Physics Overview

The present model relies on a bottom-up construction based on the gauge-gravity duality. Within this framework, the thermodynamic properties of the strongly coupled plasma, such as the $T$, $s$ $P$ , $\rho_B$ and $\chi^B_2$, are extracted from the equilibrium black hole background.In short, the equation of state is calculated for a theory living in the 3+1-dimensional boundary of a Einstein–Maxwell–Dilaton (EMD - that is, gravity + scalar field + gauge field) theory in a 5-dimensional asymptotically anti-deSitter (AdS) spacetime. In addition, by studying linear perturbations around this equilibrium, the model also provides access to the holographic transport coefficients and energy loss. Which include the baryon conductivity ($\sigma_B/T$), diffusion coefficient($TD_B$), bulk viscosity ($\zeta /s$), and other out-of-equilibrium observables.

In this description, one identifies the properties of a charged black-hole in the bulk theory with the thermodynamics of a strongly-coupled fluid in a 3+1-dimensional slice of the AdS spacetime. The dynamics are governed by the EMD action,

\[S = \frac{1}{2\kappa_5^{2}} \int_{\mathcal{M}_5} d^{5}x\sqrt{-g}\Bigg[R - \frac{1}{2}(\partial_\mu\phi)^2 - V(\phi) - \frac{f(\phi)}{4}F_{\mu\nu}F^{\mu\nu}\Bigg]\]

with $F_{\mu\nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}$, $\mu,\nu=0,\dots,4$, and $2\kappa_5^2=16\pi G_5$, where $G_5`$ denotes Newton’s constant in 5 spacetime dimensions. Here, $R$ and $\sqrt{-g}$ are the Ricci scalar and the square root of the metric determinant, $A_\mu`$ is a Maxwell $U(1)$$ gauge field dual to baryon number, and $\phi$ is a scalar dilaton field responsible for breaking conformal symmetry. Thermal quantities are encoded, for instance, in the Hawking temperature $T$ and the Bekenstein–Hawking entropy (density) $s`$ of the black brane. The bulk $U(1)$ gauge field $A_mu$ is responsible for endowing the black-brane with baryon number. Its boundary value $A_t\big|_{r\to\infty}\equiv \Phi(\infty)$ sets $\mu_B$, and the radial electric flux determines the baryon density $n_B$. The breaking of conformal symmetry by the dilaton field $\phi$ is controlled by the potential $V(\phi)$, chosen so the model can reflect the QCD running coupling, while the potential $f(\phi)$ models the coupling between the nonconformal dynamics and the baryon sector, where it is important that $V(0)=-12$ and $f(0)=1$.

In this module, we have two major models. The polynomial hyperbolic model is implemented with the following functional forms,

\[V(\phi) = -12 \cosh(\gamma \phi) + b_2 \phi^2 + b_4 \phi^4 + b_6 \phi^6 ,\]

\[f(\phi) =\frac{\operatorname{sech}\,(c_{1}\phi+c_{2}\phi^{2}+c_{3}\phi^{3})}{1+d_{1}} + \frac{d_{1}}{1+d_{1}}\operatorname{sech}\,(d_{2}\phi) .\]

The MUSES parametric model is implemented with the following functional forms,

\[V(\phi) = -12\cosh\left[\left(\frac{\gamma_1\,\Delta\phi_V^2 + \gamma_2 \,\phi^2}{\Delta \phi_V^2 + \phi^2}\right) \phi\right],\]

\[f(\phi) = 1 - (1-A_1) \left[\frac{1}{2} + \frac{1}{2}\tanh\left(\frac{\phi - \phi_1}{\delta \phi_1}\right)\right] - A_1\left[\frac{1}{2} + \frac{1}{2}\tanh\left(\frac{\phi - \phi_2}{\delta \phi_2}\right)\right].\]

The construction of the model is presented in Refs. [1] and [2] and is inspired by earlier models such as the one in Ref. [3]. Since the publication of [2], two other groups have also proposed similar models capable of reproducing lattice data, and which will also be made available in the present code: [4] and [5].

Transport Coefficients

Beyond thermodynamics, the holographic framework provides a natural way to compute real-time, out-of-equilibrium transport properties. These are extracted by studying small linear perturbations around the equilibrium black-brane background and computing the associated retarded Green’s functions via the AdS/CFT prescription [6].

For the baryon conductivity $\sigma_B$ and the baryon diffusion coefficient $D_B$, one considers fluctuations of the $U(1)$ gauge field $\delta A_x$ transverse to the momentum. The corresponding perturbation equation takes the form of a second-order ODE for a complex field $P(r)$, sourced by the background fields of the black brane. Imposing infalling boundary conditions at the horizon and reading off the near-boundary behavior, one obtains $\sigma_B$ via a Kubo formula.

For the bulk viscosity $\zeta$, one considers coupled perturbations of the metric and dilaton in the sound channel. The corresponding field $F(r)$ satisfies a similar second-order ODE. Both sets of equations are solved simultaneously with the background ODE system, sharing the same adaptive radial integration from the near-horizon region to the AdS boundary.

Energy-loss observables — the jet quenching parameter $\hat{q}$, the drag force, and the Langevin diffusion coefficients $\kappa_\perp$ and $\kappa_\parallel$ — are computed from the worldsheet fluctuations of a trailing string moving through the plasma at a given velocity, following Refs. [7] [8].

References

[1] R. Critelli, J. Noronha, J. Noronha-Hostler, I. Portillo, C. Ratti and R. Rougemont, ``Critical point in the phase diagram of primordial quark-gluon matter from black hole physics,’’ Phys. Rev. D 96 (2017) no.9, 096026
[2] J. Grefa, J. Noronha, J. Noronha-Hostler, I. Portillo, C. Ratti and R. Rougemont, ``Hot and dense quark-gluon plasma thermodynamics from holographic black holes,’’ Phys. Rev. D 104 (2021) no.3, 034002
[3] O. DeWolfe, S. S. Gubser and C. Rosen, ``A holographic critical point,’’ Phys. Rev. D 83 (2011), 086005
[4] J. Knaute, R. Yaresko and B. Kämpfer, ``Holographic QCD phase diagram with critical point from Einstein–Maxwell–dilaton dynamics,’’ Phys. Lett. B 778 (2018), 419-425
[5] R. G. Cai, S. He, L. Li and Y. X. Wang, ``Probing QCD critical point and induced gravitational wave by black hole physics,’’ arXiv:2201.02004 [hep-th]
[6] D. T. Son and A. O. Starinets, ``Minkowski-space correlators in AdS/CFT correspondence: Recipe and applications,’’ JHEP 09 (2002) 042
[7] C. P. Herzog, A. Karch, P. Kovtun, C. Kozcaz and L. G. Yaffe, ``Energy loss of a heavy quark moving through N=4 supersymmetric Yang-Mills plasma,’’ JHEP 07 (2006) 013
[8] S. S. Gubser, ``Drag force in AdS/CFT,’’ Phys. Rev. D 74 (2006) 126005