EPOS4: High-Energy Collision Simulation

• EPOS4 is an event-by-event Monte Carlo event generator that simulates hadronic and nuclear collisions using a unified approach of hard and soft QCD physics.
• It employs a parallel S-matrix framework, dynamical saturation scales, and core–corona separation to achieve realistic energy sharing and event fluctuations.
• EPOS4 accurately reproduces observables like strangeness enhancement, radial flow, and particle yield ratios across diverse collision systems.

EPOS4 Model

The EPOS4 ("Energy conserving Parton-based Off-shell Scatterings") model is a next-generation, event-by-event Monte Carlo event generator designed for high-precision simulation of hadronic and nuclear collisions across the full spectrum from small (pp) to large (AA) systems and from several GeV up to multi-TeV collision energies. EPOS4 provides a unified theoretical and computational framework for integrating hard and soft QCD physics, collective phenomena, and realistic energy-momentum conservation, aiming to address strangeness enhancement, flow, and other multi-particle observables in high-multiplicity collision environments (Singh et al., 22 Jul 2025, Werner, 2023, Werner, 13 Oct 2024, Werner, 2023).

1. Theoretical Foundations and Core Ingredients

EPOS4 implements a rigorously parallel S-matrix-based multiple scattering formalism, embedding both Gribov–Regge theory for soft multiparton processes and perturbative QCD for hard interactions:

• Parallel (Simultaneous) Primary Scatterings: The model treats all possible nucleon–nucleon (or parton–parton) subcollisions as occurring instantaneously in parallel. Each subcollision is mapped to an individual "Pomeron" (parton ladder), and all ladders share the incoming energy through exact energy–momentum conservation implemented at the amplitude level.
• Dynamical Saturation Scales: EPOS4 introduces a dynamical (event-by-event, subcollision-dependent) saturation scale, $$Q^2_{\mathrm{sat}}(N_{\mathrm{conn}},x_{\mathrm{PE}})$$, as a lower cutoff for DGLAP-evolved parton ladders. Here $N_{\mathrm{conn}}$ counts the number of Pomerons attached to the parent nucleon, and $x_{\mathrm{PE}}$ is the fraction of available energy. This scaling restores both factorization in high-$p_T$ (hard) limits and binary scaling in AA collisions, rectifying the violation of the AGK theorem (destructive interference between multiple scatterings) that arises when energy sharing is enforced (Werner, 2023, Werner, 2023).
• Core–Corona Separation: At a specified early proper time ($\tau_0 \sim 0.4$–$0.6$ fm/c), the system is decomposed into a dense "core" (regions of high overlapping string density) and a dilute "corona" (low-density regions). The core is evolved hydrodynamically; the corona hadronizes immediately via microcanonical string fragmentation.
• Event-by-Event Fluctuations: Initial conditions include event-wise fluctuations in number, spatial distribution, and energy of Pomerons/strings, sampled from a Glauber–like geometry augmented by color-fluctuation effects.

2. Dynamical Evolution: Hydrodynamics, Hadronization, and Afterburner

2.1 Hydrodynamic Evolution

The core domain, identified by local high string/energy density, is evolved with fully (3+1)D viscous hydrodynamics, employing:

• Conservation Laws:

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

with $T^{\mu\nu}$ containing energy density $\epsilon$, pressure $p(\epsilon, n_B)$ (from a lattice-QCD hadron–resonance–gas-matched EoS), bulk $\Pi$, and shear stress $\pi^{\mu\nu}$ tensors. Shear viscosity is typically set to $\eta/s \approx 0.08$–$0.2$ with a minimal bulk viscosity near $T_c$ (Singh et al., 22 Jul 2025, Werner, 2023).

2.2 Hadronization: Cooper–Frye and Microcanonical Methods

• Cooper–Frye Freeze-out: Once core fluid cells cool to $T_{\mathrm{fo}} \approx 155$–$165$ MeV or energy density $\epsilon_{\mathrm{sw}} \simeq 0.3$ GeV/fm$^3$, conversion to hadrons is performed using the Cooper–Frye formula:

$dN_i = g_i \int_\Sigma p^\mu d\sigma_\mu \, f(x,p)$

with $f$ a locally boosted equilibrium distribution plus viscous corrections.

