Quark-hadron continuity: establishing (or refuting) the continuous connection between hadronic matter and the color-flavor-locked phase of QCD

Schaefer and Wilczek (Phys. Rev. Lett. 82, 3956 (1999)) proposed that, for three (nearly) degenerate quark flavors, strongly coupled hadronic matter at low baryon density may go over into the calculable color-flavor-locked (CFL) phase of quark matter at high density without a phase transition ('quark-hadron continuity'). The paper itself acknowledged five limitations delimiting this claim: the model character of the gap calculation (one-gluon-exchange-inspired four-fermion kernel, mean-field/cactus approximation, sharp cutoff); the need for a controlled treatment of singular near-forward magnetic gluon exchange; the omission of unequal quark masses, electromagnetism, and neutrality constraints; the uncertain status of the H dibaryon on which the hadronic-side superfluidity rests; and the dependence of numerical gap and energy estimates on regularization and parameter choices. The asymptotic weak-coupling program (controlled CFL gap including the long-range magnetic interaction) and the neutrality/strange-mass framework are now established, so those acknowledged gaps have been substantially filled at asymptotic density. The surviving open core is the hadronic side and the realistic scenario: whether the H dibaryon is bound or near-threshold in QCD with physical quark masses, whether dense hadronic matter realizes the superfluidity with the quantum numbers that continuity requires (H condensation or baryon pairing near the Fermi surface, including in the three-degenerate-flavor idealization where condensation at any finite density is unverified from first principles), and whether continuity survives realistic quark masses, electromagnetism/neutrality, meson condensation, and anomaly effects, or a phase transition or critical point intervenes at neutron-star densities.

Background

Quantum chromodynamics (QCD), the theory of the strong interaction, is asymptotically free: its coupling becomes weak at high energy, and correspondingly at large baryon chemical potential $\mu$ (large density) the relevant excitations are quarks near a Fermi surface (the surface in momentum space separating filled from empty quark states at zero temperature). By the Bardeen-Cooper-Schrieffer (BCS) mechanism familiar from superconductivity, even weak attractions between quarks near the Fermi surface destabilize that surface: quarks form Cooper pairs and open an energy gap $\Delta$ in the fermionic spectrum. Because quarks carry color, the attractive one-gluon-exchange channel (antisymmetric in color) leads to pairing that breaks color gauge symmetry ('color superconductivity'); the dominant condensate is a scalar diquark (two-quark) operator. For three degenerate quark flavors the ground state at asymptotically large $\mu$ is the color-flavor-locked (CFL) phase (Alford, Rajagopal and Wilczek, Nucl. Phys. B 537, 443 (1999)): the diquark condensate locks color rotations to flavor rotations, all eight gluons acquire masses by the Anderson-Higgs (Meissner) mechanism, chiral symmetry (the symmetry under independent rotations of left- and right-handed light quarks) is spontaneously broken, baryon number is broken modulo two (a superfluid, with a massless scalar Nambu-Goldstone mode - the generic massless excitation produced whenever a continuous symmetry is spontaneously broken), and a linear combination of the photon and a color generator remains massless - the 'modified photon', with respect to which the gapped quark excitations carry exactly the integer charges of the baryon octet plus one singlet. The CFL pattern $SU(3)_c \times SU(3)_L \times SU(3)_R \times U(1) \to SU(3)_{c+L+R}$ is, remarkably, the symmetry pattern of dense hadronic matter with chiral symmetry breaking, confinement, and superfluidity.

Schaefer and Wilczek (Phys. Rev. Lett. 82, 3956 (1999)) sharpened this into the 'quark-hadron continuity' proposal: that for three (nearly) degenerate flavors, strongly coupled hadronic matter at low density and the weakly coupled CFL state at high density might be two ends of one phase, with no phase transition between them - making the CFL calculation a controlled weak-coupling realization of confinement and chiral symmetry breaking. The apparent mismatches (an extra massless singlet scalar, eight rather than nine vector mesons, nine rather than eight baryons) are resolved on the hadronic side by superfluidity: for three degenerate flavors the lightest object per unit baryon number is expected to be the H dibaryon, the flavor-singlet spin-zero six-quark state $uuddss$ predicted by Jaffe (Phys. Rev. Lett. 38, 195 (1977)) to be anomalously light, so that any finite baryon density should form an H condensate - or, if H lies above threshold, baryons near their Fermi surface should pair via BCS attraction in the H channel - supplying exactly the superfluidity and the instability of the singlet vector meson that the CFL side demands. In the same paper the authors verified, in a mean-field model with the leading attractive interaction (a one-gluon-exchange-inspired four-fermion contact vertex of strength $K$), that the CFL order parameter minimizes the effective potential for three (and more) flavors; the calculation resummed 'cactus' diagrams, used a sharp three-momentum cutoff $\Lambda$ with illustrative parameters ($\mu = 0.5$ GeV, $\Lambda = 0.6$ GeV, $K = 3.33/\Lambda^2$), and found the order parameter antisymmetric in color and flavor with many 'trivial flat directions' that subleading interactions and six-quark vertices induced by instantons (nonperturbative classical tunneling solutions of the gauge-field equations; the corresponding 't Hooft vertex) would lift.

