Background: The SMASH-vHLLE-hybrid model integrates the SMASH transport code for the hadronic phase and the vHLLE hydrodynamic model for the quark-gluon plasma (QGP) phase in heavy-ion collisions. Bayesian analysis offers a robust statistical method to constrain model parameters of a hybrid approach and evaluate uncertainties by comparing predictions with experimental data. Traditional modeling often relies on parametric assumptions for initial conditions, which may limit the ability to constrain the parameter space due to overfitting.

Purpose: This study aims to apply statistical learning, specifically Bayesian inference, to the $(3+1)\text{−}\mathrm{dimensional}$ [(3+1)D] SMASH-vHLLE-hybrid model using initial conditions generated by the SMASH transport code itself, with the objective of constraining model parameters and gaining deeper insight on the temperature and baryochemical potential dependence of both the shear and the bulk viscosity.

Method: This study is performed in the hybrid approach SMASH-vHLLE, composed of the hadronic transport approach SMASH and the $(3+1)\mathrm{D}$ viscous hydrodynamic code vHLLE. A Bayesian framework is employed, utilizing Markov-chain Monte Carlo sampling to explore the parameter space. The analysis compares model predictions against experimental observables in the range of $\sqrt{s_{NN}}=7.7\text{–}200$ GeV, including particle yields, momentum, and flow coefficients both at midrapidity as well as in forward and backward direction.

Results: We find that the SMASH-vHLLE-hybrid framework, using hadronic initial conditions for $\mathrm{Au}+\mathrm{Au}$ collisions at different beam energies, can reproduce a variety of experimental observables at midrapidity and forward/backward rapidities. Notably, the preferred posterior distribution suggests a near-vanishing specific shear viscosity in the high-temperature QGP phase, combined with moderate-to-large bulk viscosity around the phase transition region, although the constraints on baryochemical potential dependence are weak.

Conclusions: Our findings reveal that a hadronic initial condition constrains the evolution more strictly at intermediate energies, making parameters such as the hydrodynamic onset time highly sensitive. Intriguingly, the extracted shear viscosity differs substantially from previous Bayesian analyses, motivating further systematic studies with higher-statistics data sets and refined modeling assumptions.

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• Collective flow
• Relativistic heavy-ion collisions
• Viscosity
• Bayesian methods
• Data analysis