SMASH-vHLLE-Hybrid Framework

• SMASH-vHLLE-hybrid is a modular heavy-ion collision framework that couples microscopic hadronic transport with 3+1D viscous hydrodynamics to capture multi-stage nuclear dynamics.
• It integrates dynamic initialization, consistent equations of state, and a precisely matched transport–hydrodynamics interface to minimize systematic uncertainties across beam energies.
• The framework has been validated against experimental data from RHIC, FAIR, and NICA and is adapted for applications like light-nucleus production and small-system collectivity studies.

SMASH-vHLLE-hybrid is a modular heavy-ion collision framework that couples the hadronic transport code SMASH to the $3+1$D viscous hydrodynamics code vHLLE, with SMASH used again as a hadronic afterburner. It was introduced to describe relativistic nucleus-nucleus collisions from the high-baryon-density regime relevant for the RHIC Beam Energy Scan, FAIR, and NICA up to top RHIC energies, and in later work was applied from $\sqrt{s_{NN}}=3$ GeV to $\sqrt{s_{NN}}=5.02$ TeV. Its defining feature is a transport $\rightarrow$ hydrodynamics $\rightarrow$ transport evolution in which baryon stopping, finite-$\mu_B$ dynamics, viscous fluid expansion, and late hadronic rescattering are treated within a single framework (Schäfer et al., 2021, Schäfer et al., 2021).

1. Constituent models and hybrid architecture

The hybrid combines two descriptions with different domains of applicability. SMASH is a microscopic hadronic transport approach designed to explore the QCD phase diagram at low and intermediate beam energies, with hadrons and resonances as the relevant degrees of freedom. vHLLE is a $3+1$D viscous hydrodynamic model that evolves the dense stage of the collision as a relativistic fluid (Bailung et al., 2024, Schäfer et al., 2021).

In its standard form, the collision is modeled in three stages. First, SMASH runs in pure transport mode and generates the early non-equilibrium dynamics. Second, the system is converted to hydrodynamic fields and evolved with vHLLE. Third, when the fluid becomes dilute, it is converted back to hadrons and propagated in SMASH as a hadronic afterburner (Bailung et al., 2024, Schäfer et al., 2021).

The hydrodynamic stage solves conservation equations for energy-momentum and conserved charges. In the ideal limit used for part of the conservation-law validation, these are

$$\partial_\mu T^{\mu\nu} = 0, \qquad \partial_\mu N_B^{\mu} = 0, \qquad \partial_\mu N_Q^{\mu} = 0,$$

with

$T^{\mu\nu} = (e + p)u^\mu u^\nu - p g^{\mu\nu},$

while viscous studies use the full second-order viscous formulation available in vHLLE (Schäfer et al., 2021, Schäfer et al., 2021).

A standard initialization in the published SMASH-vHLLE-hybrid uses a constant-proper-time switch from SMASH to hydrodynamics. The reference proper time is set by the nuclear passing time,

$$\tau_0 = \frac{R_p + R_t}{\sqrt{\left(\frac{\sqrt{s_\mathrm{NN}}}{2 m_\mathrm{N}}\right)^2 - 1}},$$

with a lower bound $\sqrt{s_{NN}}=3$0 in the high-energy implementation (Schäfer et al., 2021). In the standard particlization setup, the fluid is converted back to particles on a constant-energy-density hypersurface, with published studies using $\sqrt{s_{NN}}=3$1, $\sqrt{s_{NN}}=3$2, or $\sqrt{s_{NN}}=3$3, depending on the analysis (Schäfer et al., 2021, Götz et al., 2022, Góes-Hirayama et al., 25 Jul 2025).

2. Initial-state construction and dynamic fluidization

The original SMASH-vHLLE implementation constructs hydrodynamic initial conditions by evolving the nuclei in SMASH until $\sqrt{s_{NN}}=3$4, removing particles when they cross the iso-$\sqrt{s_{NN}}=3$5 hypersurface, and coarse-graining their energy-momentum tensor and conserved currents with Gaussian smearing in transverse and longitudinal directions (Schäfer et al., 2021). In that setup, the smearing widths $\sqrt{s_{NN}}=3$6 and $\sqrt{s_{NN}}=3$7 are beam-energy dependent and are part of the tuned hybrid parameter set (Schäfer et al., 2021).

