neutral without any electrons (Alford, Rajagopal and Wilczek, Nucl. Phys. B 537, 443 (1999)). Its lightest charged excitations are pseudo-Goldstone mesons, so the low-energy physics is described by a chiral effective theory with parameters such as the pion decay constant $f_\pi$, the meson velocity $v_\pi$, and the in-medium kaon mass $m_K$. In a compact star these two guaranteed phases could be joined with increasing depth either directly or through intervening phases such as two-flavor color superconductivity (2SC, pairing only $u$ and $d$ quarks) or crystalline (LOFF) color-superconducting phases; the minimal possibility is a single first-order transition from nuclear matter to CFL matter.

A first-order transition between phases with different conserved-charge compositions can be realized spatially in two limiting ways. In the Maxwell construction the two locally neutral phases meet at one sharp interface across which the pressure is constant and the density jumps. In the Gibbs construction (Glendenning, Phys. Rev. D 46, 1274 (1992)) local charge neutrality is relaxed: positively charged nuclear domains and negatively charged CFL domains interpenetrate in a mixed phase, with the domain geometry (droplets, rods, slabs, tubes, bubbles, i.e. nuclear pasta, usually treated in a Wigner-Seitz approximation) set by the balance of Coulomb energy against surface energy. Which alternative is favored therefore hinges on the interface surface tension.

Alford, Rajagopal, Reddy and Wilczek, 'The Minimal CFL-Nuclear Interface' (Phys. Rev. D 64, 074017 (2001)), constructed the first quantitative model of the sharp nuclear-CFL interface. Because bulk CFL matter contains no electrons while bulk neutral nuclear matter is electron-rich, the sharp interface must be dressed by charged boundary layers extending over the electron Debye screening length (tens of fm): a pileup of protons on the nuclear side and, on the CFL side, a condensate of negatively charged kaons, which forms wherever the effective electron chemical potential exceeds the in-medium kaon mass. They computed the boundary-layer contribution to the surface tension, $\sigma_{\rm boundary\ layer}\simeq 420$ MeV/fm$^2$, which dominates the single-interface energy and is calculable within their effective-theory treatment (a Walecka-type relativistic mean-field model for the nuclear side). The remaining piece is the fundamental tension $\sigma_{\rm QCD}$ of the underlying QCD-scale micro-boundary, the $\sim 1$ fm thick region in which quark pairing and nuclear order actually interconvert, for which no microscopic calculation existed. Dimensional analysis, using the energy-density mismatch of about 325 MeV/fm$^3$ across roughly 1 fm, gives $\sigma_{\rm QCD}\sim 300$ MeV/fm$^2$, an estimate the authors themselves warn could be a significant overestimate. Comparing the sharp interface with the mixed phase, whose bulk free-energy gain from relaxing local charge neutrality must exceed its Coulomb plus surface costs, they concluded that the mixed phase is favored only if $\sigma_{\rm QCD}\lesssim 40$ MeV/fm$^2$ in their model, while for $\sigma_{\rm QCD}\gtrsim 40$ MeV/fm$^2$ the single sharp interface is free-energetically favored. The unresolved target is thus the microscopic determination of $\sigma_{\rm QCD}$: its magnitude, its dependence on the local chemical potentials $(\mu,\mu_e)$, and its microscopic origin in the internal structure of the micro-boundary. Lattice QCD cannot currently evaluate this quantity because at nonzero baryon density the Monte Carlo sampling weight is no longer positive (the fermion sign problem), which blocks direct simulation; the model frameworks used for adjacent interface problems include the MIT bag model (deconfined quark matter with a confining vacuum-energy parameter) and the Nambu-Jona-Lasinio (NJL) model, a chiral quark model with diquark pairing channels, with finite-size effects often treated in the multiple reflection expansion (MRE).

Problem Statement

Determine, from a microscopic QCD-based description that contains both electrically neutral, beta-equilibrated nuclear matter and CFL quark matter and is controlled to a fidelity sufficient to resolve the microscopic structure of the transition, the QCD-scale surface tension $\sigma_{\rm QCD}$ of the $\sim 1$ fm micro-boundary between these two phases at their bulk coexistence point: its magnitude, its dependence on the local chemical potentials $(\mu,\mu_e)$, and its microscopic origin, that is, the internal structure (order-parameter, charge, and electrostatic profiles) of the micro-boundary through which the two ground states mesh. Using the computed $\sigma_{\rm QCD}$, decide whether the minimal nuclear-CFL transition inside a compact star is realized as a single sharp interface (Maxwell construction, dressed by the calculable charged boundary layers of the 2001 analysis) or as a charge-separated mixed phase (Gibbs construction), by comparing $\sigma_{\rm QCD}$ against the mixed-phase critical tension, the value at which the bulk free-energy gain of the mixed phase equals its Coulomb plus surface energy costs for the favored domain geometry. The determination is quantitative: the values of $\sigma_{\rm QCD}(\mu,\mu_e)$, the critical tension, the resulting sharp-versus-mixed decision, and how that decision depends on the model uncertainties (nuclear equation of state, CFL effective-theory parameters such as $f_\pi$, $v_\pi$ and the in-medium kaon mass).

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