Mimicking the QCD equation of state with a dual black hole (0804.0434v1)

Abstract: We present numerical and analytical studies of the equation of state of translationally invariant black hole solutions to five-dimensional gravity coupled to a single scalar. As an application, we construct a family of black holes that closely mimics the equation of state of quantum chromodynamics at zero chemical potential.

• The paper constructs a dual gravitational model using a five-dimensional black hole with a scalar field to mimic QCD's thermodynamics near the crossover temperature Tc.
• It employs both analytical and numerical methods to relate scalar field potential tuning with the speed of sound in the dual field theory.
• The study bridges holographic duality with QCD properties, suggesting future extensions to include finite chemical potential and shear viscosity effects.

Summary: Mimicking the QCD Equation of State with a Dual Black Hole

The paper by Gubser and Nellore addresses the challenging task of replicating the equation of state (EoS) of quantum chromodynamics (QCD) using a gravitational dual, specifically, a five-dimensional black hole model. The researchers explore the thermodynamic properties of translationally invariant black hole solutions derived from five-dimensional gravity coupled with a scalar field. Their primary focus is to model the EoS of QCD at zero chemical potential via a family of black hole solutions.

Key Contributions

The authors leverage the holographic duality between gravity theories in higher-dimensional Anti-de Sitter (AdS) spaces and strongly coupled field theories to construct a gravitational dual whose thermodynamics matches that of QCD, specifically around the cross-over temperature $T_c$. This duality offers a powerful technique to explore the properties of strongly interacting field theories, like QCD, particularly in regimes where perturbative approaches fail.

The paper demonstrates both numerical and analytical methods for establishing a connection between the scalar field potential $V(\phi)$ in the five-dimensional gravitational action and the speed of sound in the dual field theory, $c_s^2$. The central result is that by carefully tuning $V(\phi)$, the speed of sound as a function of temperature can be made to resemble that of QCD.

Methodology and Results

1. Model Construction: The researchers utilize a simplified model involving a gravity action with a scalar field. The gravity action is given by:

$$S = \frac{1}{2\kappa_5^2} \int d^5 x \, \sqrt{-g} \left[ R - \frac{1}{2} (\partial\phi)^2 - V(\phi) \right]$$

This setup does not attempt to include effects like chemical potentials, chiral symmetry breaking, or asymptotic freedom. Instead, it focuses on mimicking essential thermal features of QCD near $T_c$.

1. Thermodynamic Quantities: The authors derive expressions for entropy density $s$ and temperature $T$ from the geometry of the black hole solutions. They further compute the speed of sound $c_s^2$, a key quantity capturing the EoS characteristics:

$c_s^2 = \frac{d \log T}{d \log s}$

1. Master Equation: A significant analytical tool is the "nonlinear master equation," which relates the function $G(\phi)$, the scalar field potential $V(\phi)$, and ultimately the thermodynamic behavior of the black hole, including the speed of sound.
2. Examples and Implications: Several potential forms for $V(\phi)$ are examined to illustrate different thermodynamic behaviors. One particular choice closely mimics the QCD behavior, showing a rapid change near $T_c$, while others lead to interesting features like first-order transitions or mixed phases.

Implications and Future Directions

The work provides a framework connecting gravitational physics to high-energy physics, particularly through the holographic duality. It suggests a pathway to understanding non-perturbative QCD dynamics using black hole thermodynamics but also highlights the limitations of the dual construction, including its inability to model phenomena like confinement or the high-temperature limit fully.

Future research might explore incorporating additional fields or higher curvature corrections to model the shear viscosity of QCD more accurately. Additionally, extending these methods to include finite chemical potential or other QCD properties related to the confined phase could provide even richer insights.

By effectively capturing some key aspects of QCD's EoS, this research builds a bridge between abstract theoretical physics and the real-world complexities of strong interactions, hinting at broader applications within holographic frameworks.