A later extension pushed the framework to lower beam energies by replacing the single-time initialization with dynamic initial conditions from hadronic transport. In that scheme, SMASH is run from $\sqrt{s_{NN}}=3$8 to 20 fm, the local rest-frame energy density is evaluated on a Cartesian lattice, and hadrons in cells above a fluidization threshold $\sqrt{s_{NN}}=3$9 are designated as core hadrons that act as source terms for vHLLE. Fluidization is continuous in time and local in space, rather than being tied to one global hypersurface (Góes-Hirayama et al., 25 Jul 2025).

The dense core is identified through a local energy-density criterion. Using a covariant Gaussian kernel,

$\sqrt{s_{NN}}=5.02$0

with

$\sqrt{s_{NN}}=5.02$1

the local Landau rest frame is determined and the hadron is fluidized if

$\sqrt{s_{NN}}=5.02$2

The final low-energy setup uses $\sqrt{s_{NN}}=5.02$3, $\sqrt{s_{NN}}=5.02$4, and a string fluidization time $\sqrt{s_{NN}}=5.02$5 (Góes-Hirayama et al., 25 Jul 2025).

This dynamic source-based initialization was introduced because, at $\sqrt{s_{NN}}=5.02$6 GeV, different longitudinal slices interact at different times and local equilibration is strongly space-time dependent. The low-energy study reports good agreement with measured bulk observables between $\sqrt{s_{NN}}=5.02$7 and $\sqrt{s_{NN}}=5.02$8 GeV and presents the resulting framework as a baseline for future studies of phase-transition effects (Góes-Hirayama et al., 25 Jul 2025).

At higher energies, the status of SMASH-generated initial conditions changes. A comparative study of longitudinal deposition in SMASH and the saturation-based McDipper model finds that the two are in good agreement at lower energies, while their energy and baryon deposition differs substantially at higher center-of-mass energies. In that comparison, SMASH retains more baryon number at midrapidity and underestimates $\sqrt{s_{NN}}=5.02$9 relative to the saturation-based description at high energies, indicating that the initial-state model itself becomes a major systematic uncertainty outside the range where string-based stopping is adequate (Constantin et al., 16 May 2026).

3. Equation of state, viscosities, and particlization consistency

A central structural requirement of the hybrid is consistency between the hydrodynamic equation of state and the hadronic content used at particlization and in the afterburner. The original validation study introduced a SMASH hadron resonance gas equation of state tailored to the SMASH particle content and showed that the mapping

$\rightarrow$0

is of fundamental importance for conserving energy, baryon number, and electric charge across the hydro-to-transport interface (Schäfer et al., 2021).

With an unmodified hadron resonance gas table, conservation is badly violated at particlization, with up to $\rightarrow$1 deviations in total $\rightarrow$2, $\rightarrow$3, and $\rightarrow$4. Using the improved SMASH hadron resonance gas equation of state removes those large jumps, and the residual violations over the entire hybrid evolution are no more than $\rightarrow$5 in an ideal hydrodynamic setup over $\rightarrow$6–200 GeV (Schäfer et al., 2021).

The shear-viscosity sector has evolved from collision-energy-dependent constants to generalized finite-density parametrizations. A key form introduced for BES energies is a piecewise-linear $\rightarrow$7,

$\rightarrow$8

augmented by a linear dependence on net baryon density $\rightarrow$9. In that construction, $\rightarrow$0 and $\rightarrow$1, with the low-$\rightarrow$2 branch matched to SMASH box calculations and the high-$\rightarrow$3 branch constrained by pQCD estimates (Götz et al., 2022).

Within the hybrid, this generalized viscosity has two notable consequences. First, the explicit net-baryon dependence was found to have negligible impact on final elliptic flow and to matter mainly in the early stages of the collision. Second, the energy-density-dependent parametrization makes integrated $\rightarrow$4 nearly independent of the chosen switching energy density, which supports its interpretation as a good proxy for the effective shear viscosity of the non-equilibrium hadronic transport stage (Götz et al., 2022).

