Color superconductivity in dense quark matter

TL;DR

This work provides a comprehensive, multi-scale account of color superconductivity in dense quark matter, establishing CFL as the asymptotic ground state and detailing how stresses from finite strange-quark mass and neutrality constraints drive a rich phase structure at lower densities. It combines rigorous weak-coupling QCD results with effective field theories and NJL-model analyses to map the high- to moderate-density regime, including CFL, 2SC, gCFL, kaon-condensed states, and crystalline color superconductivity, and links these phases to transport, neutrino processes, and neutron-star phenomenology. The key contributions include a detailed derivation of the gap equation and Meissner effects in the CFL phase, a systematic EFT framework for low-energy excitations, and robust predictions for the rigidity and transport of crystalline color superconducting quark matter with potential observational consequences for pulsar glitches, cooling, and gravitational waves. Overall, the results establish a coherent, testable picture of how QCD at high density manifests in observable astrophysical phenomena and provide a foundation for confronting neutron-star data with the physics of color-superconducting quark matter.

Abstract

Matter at high density and low temperature is expected to be a color superconductor, which is a degenerate Fermi gas of quarks with a condensate of Cooper pairs near the Fermi surface that induces color Meissner effects. At the highest densities, where the QCD coupling is weak, rigorous calculations are possible, and the ground state is a particularly symmetric state, the color-flavor locked (CFL) phase. The CFL phase is a superfluid, an electromagnetic insulator, and breaks chiral symmetry. The effective theory of the low-energy excitations in the CFL phase is known and can be used, even at more moderate densities, to describe its physical properties. At lower densities the CFL phase may be disfavored by stresses that seek to separate the Fermi surfaces of the different flavors, and comparison with the competing alternative phases, which may break translation and/or rotation invariance, is done using phenomenological models. We review the calculations that underlie these results, and then discuss transport properties of several color-superconducting phases and their consequences for signatures of color superconductivity in neutron stars.

Figures (12)

• Figure 1: (Color online) A schematic outline for the phase diagram of matter at ultra-high density and temperature. The CFL phase is a superfluid (like cold nuclear matter) and has broken chiral symmetry (like the hadronic phase).
• Figure 2: (Color online) Illustration of the splitting apart of the Fermi momenta of the various colors and flavors of quarks (exaggerated for easy visibility). In the unpaired phase, requirements of neutrality and weak interaction equilibration cause separation of the Fermi momenta of the various flavors. The splittings increase with decreasing density, as $\mu$ decreases and $M_s(\mu)$ increases. In the 2SC phase, up and down quarks of two colors pair, locking their Fermi momenta together. In the CFL phase, all colors and flavors pair and have a common Fermi momentum.
• Figure 3: (Color online) Free energy of various phases of dense 3-flavor quark matter, assuming $\Delta_{\rm CFL}=25~{\rm MeV}$. The homogeneous phases are CFL and 2SC, their gapless analogs gCFL and g2SC, and the kaon-condensed phase CFL-$K^0$. The true ground state must have a free energy below that of the gCFL phase, which is known to be unstable. The inhomogeneous phases are curCFL-$K^0$, which is CFL-$K^0$ with meson supercurrents, and 2PW, CubeX, and 2Cube45z, which are crystalline color superconducting phases. The transition from CFL-$K^0$ to curCFL-$K^0$ is marked with a dot. In 2PW the condensate is a sum of only two plane waves. CubeX and 2Cube45z involve more plane waves, their condensation energies are larger but less reliably determined, so their curves should be assumed to have error bands comparable in size to the difference between them.
• Figure 4: Upper panel: Diagrammatic representation of the quark self-energy in Nambu-Gorkov space. Curly lines correspond to the gluon propagator $D$. The quasiparticle propagators $G^+$ and $G^-$ are denoted by double lines with an arrow pointing to the left and right, respectively. The anomalous propagators $F^\pm$ in the off-diagonal entries are drawn according to their structure given in Eq. (\ref{['Xpm']}): thin lines correspond to the term $([G_0^\mp]^{-1} + \Sigma^\mp)^{-1}$, while the cross-hatched and hatched circles denote the gap matrices $\Phi^+$ and $\Phi^-$, respectively. Lower panel: The QCD gap equation (\ref{['gapeq']}) is obtained by equating $\Phi^+$ with the lower left entry of the self-energy depicted in the upper panel (the other off-diagonal component yields an equivalent equation for $\Phi^-$).
• Figure 5: Mass terms in the high density effective theory. The first diagram shows a ${\cal O}(MM^\dagger)$ term that arises from integrating out the $\psi_-$ field in the QCD Lagrangian. The second diagram shows a ${\cal O}(M^2)$ four-fermion operator which arises from integrating out $\psi_-$ and hard gluon exchanges.