Application of the holographic equations of state for modeling experiments on heavy ion collisions

In this paper, we propose a method for numerical modeling of the nuclear matter properties within the framework of relativistic heavy-ion collisions using a holographic equation of state. Machine learning methods were applied to address the regression and optimization issues during the calibration of the relevant parameters using the LQCD results for quark masses that approximate the physical values. Numerical simulations are performed using the iEBE-MUSIC and vHLLE-SMASH frameworks, which incorporate certain relativistic hydrodynamics solvers. We modify the code by implementing a tabulated holographic equation of state, enabling simulations of quark-gluon plasma evolution with dynamically generated initial conditions via the 3D Monte Carlo Glauber Model and SMASH. Finally, the spectra of produced hadrons are computed using a hybrid iSS+UrQMD and Hadron Sampler+SMASH approaches at the freeze-out stage.12 p

💡 Research Summary

The manuscript presents a comprehensive framework that brings together holographic gauge‑gravity duality, lattice QCD data, and state‑of‑the‑art relativistic hydrodynamic simulations of heavy‑ion collisions. The authors start from a “bottom‑up” soft‑wall AdS/QCD model in five dimensions, introducing two dilaton scalar fields and two Maxwell fields. The bulk metric is deformed by a function b(z) and an anisotropy parameter ν, allowing the construction of both isotropic (ν = 1) and anisotropic (ν ≠ 1) versions of the equation of state (EoS). Two functional forms for the deformation factor are examined: the traditional logarithmic form A(z)=−a ln(bz²+1) and a newer logarithmic‑quadratic form A(z)=d ln(az²+1)+d ln(bz⁴+1). By solving the Einstein‑Maxwell‑dilaton equations of motion, the authors obtain analytic expressions for temperature T, entropy density s, pressure p, and energy density ε as functions of the horizon coordinate z_h and the model parameters a, b, d, G.

Calibration of these free parameters is performed against lattice QCD results for the dimensionless quantity s/T³ and the quark susceptibility χ = ∂ρ_q/∂μ_q. Recognizing that a straightforward least‑squares fit suffers from a highly non‑convex χ² landscape—especially in the anisotropic case—the authors introduce a two‑stage machine‑learning pipeline. In the first stage, a four‑layer feed‑forward neural network (64‑128‑64‑1 ReLU units) is trained on the lattice data using mean‑squared‑error loss and the Adam optimizer (learning rate 0.001). The trained network provides a smooth surrogate for the lattice thermodynamics over the entire temperature range. In the second stage, the surrogate predictions are used as a reference to define a loss function that measures the discrepancy between the holographic model’s thermodynamic outputs (computed from the current set of parameters) and the neural‑network predictions. The same Adam optimizer is then employed to minimize this loss, yielding calibrated values of a, b, d, G that achieve substantially lower χ² than a direct fit. This approach stabilizes the optimization even when the anisotropy parameter ν is varied.

Having obtained calibrated holographic EoS tables, the authors embed them into two widely used hybrid simulation frameworks: iEBE‑MUSIC and vHLLE‑SMASH. Initial conditions are generated with a three‑dimensional Monte‑Carlo Glauber model (implemented in SMASH) for iEBE‑MUSIC, and directly with SMASH for the vHLLE‑SMASH chain. The hydrodynamic evolution proceeds with MUSIC (MUSCL‑type solver) or vHLLE, respectively, using linear interpolation of the tabulated EoS. Freeze‑out is performed at a fixed energy‑density hypersurface; particle sampling is carried out by iSS (followed by UrQMD rescattering) in the iEBE‑MUSIC chain, and by a Hadron Sampler plus SMASH in the vHLLE‑SMASH chain.

The authors compare the resulting transverse‑mass (m_T) spectra of positively charged kaons (K⁺) at √s = 8.9 GeV (central NA49 collisions, impact parameter b < 2 fm) for four model variants: isotropic vs. anisotropic, each with the standard logarithmic deformation factor or the newer logarithmic‑quadratic factor. As a benchmark, they also show results obtained with the NEOS EoS (for MUSIC) and the AZ‑Hydro EoS (for vHLLE). In both simulation chains, the holographic EoS reproduces the experimental m_T spectra at least as well as the conventional EoS. Notably, the alternative deformation factor (log‑quadratic) yields a markedly better agreement with lattice thermodynamics (lower χ² for s/T³ and χ) and consequently improves the description of the kaon spectra, especially in the low‑m_T region where the QGP equation of state is most influential. The anisotropic holographic model shows a modest enhancement of the high‑m_T tail, hinting at sensitivity to spatial anisotropy in the bulk metric.

The paper’s contributions can be summarized as follows:

1. Physical Realization of Holographic EoS – The authors translate a sophisticated AdS/QCD construction into a practical tabulated EoS suitable for heavy‑ion hydrodynamics.
2. Machine‑Learning‑Assisted Calibration – By coupling a neural‑network surrogate with gradient‑based optimization, they overcome the non‑convexity of the parameter‑fit problem and achieve a more reliable extraction of the holographic parameters.
3. Integration into Established Simulation Suites – The holographic EoS is successfully incorporated into iEBE‑MUSIC and vHLLE‑SMASH, demonstrating compatibility with existing pre‑equilibrium, hydrodynamic, and hadronic afterburner modules.
4. Phenomenological Validation – Comparison with NA49 K⁺ m_T spectra shows that the holographic approach can match, and in some aspects surpass, traditional EoS models, especially when the newer deformation factor is employed.

The study also acknowledges limitations. The calibration is performed at vanishing baryon chemical potential (μ_B ≈ 0); extending the method to finite μ_B—where the sign problem hampers lattice calculations—remains an open challenge. Moreover, the χ² landscape’s complexity suggests a risk of over‑fitting, and the current validation focuses on a single observable (kaon m_T spectra) at one collision energy. Future work could explore a broader set of observables (flow coefficients, femtoscopic radii, higher‑order cumulants) and test the holographic EoS at lower beam energies where μ_B is sizable.

In conclusion, this work bridges a gap between abstract holographic QCD models and concrete heavy‑ion phenomenology. By demonstrating that a holographically derived EoS can be calibrated against lattice data with modern machine‑learning tools and can be embedded into realistic dynamical simulations, the authors open a promising avenue for incorporating strong‑coupling insights into the quantitative description of the quark‑gluon plasma.