A note on the Holographic model for color superconductivity in d-dimension without confinement phase

TL;DR

The paper investigates whether color superconductivity (CSC) can be realized holographically in a $d$-dimensional AdS spacetime without confinement by using a $d$-dimensional Einstein–Maxwell theory with a charged scalar on a planar RN–AdS background. The onset of CSC is analyzed through the Breitenlohner–Freedman (BF) bound, with an effective mass $m_{\text{eff}}^2=m^2-\frac{q^2\phi^2}{r^2 f}$, leading to a dimensionless condition involving a function $F(\hat\mu,z,d)$ and a threshold $G(d,\tilde\mu)$; a Sturm–Liouville analysis yields the equation of state in $d=4$ and identifies the allowed color number. The main findings show that, within this minimal setup, CSC is possible for $N_c=2$ only when $d=4$; CSC with $N_c\ge3$ or in other dimensions requires modifications to the gravity or gauge sector, and there is no CSC in $2d$ in agreement with the Coleman–Mermin–Wagner theorem. These results delineate dimensional and color-count constraints for holographic CSC in non-confining theories and guide future work toward confinement, magnetic fields, backreaction, and holographic entanglement entropy calculations.

Abstract

In this note, we will generalize the concept of the holography for the color superconductivity (CSC) phase becomes to d-dimension AdS instead of 6d. The dual field theory live in (d-1)-dimension $SU(N_c)$ and have no confinement phase contradiction with the QCD color superconductivity. And we will try to use holographic model with Einstein-Maxwell gravity in d dimension AdS and we study this phase with $N_c\geq 2$. And after, we will discuss the equation of state of the color superconductivity in $d=4$ case without confinement via holography

Figures (1)

• Figure 1: Our numerical investigation for $G(d,\tilde{\mu})$ in Eq. \ref{['G_man']}. This calculation was done using MatLab R2023a. (A) The surface function $G(d,\tilde{\mu})$ inside the region of interests i.e. $(d,\tilde{\mu}) \in [4,11] \times [0,1]$. (B) We zoom into the small corner where $G(d,\tilde{\mu})>2$ can be realized.