Mimicking the QCD equation of state with a dual black hole

TL;DR

The paper develops a holographic approach to mimic the QCD equation of state at zero chemical potential by engineering a five-dimensional gravity theory with a single scalar field. It introduces a nonlinear master equation and adiabatic approximations to map scalar potentials V(φ) onto thermodynamic observables, enabling construction of black-hole solutions with QCD-like thermodynamics. Through explicit potentials, notably V(φ) = -12/L^2 cosh(γφ) + bφ^2, the authors demonstrate a smooth cross-over with a deep dip in the speed of sound and discuss stability, transitions, and mixed phases. They also acknowledge limitations of the supergravity setup and highlight avenues for extension to achieve a broader and more realistic QCD-like regime.

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.

Figures (7)

• Figure 1: A comparison of the exact ${d(\log s) \over d\phi_H}$ and ${d(\log T) \over d\phi_H}$ for $V(\phi)=-{12\over L^2}\cosh{\phi\over 2}$ with the adiabatic approximation, (\ref{['sAndTguess']}), and the improved approximation scheme, (\ref{['Fudgy']}) with the choice (\ref{['GuessedInterpolator']}).
• Figure 2: The speed of sound for $V(\phi) = -{12 \over L^2} \cosh {\phi \over \sqrt{6}}$.
• Figure 3: The equation of state of a black hole (red) compared to the lattice equation of state for pure glue (blue) and $2+1$ QCD. The pure glue curve is based on Boyd:1996bx and private communications from F. Karsch. The $2+1$ QCD points are based on Cheng:2007jq.
• Figure 4: Left: The potential (\ref{['bPotential']}) with the parameter choices (\ref{['bFromDelta']}) that give an equation of state resembling QCD's. Right: Although $V(\phi)$ is relatively featureless, the adiabatic formula $c_s^2 \approx {1 \over 3} - {1 \over 2} {V'(\phi_H)^2 \over V(\phi_H)^2}$ suggests that the equation of state resulting from it will indeed exhibit a low minimum for the speed of sound.
• Figure 5: The equation of state for $V(\phi)=-{12 \over L^2} \cosh \sqrt{3 \over 4} \phi + {3 \over L^2} \phi^2$.