Backreaction, color superconductivity, and the fate of the quarkyonic phase in the holographic QCD phase diagram

A widely used bottom-up holographic QCD model based on a five-dimensional Einstein–Maxwell–Scalar system with probe matter fields (a chiral scalar and open strings) reproduces a QCD phase diagram in which, beyond a reduced triple point, the deconfinement transition is forced onto the chiral transition line by a 'bypass mechanism' involving the bulk black hole phase transition, eliminating the dynamically stable quarkyonic region. The authors explicitly flag what the analysis omits: the matter fields' backreaction on the background, color-superconducting (CSC/CFL) phases at low temperature and large chemical potential, and the model-independence of the quarkyonic conclusion. The open problem is to complete and stress-test this holographic phase diagram: determine whether the coincidence/bypass mechanism and the disappearance of the quarkyonic phase survive full backreaction of the matter fields, how CSC/CFL pairing at low temperature and large baryon chemical potential reshapes the diagram, and whether a stable quarkyonic region exists for finite number of colors $N_c$ beyond model-dependent constructions.

Background

Quantum chromodynamics (QCD) at finite temperature $T$ and baryon chemical potential $\mu$ is expected to exhibit a rich phase structure: a confined hadronic phase at low $T$ and small $\mu$, a deconfined quark–gluon plasma at high $T$, and, at low $T$ and large $\mu$, exotic dense phases including color superconductivity (CSC), in which quarks near the Fermi surface pair; in the color–flavor-locked (CFL) variant pairing locks color and flavor symmetries. McLerran and Pisarski argued that at large number of colors $N_c$ a fourth phase should appear: 'quarkyonic' matter, which confines color yet has a Fermi sea of deconfined-like quarks, with baryon density as its conjectured order parameter. Whether such a phase survives at the physical value $N_c=3$ is unsettled: lattice QCD (first-principles simulation) is blocked at large $\mu$ by the sign problem (the fermion determinant becomes complex, defeating Monte Carlo sampling), so model studies carry the burden there.

The gauge/gravity duality (holography) addresses strongly coupled QCD by mapping it to a classical gravitational theory in one higher dimension. A popular bottom-up construction is the Einstein–Maxwell–Scalar (EMS) or Einstein–Maxwell-dilaton (EMD) model: a five-dimensional black-hole spacetime with a neutral scalar and a Maxwell field encoding the gluon sector, with parameters calibrated to lattice thermodynamics at $\mu=0$. Chiral symmetry breaking ($\chi$SB, the formation of the quark condensate $\langle\bar\psi\psi\rangle$) is probed by a bulk scalar $X$, and confinement by open strings in the Nambu–Goto action (the string-frame analogue of the Wilson loop): strings fail to reach the boundary when the string-frame geometry develops a 'dynamical wall'.

In this framework, Yi Yang and Pei-Hung Yuan (Phys. Lett. B 832 (2022) 137212; arXiv:2011.11941) proposed to interpret the bulk black hole (Hawking–Page-type) phase transition — a first-order jump between small and large black holes — as the quarkyonic transition in the dual QCD, justified by a jump in baryon density at the transition. They found that the quarkyonic line coincides with the chiral line, and introduced a 'bypass mechanism': for $\mu$ above a reduced triple point $(\mu_t,T_t)\simeq(0.607,0.150)$ (model units), the black hole horizon jumps past the dynamical wall, forcing deconfinement to occur at the same temperature, so deconfinement, $\chi$SB, and quarkyonic transitions all coincide and the quarkyonic region beyond the triple point disappears. This explained the observed coincidence of chiral and deconfinement crossovers near $\mu=0$ and matched chemical freeze-out data from heavy-ion experiments — but only within two deliberate simplifications the authors themselves flag: all matter fields (the chiral scalar $X$ and the probe strings) were treated as probes whose gravitational backreaction on the EMS background was ignored, and CSC/CFL phases at low $T$ and large $\mu$ were excluded by hand ('We will not consider CSC/CFL phase in this work'). Moreover, other holographic constructions reach opposite conclusions — e.g. a quenched dynamical holographic model by Chen, Li, Hou, and Huang finds a quarkyonic phase — so whether the absence of a dynamically stable quarkyonic region is a generic feature of QCD at finite $N_c$ or an artifact of one model's probe approximation remains unknown.

Problem Statement

Determine the complete and robust holographic QCD phase diagram at finite temperature $T$ and baryon chemical potential $\mu$, and thereby the fate of the quarkyonic phase at finite $N_c$, by resolving the three limitations that Yang and Yuan explicitly flagged in their probe EMS model:

1. Backreaction. Include the full gravitational backreaction of the matter fields — the chiral scalar $X$ dual to the quark condensate and the open-string sector dual to the confinement probe — on the Einstein–Maxwell–Scalar background, and determine how the background thermodynamics, the string-frame dynamical wall, and the coincidence/bypass mechanism (in which the black hole horizon jump forces the deconfinement transition onto the quarkyonic/chiral transition line for $\mu>\mu_t$) are modified. In particular: does the elimination of the dynamically stable quarkyonic region beyond the reduced triple point survive, or is it an artifact of the probe approximation?

2. Color superconductivity. Extend the holographic phase diagram to the low-temperature, large-$\mu$ region in which QCD is conjectured to enter color-superconducting (CSC/CFL) phases, and determine how the pairing transitions reshape the diagram and interact with the quarkyonic and deconfinement/chiral transition structure.

3. Universality of the quarkyonic conclusion. Assess whether the absence (or presence) of a dynamically stable quarkyonic region is generic across distinct holographic constructions — given that other holographic models, e.g. the quenched dynamical model of Chen et al., do find a quarkyonic phase — and whether the conclusion is consistent with QCD proper at finite $N_c$, where lattice simulation at large $\mu$ is blocked by the sign problem.

These three subgoals jointly serve one objective: establishing whether the coincidence of deconfinement and chiral symmetry restoration and the disappearance of the quarkyonic phase found in the probe model are robust features of dense QCD rather than model artifacts. The corrected phase diagram, together with an argued conclusion on quarkyonic-matter existence that holds beyond a single model choice, bears on this objective.

References

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