Rho meson condensation at finite isospin chemical potential in a holographic model for QCD

TL;DR

The paper analyzes finite isospin density in a holographic QCD model (the Sakai-Sugimoto construction) at temperatures below chiral restoration. Using a combination of chiral Lagrangian reasoning and full five-dimensional gauge dynamics on D8-branes, it shows that small $\mu_I$ drives a charged pion condensate, while at a larger threshold $\mu_{\text{crit}} \approx 1.7\, m_\rho$ the rho meson becomes unstable and condenses, yielding a new ground state that also breaks rotational symmetry. The rho-condensed phase coexists with the pion condensate and emerges through a second-order transition with $\langle\rho\rangle \propto \sqrt{\mu_I-\mu_{\text{crit}}}$. The work provides a concrete holographic realization of rho condensation in QCD-like theories, discusses the stability and generalizations of the new phase, and outlines how corrections and higher-density regimes might modify the phase structure.

Abstract

We analyze the effect of an isospin chemical potential μ_I in the Sakai-Sugimoto model, which is the string dual of a confining gauge theory related to large N_c QCD, at temperatures below the chiral symmetry restoration temperature. For small chemical potentials we show that the results agree with expectations from the low-energy chiral Lagrangian, and the charged pion condenses. When the chemical potential reaches a critical value μ_I = μ_{crit} ~ 1.7 m_ρ, the lowest vector meson (the "rho meson") becomes massless, and it condenses (in addition to the pion condensate) for μ_I > μ_{crit}. This spontaneously breaks the rotational symmetry, as well as a residual U(1) flavor symmetry. We numerically construct the resulting new ground state for μ_I > μ_{crit}.

Figures (7)

• Figure 1: The topologies of the background and of the D8-branes in the three phases of the Sakai-Sugimoto model. For $L > 0.97 R$, the model jumps directly from the first phase to the third phase at $T=1/2\pi R$.
• Figure 2: Introducing a vectorial isospin chemical potential leads to an instability for two of the pions (top left), and the theory wants to sit at a new minimum $U=U_{\text{min}}$ (bottom). A global symmetry transformation brings us back to the vacuum $U=I$; this rotation changes the vector chemical potential to an axial one (top right).
• Figure 3: The frequencies of zero spatial momentum solutions for the transverse vectors (left) and scalars (right), as a function of the isospin chemical potential, in units of $\sqrt{u_{\Lambda}/R_{\text{D4}}^3}=2/(3R)$. The frequencies at $\mu_I=0$ reproduce the spectrum found in Sakai:2004cn.
• Figure 4: The energy $\omega$ as a function of the isospin chemical potential $\mu_I$, in units of $\sqrt{u_{\Lambda}/R_{D4}^3}=2/(3R)$, for the sector consisting (at $\mu_I=0$) of the pion $\pi$ and the first two longitudinal vector mesons $\rho_L$ and $a_1$. At $\mu_I\approx 2.1*2/(3R)$ the first longitudinal vector meson becomes unstable. This coincides with the instability of the transverse vector meson depicted in figure \ref{['f:transverse_and_scalars']}.
• Figure 5: Appearance of the new ground state in which the $\rho$ meson has condensed. This new ground state also still has a non-zero pion condensate. We plot $(-c_3)$ (in units of $u_{\Lambda}$) and $(-d_3)$ (in units of $(u_{\Lambda}/R_{D4})^{3/2}$) for both solutions, where $c_3$ and $d_3$ are defined by $A_0^{(3)} = \mu_I (1/2 + c_3 / \pi z + \cdots)$, $A_3^{(1)} = d_3/z + \cdots.$