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 :math:`T`, :math:`s` :math:`P` , :math:`\rho_B` and :math:`\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 (:math:`\sigma_B/T`), diffusion coefficient(:math:`TD_B`), bulk viscosity (:math:`\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, .. math:: 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 :math:`F_{\mu\nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}`, :math:`\mu,\nu=0,\dots,4`, and :math:`2\kappa_5^2=16\pi G_5`, where :math:`G_5`` denotes Newton's constant in 5 spacetime dimensions. Here, :math:`R` and :math:`\sqrt{-g}` are the Ricci scalar and the square root of the metric determinant, :math:`A_\mu`` is a Maxwell :math:`U(1)$` gauge field dual to baryon number, and :math:`\phi` is a scalar dilaton field responsible for breaking conformal symmetry. Thermal quantities are encoded, for instance, in the Hawking temperature :math:`T` and the Bekenstein--Hawking entropy (density) :math:`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 :math:`A_t\big|_{r\to\infty}\equiv \Phi(\infty)` sets :math:`\mu_B`, and the radial electric flux determines the baryon density $n_B$. The breaking of conformal symmetry by the dilaton field :math:`\phi` is controlled by the potential :math:`V(\phi)`, chosen so the model can reflect the QCD running coupling, while the potential :math:`f(\phi)` models the coupling between the nonconformal dynamics and the baryon sector, where it is important that :math:`V(0)=-12` and :math:`f(0)=1`. In this module, we have two major models. The polynomial hyperbolic model is implemented with the following functional forms, .. math:: V(\phi) = -12 \cosh(\gamma \phi) + b_2 \phi^2 + b_4 \phi^4 + b_6 \phi^6 , .. math:: 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, .. math:: 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], .. math:: 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 :math:`\sigma_B` and the baryon diffusion coefficient :math:`D_B`, one considers fluctuations of the :math:`U(1)` gauge field :math:`\delta A_x` transverse to the momentum. The corresponding perturbation equation takes the form of a second-order ODE for a complex field :math:`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 :math:`\sigma_B` via a Kubo formula. For the bulk viscosity :math:`\zeta`, one considers coupled perturbations of the metric and dilaton in the sound channel. The corresponding field :math:`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 :math:`\hat{q}`, the drag force, and the Langevin diffusion coefficients :math:`\kappa_\perp` and :math:`\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 `__