Sakai-Sugimoto Brane System at High Density

TL;DR

The paper analyzes high-density behavior in the Sakai–Sugimoto brane model within weak coupling, using an effective one-gluon-exchange picture recast as four-fermion interactions and studied via renormalization group and Dyson–Schwinger equations. It demonstrates that the usual high-density instabilities—color superconductivity (BCS) for $N_c\sim 3$ and the chiral density wave (DGR) for $N_c\to\infty$—disappear when the product $\mu L$ is sufficiently large, implying a Fermi-liquid ground state in that regime. The work provides explicit gap formulas for both diquark and χDW channels and shows their dependence on the extra dimension through the scale $L$, with thresholds around $\mu L \sim 1/g$ (BCS) and $\mu L \sim e^{1/\sqrt{\lambda}}/\sqrt{\lambda}$ (χDW). It also discusses the strong-coupling holographic perspective, noting that the standard probe-limit analysis does not realize these phases and that backreaction or alternative setups may be required to capture color superconductivity or χDW in holography. Overall, the results illuminate how finite-density dynamics and extra-dimensional effects shape the QCD-like phase structure in a holographic model and, by extension, inform the QCD phase diagram at intermediate densities.

Abstract

The D4-D8 brane system of Sakai-Sugimoto model at high quark density is studied in the weak coupling regime. We show that the color superconducting phase (for Nc~3) or the chiral density wave (for Nc->infinity) disappears at very large chemical potential, or equivalently at very large compactified dimension that the model possesses. We also comment on the prospects in the strong coupling regime along with the QCD phase diagram.

Figures (12)

• Figure 1: One loop diagram that contributes to the finite density screening.
• Figure 2: Sakai-Sugimoto brane configuration. The $x_4$-direction is compactified with period $L$. The D4 world-volume fermions have the anti-periodic boundary condition. We locate the $N_f$ D8-branes and $N_f$$\overline{\text{D8}}$-branes at $x_4=0=L$ and $x_4=L/2$, respectively.
• Figure 3: A schematic phase diagram of the theory at weak coupling, high density and zero temperature. Being schematic, the straight lines may not be straight nor sharp transitions in reality.
• Figure 4: Replacing the one-gluon exchange to an effective one point interaction.
• Figure 5: The double line notations of the one-gluon exchange. The top and bottom sets represent the symmetric (or antisymmetric) and singlet channels, respectively. If we fix the colors of the incoming quarks, the symmetric channel has the fixed colors for the scattered quarks, while the singlet channel has $N_c$ choices.