A later Bayesian analysis of the $\rightarrow$5D SMASH-vHLLE-hybrid with SMASH-generated initial conditions constrained the transport sector directly from data. That study finds that the framework can reproduce a variety of experimental observables at midrapidity and forward/backward rapidities, and that the preferred posterior suggests a near-vanishing specific shear viscosity in the high-temperature QGP phase together with moderate-to-large bulk viscosity around the phase transition region, while the constraints on baryochemical-potential dependence are weak (Götz et al., 13 Mar 2025).

The same consistency problem extends to the particle list itself. The PDG21+ study argues explicitly that the hadronic afterburner and the underlying EoS used in hydrodynamics must employ the same particle list and decays. It provides a SMASH-compatible implementation of a 418-particle PDG21+ list, together with a ThermalFIST-compatible format for building matching HRG tables, precisely to reduce species, mass-spectrum, and feed-down mismatches between hydro and afterburner (Martin et al., 2022).

4. Bulk observables, stopping, and flow systematics

The framework was first validated systematically for Au+Au and Pb+Pb collisions between $\rightarrow$6 GeV and $\rightarrow$7 GeV. In that beam-energy scan study, the SMASH-vHLLE-hybrid gives good agreement with experimental measurements of rapidity and transverse-mass distributions of identified particles, midrapidity yields, and $\rightarrow$8, including baryon stopping dynamics. In particular, it reproduces the transition from a Gaussian rapidity spectrum of protons at lower energies to the double-hump structure at high energies, and the centrality and energy dependence of charged-particle $\rightarrow$9 is described reasonably well (Schäfer et al., 2021).

The same study also clarifies where the model is less successful. Charged-hadron $\mu_B$0 is underestimated in more peripheral collisions at lower energies, and $\mu_B$1 is underpredicted across energies and centralities, which is attributed to the strong Gaussian smearing of the initial state and the short hydrodynamic lifetime at low energies (Schäfer et al., 2021).

A separate analysis of momentum-anisotropy generation argues that all three stages of the hybrid contribute to the final momentum anisotropy. Using a generalized $\mu_B$2 decreases the effect of the point of particlization, and the study quantifies uncertainties from different initial-state profiles, including SMASH initial conditions, TrENTo, and IP-Glasma. It concludes that the initial-state transverse momentum impacts final-state momentum anisotropy (Götz et al., 2023).

A more focused follow-up on exchanged initial-condition models sharpens that statement. For Au+Au at $\mu_B$3 GeV, it finds that although averaged eccentricities are similar across SMASH IC, TrENTo, and IP-Glasma, substantial differences exist in the distributions and correlations of initial-state quantities. Notably, initial-state momentum anisotropy is shown not to affect the final-state flow, whereas the presence of radial flow affects the emergence of final-state momentum anisotropies, and inclusion of radial flow in the linear fit improves the prediction of final-state flow from initial-state properties (Götz et al., 2023).

The recent Bayesian calibration of the hybrid reinforces the view that using hadronic initial conditions constrains the subsequent evolution more strictly at intermediate energies. In that analysis, the hydrodynamic onset time is highly sensitive, and the resulting shear-viscosity extraction differs substantially from previous Bayesian analyses that used more parametric initial-state models (Götz et al., 13 Mar 2025).

5. Specialized applications of the framework

One of the most developed specialized uses of the hybrid is light-nucleus production and femtoscopy. In Au+Au collisions at $\mu_B$4 GeV, SMASH-vHLLE is combined with a Wigner-function coalescence treatment and the CRAB correlation afterburner to study deuterons, tritons, and $\mu_B$5He. In that setup, the hybrid background performs well for $\mu_B$6–$\mu_B$7 and $\mu_B$8–$\mu_B$9 correlation functions and extracted source radii, but for azimuthal anisotropies it “overestimates the measured $3+1$0 of all light nuclei species by almost factor 2” and “fails to describe the negative values of $3+1$1” at central rapidity. The comparison with a pure transport hard-EoS mean-field SMASH calculation leads to the conclusion that hydrodynamics is not sufficient to describe low-energy elliptic flow of light nuclei, whereas coalescence plus either background reproduces the femtoscopic observables rather well (Bailung et al., 2024).