• Microcanonical Hadronization: For both small and large systems, EPOS4 employs a microcanonical statistical prescription, exactly conserving energy–momentum and all quantum numbers ($B$, $S$, $Q$) within each freeze-out region:

$$\Omega(E, V, \{Q_A\}) \propto \int \delta\left(E-\sum_i E_i\right) \delta^3\left(\sum_i \vec{p}_i\right) \prod_i d^3p_i \prod_A \delta(Q_A-\sum_i q_{A,i})$$

This procedure recovers canonical suppression for strangeness in small volumes and is essential for the observed smooth rise of multi-strange-to-pion ratios across system sizes and multiplicities (Werner, 2023, Koley et al., 28 Aug 2025).

2.3 Hadronic Cascade (Afterburner)

All hadrons (from both core and corona) undergo further rescattering and resonance decays in an afterburner based on the UrQMD transport model, which treats elastic/inelastic interactions, regeneration, and annihilation, and is necessary for proper modeling of resonance yields and final-state spectra (Sumberia et al., 6 Dec 2024).

3. Key Algorithmic and Model Parameters

Parameter	Typical Value/Implementation	Role
$\tau_0$ (hydro start)	$0.4$–$0.6$ fm/c	Core–corona separation time
$\eta/s$	$0.08$–$0.2$ (globally tuned)	Shear-viscosity/entropy ratio
$T_{\mathrm{fo}}$	$155$–$165$ MeV	Chemical (hadronization) temp.
$\epsilon_{\mathrm{sw}}$	$\simeq 0.3$ GeV/fm$^3$	Hydro→cascade switch
$\sigma_\perp$ (flux width)	$0.2$–$0.3$ fm	Transverse string smearing
$\sigma_\eta$ (long. width)	$1.0$–$1.5$	Longitudinal density profile
Core density threshold	$1.0$ fm$^{-3}$ (string-segment count)	Core–corona discrimination

The formalism for energy-momentum sharing and saturation is fully event-by-event and tabulated for all relevant parton–parton subcollisions, ensuring consistency with global cross sections and PDFs (Werner et al., 2023, Werner, 2023).

4. Strangeness Enhancement, Radial Flow, and Collectivity

• Strangeness Production: EPOS4 quantitatively reproduces the enhancement of strange- and multi-strange-to-pion ratios as a function of final state multiplicity (e.g., $(\Xi + \overline{\Xi})/(\pi^+ + \pi^-)$) across $pp$, $p$Pb, and PbPb systems. The model attributes this to both a rising core fraction (leading to chemical equilibrium in larger systems) and exact quantum number conservation in microcanonical sampling, rather than purely canonical suppression. The enhancement factors for $\Xi/\pi$ reach $\sim5\times$, and for $\Omega/\pi$ $\sim10\times$ from low to high multiplicity, in agreement with ALICE data (Werner, 2023, Singh et al., 22 Jul 2025, Koley et al., 28 Aug 2025).
• Radial and Anisotropic Flow: Hydrodynamic expansion of the core generates strong, mass-dependent radial flow. Identified hadron $p_T$ spectra and mean $\langle m_T \rangle$ values show characteristic mass ordering and flattening in central collisions, with radial-flow velocities $\langle \beta_T \rangle \approx 0.6$–$0.65$ ($\sim10\%$ above AMPT predictions). Multi-particle cumulants ($v_n\{2\}$, $v_n\{4\}$, etc.) and flow harmonics display realistic scaling with multiplicity, system size, and centrality, a feature absent in transport-only models (Werner, 10 Aug 2025, Singh et al., 22 Jul 2025).
• Resonance Dynamics: The time duration of the hadronic phase ($\tau$) estimated from resonance suppression (e.g., $K^*/K$) increases with multiplicity and system size. Early freeze-out and annihilation effects (e.g., $\bar{p} p$) are described by the UrQMD module (Sumberia et al., 6 Dec 2024).

