A holographic model of deconfinement and chiral symmetry restoration

TL;DR

This work analyzes the finite-temperature behavior of the Sakai-Sugimoto holographic model, connecting confinement and chiral symmetry breaking to a gravity dual. By comparing two Euclidean backgrounds and D8-brane embeddings, it derives a phase diagram characterized by a deconfinement temperature $T_d = 1/(2\\pi R)$ and a chiral-restoration temperature $T_{\\chi SB} \\simeq 0.154 / L$, with a critical separation $L_c \\simeq 0.97 R$ separating regimes where the chiral transition coincides with or follows deconfinement. All phase transitions are first order in the gravity limit, and the analysis reveals a clear separation between confinement and chiral-symmetry-breaking scales when $L \\lows0 L$ is small compared to $R$. The results provide holographic insights into large-$N_c$ QCD behavior and offer a framework for comparing confinement and chiral dynamics to lattice and field-theory expectations.

Abstract

We analyze the finite temperature behavior of the Sakai-Sugimoto model, which is a holographic dual of a theory which spontaneously breaks a U(N_f)_L x U(N_f)_R chiral flavor symmetry at zero temperature. The theory involved is a 4+1 dimensional supersymmetric SU(N_c) gauge theory compactified on a circle of radius R with anti-periodic boundary conditions for fermions, coupled to N_f left-handed quarks and N_f right-handed quarks which are localized at different points on the compact circle (separated by a distance L). In the supergravity limit which we analyze (corresponding in particular to the large N_c limit of the gauge theory), the theory undergoes a deconfinement phase transition at a temperature T_d = 1 / 2 πR. For quark separations obeying L > L_c = 0.97 * R the chiral symmetry is restored at this temperature, but for L < L_c = 0.97 * R there is an intermediate phase which is deconfined with broken chiral symmetry, and the chiral symmetry is restored at T = 0.154 / L. All of these phase transitions are of first order.

Figures (7)

• Figure 1: The dominant configurations of the D8 and anti-D8 probe branes in the Sakai-Sugimoto model at zero temperature, which break the chiral symmetry. The same configurations will turn out to be relevant also at low temperatures. On the left a generic configuration with an asymptotic separation of $L$, that stretches down to a minimum at $u=u_0$, is drawn. The figure on the right describes the limiting antipodal case $L=\pi R$, where the branes connect at $u_0=u_{\Lambda}$.
• Figure 2: The high-spin "stringy" meson is a string starting at the lowest point of the probe branes $u=u_0$, going down to the wall, stretching horizontally in space along the wall, and then going back up vertically to the probe branes at $u=u_0$.
• Figure 3: The topology of the solutions which dominate in the low temperature (confined) and high temperature (deconfined) phases of the Sakai-Sugimoto model, as reflected in the $(t, u)$ and $( x_4, u)$ submanifolds.
• Figure 4: A minimal conjecture for the schematic form of the phase diagram of the $4+1$ dimensional maximally supersymmetric $SU(N_c)$ gauge theory compactified on a circle with anti-periodic boundary conditions for the fermions, as a function of the dimensionless coupling constant $\lambda_5/R$ and the dimensionless temperature $TR$, based on the known results in the various limits and on the $T \leftrightarrow 1/2\pi R$ symmetry. All solid lines in the diagram denote phase transitions; the dashed line may or may not be a phase transition line.
• Figure 5: The possible configurations of the D8 and anti-D8 probe branes in the high temperature (deconfined) phase. A generic connected configuration with an asymptotic separation of $L$, that stretches down to a minimum at $u=u_0$, is drawn in (a). This corresponds to the chiral symmetry broken phase. Figure (b) depicts a chiral-invariant solution, with separate D8 and anti-D8-branes.