The hybrid has also been used as a controlled environment for small-system collectivity studies at the LHC. In O–O and Ne–Ne collisions at $3+1$2 TeV, SMASH-vHLLE-hybrid, pure SMASH, and Angantyr were compared on an equal basis. The hybrid predicts the largest $3+1$3 and $3+1$4 in central collisions, a strong enhancement in $3+1$5 at intermediate $3+1$6, and mass ordering with a larger enhancement for baryons than for mesons, consistent with radial flow. By contrast, pure SMASH and Angantyr show patterns attributed mainly to non-flow, and the authors stress that the parameter set was not tuned for LHC light ions (Constantin et al., 6 Sep 2025).

Beyond hadronic spectra and flow, the SMASH review identifies further hybrid-relevant applications. SMASH can be embedded in Bayesian analyses of transport coefficients as a function of temperature and chemical potential, can be used in coarse-grained dilepton studies with medium-modified spectral functions, and can be coupled to recent dynamical fluidization schemes that extend the hybrid approach to lower beam energies (Elfner et al., 29 Aug 2025). This suggests a broader role for SMASH-vHLLE-hybrid as an inference and model-comparison platform rather than only a fixed phenomenological code.

6. Limitations, controversies, and current directions

A persistent technical issue is that hybrid results depend on how the hydro–transport interface is defined unless the transport coefficients are matched carefully. The generalized $3+1$7 program was motivated precisely by the observation that constant or purely temperature-dependent viscosities produce a sizable dependence of integrated $3+1$8 on the switching energy density, whereas the matched energy-density-dependent form largely removes this ambiguity (Götz et al., 2022).

Another systematic issue concerns the initial state. The 2026 comparison of SMASH and McDipper shows that while the two descriptions overlap reasonably at lower energies, their longitudinal energy and baryon deposition differ substantially at higher energies. SMASH provides a realistic baryon-rich three-dimensional initial state for RHIC BES and lower-energy applications, but at LHC energies it underestimates transverse-energy deposition and over-stops baryons at midrapidity relative to the saturation-based model. This suggests that interfacing string-based and saturation-based descriptions is likely necessary in the intermediate-to-high-energy transition region (Constantin et al., 16 May 2026).

Low beam energies present a different challenge: a fixed-$3+1$9 initialization becomes increasingly artificial when stopping is strong and the overlap region has substantial temporal extent. Dynamic fluidization addresses this by allowing only locally dense core hadrons to source hydrodynamics, while corona and spectators remain in transport. The low-energy DynFlu study presents this as the appropriate extension of the hybrid paradigm below the conventional BES regime and explicitly frames the resulting calculation as a baseline with respect to future studies of the phase transition (Góes-Hirayama et al., 25 Jul 2025).

Current extensions of the framework also explore alternative equations of state. One line of work implemented tabulated holographic equations of state in both iEBE-MUSIC and vHLLE-SMASH chains and reported that the resulting $$\partial_\mu T^{\mu\nu} = 0, \qquad \partial_\mu N_B^{\mu} = 0, \qquad \partial_\mu N_Q^{\mu} = 0,$$0 transverse-mass spectra are very similar to those obtained with standard reference equations of state, although the hydrodynamic parameters were not systematically tuned and the focus was on EoS substitution rather than a full phenomenological recalibration (Anufriev et al., 3 Oct 2025). A plausible implication is that SMASH-vHLLE-hybrid is now mature enough to function as a common backend for systematic equation-of-state studies at finite baryon density.

Taken together, the literature defines SMASH-vHLLE-hybrid not as a single immutable setup but as a family of closely related transport–hydrodynamics–transport couplings. Its stable core is the use of SMASH for the non-equilibrium entrance and exit channels and vHLLE for the viscous dense stage; its active research frontier lies in dynamic initialization, finite-$$\partial_\mu T^{\mu\nu} = 0, \qquad \partial_\mu N_B^{\mu} = 0, \qquad \partial_\mu N_Q^{\mu} = 0,$$1 transport coefficients, resonance-list consistency, specialized observables such as light nuclei and dileptons, and the controlled matching of hadronic and saturation-based descriptions across beam energies (Schäfer et al., 2021, Elfner et al., 29 Aug 2025).