5. Quantitative Model Performance and Comparison to Other Frameworks

• Charged-particle multiplicity: For central $O+O$ at 7 TeV, EPOS4 predicts $\langle dN_{ch}/d\eta \rangle \approx 230$ in 0–5% centrality, above AMPT (string melting and default) by $\sim10$–$20\%$ (Singh et al., 22 Jul 2025, Bashir et al., 12 May 2025).
• Spectra Hardness and Yields: EPOS4 consistently generates harder $p_T$ spectra for strange and multi-strange hadrons, and predicts absolute yields for $\Lambda$, $\Xi$, and $\Omega$ higher by $15$–$30\%$ compared to AMPT (Singh et al., 22 Jul 2025, Ashraf et al., 6 Jun 2024).
• Integrated Yield Ratios: $(\Xi + \overline{\Xi})/(\pi^+ + \pi^-)$ rises from $0.03 \to 0.08$ as $\langle dN_{ch}/d\eta \rangle$ increases, matching data trends from small to large systems. AMPT and string-melting models do not reproduce these ratios quantitatively (Singh et al., 22 Jul 2025).
• Collective Flow Signatures: Multi-particle $v_n$ cumulants, mean $\langle p_T \rangle$, and charge balance function widths in EPOS4 match LHC data, capturing narrowing with multiplicity and yield suppression of broad charge correlations (e.g., for protons).

6. Conceptual Advances Over Prior Generations and Limitations

EPOS4 represents a significant advance over EPOS3, EPOSLHC, AMPT, and pure string models:

• Core–corona identification is now local and density-based, with microcanonical hadronization for both small and large systems, capturing canonical suppression effects and correlation volumes self-consistently.
• Dynamical saturation scaling and exact energy sharing restore factorization and high-$p_T$ binary scaling across all multiplicities, as validated against global PDFs and inclusive jet data (Werner, 2023, Werner, 13 Oct 2024, Werner, 2023).
• Improvements over earlier versions include explicit strangeness-dependent viscous corrections at freeze-out and a modern vHLLE hydrodynamic solver for robust 3D evolution.

Limitations and Forthcoming Developments:

• Bulk viscosity is presently implemented as small and only near $T_c$. Future versions anticipate a dynamical $\zeta/s(T)$.
• The p/$\pi$ and K/$\pi$ ratios are systematically overestimated in small systems, suggesting the need for improved retuning of fragmentation and microcanonical parameters for $pp$ and $pA$.
• The handling of strangeness correlation volumes, differential freeze-out for multi-strange hadrons, and potential non-equilibrium production mechanisms are future avenues for increased fidelity (Koley et al., 28 Aug 2025).

7. Phenomenological Applications

EPOS4 provides predictive power for a broad range of observables:

• Minimum-Bias and High-Multiplicity $pp$, $pA$, and $AA$ Collisions: Reproduction of bulk hadron yields, identified particle spectra, multi-strange baryon enhancement, radial and anisotropic flow, jet cross sections, and heavy flavor production (Werner, 2023, Werner et al., 2023, Zhao et al., 29 Sep 2025).
• Resonance Production: Modeling of suppression and regeneration effects, direct connection to hadronic phase lifetime and detailed comparison with LHC measurements (Sumberia et al., 6 Dec 2024).
• Novel Fluctuation and Correlation Observables: The framework is uniquely suited to paper strangeness yield correlations (e.g., $\phi$-triggered yields), charge balance functions, and non-trivial event-by-event fluctuation observables that cannot be addressed by transport or grand-canonical-only models (Bierlich et al., 1 Mar 2024, Manea et al., 17 Nov 2024).
• Unified Description Across Energies: The model interpolates smoothly from full fluid-dominated regimes ($\sqrt{s_{NN}} > 24$ GeV) to sequential cascade behavior ($<4$ GeV) (Werner et al., 20 Jan 2024).

In summary, EPOS4 constitutes an explicit, modular, and universal event generator architecture, synthesizing principles of QCD factorization, dynamical saturation, energy-momentum conservation, and event-by-event collective dynamics. Its predictive successes in O+O, PbPb, $p$Pb, and $pp$ collisions at LHC/RHIC energies support its adoption for current and future high-energy nuclear physics analyses (Singh et al., 22 Jul 2025, Werner, 2023, Werner, 13 Oct 2024, Werner, 2023).