The paper was explicit about what remained open. (i) The gap calculation used only the leading attractive interaction at mean-field level with an uncontrolled cutoff, so gaps and condensation energies were illustrative. (ii) A fully rigorous weak-coupling treatment must handle extremely near-forward quark scattering mediated by magnetically polarized gluon exchange, which is singular in straightforward perturbation theory because static magnetic gluons receive no perturbative screening mass ('absence of a magnetic mass'); in the CFL phase this is ameliorated self-consistently because all gluons acquire Higgs masses. (iii) Unequal quark masses and electromagnetism - including overall electric neutrality and Coulomb energy, essential for neutron-star interiors - were left for the future, and could alter pairing patterns, gap magnitudes, and the modified photon. (iv) The hadronic-side conjecture that the H dibaryon is light enough to condense (or that baryon pairing supplies the superfluidity) was, for QCD with realistic quark masses, 'unclear, both theoretically and experimentally'. (v) A systematic renormalization and matching of the model parameters to QCD dynamics was lacking. The common scientific objective uniting these five items is the paper's own: to decide whether the quark-hadron continuity scenario is actually realized in QCD.

Subsequent developments defined the modern landscape. Son (Phys. Rev. D 59, 094019 (1999)) incorporated the long-range, unscreened magnetic interaction into the gap equation and obtained the controlled asymptotic CFL gap $\Delta \sim \mu\, g^{-5} \exp(-3\pi^2/(\sqrt{2}\, g))$; Pisarski and Rischke (Phys. Rev. D 61, 051501 (2000)) established the resulting $\exp(-1/g)$ scaling, and Hong's high-density effective theory (Nucl. Phys. B 582, 451 (2000)) organized the expansion in powers of the small coupling $g$ at the Fermi surface - so the asymptotic weak-coupling program is now considered controlled. Electric/color neutrality and strange-quark-mass effects became a mature subfield: enforced electrical neutrality of CFL (Rajagopal and Wilczek, Phys. Rev. Lett. 86, 3492 (2001)), gapless CFL phases (Alford, Kouvaris and Rajagopal, Phys. Rev. Lett. 92, 222001 (2004)), kaon condensation and effects of the axial anomaly (the quantum-mechanical breaking of part of the light-quark axial symmetry) in Ginzburg-Landau (symmetry-based order-parameter) analyses (Schmitt, Stetina and Tachibana, Phys. Rev. D 83, 045008 (2011)), all synthesized in the review of Alford, Schmitt, Rajagopal and Schaefer (Rev. Mod. Phys. 80, 1455 (2008)). On the hadronic side, lattice QCD has studied the H dibaryon: it is bound at heavy pion masses and in the SU(3) flavor limit, but at the physical point the evidence is inconclusive or disfavors binding, while the continuity question itself has been re-examined topologically (Hirono and Tanizaki, arXiv:1811.10608), extended to two-flavor pairing analogues (Fujimoto, Fukushima and Weise, Phys. Rev. D 101, 094009 (2020)), and pursued phenomenologically in neutron-star equations of state (Baym et al., Rep. Prog. Phys. 81, 056902 (2018); Kojo, arXiv:2412.20442).

Problem Statement

Decide whether the quark-hadron continuity scenario of Schaefer and Wilczek (PRL 82, 3956 (1999)) is realized in QCD: is hadronic matter at low baryon density continuously connected, without a phase transition, to the color-flavor-locked (CFL) phase of quark matter at high density - for the idealized theory with three degenerate quark flavors, and for real QCD with unequal quark masses and electromagnetism at neutron-star-relevant densities? The components of this question are: (1) the hadronic-side superfluidity on which the whole matching rests - determine the status of the H dibaryon (bound, near-threshold resonance, or unbound) in QCD with physical quark masses, and establish whether dense hadronic matter develops the required superfluidity, whether via condensation of H (expected in the three-degenerate-flavor theory, where H should be the state of smallest energy per baryon number) or via baryon pairing near the Fermi surface in the H channel; (2) the quark-side counterpart under realistic conditions - the phase structure of color-superconducting quark matter subject to unequal quark masses, electromagnetism and overall electric/color neutrality, including whether kaon condensation, gapless phases or axial-anomaly effects disrupt the continuity or induce a critical point; (3) the controlled foundation - gap and condensation-energy predictions that go beyond the leading-interaction mean-field (cactus) treatment with a sharp cutoff, incorporating subleading interactions and instanton-induced vertices (including their role in lifting the flat directions of the order parameter) and a systematic renormalization/matching to QCD, so that quantitative statements at intermediate densities, not only asymptotically, become possible.

References

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