PACIAE 4.0: Advanced Collision Simulation

• The PACIAE 4.0 model is a multipurpose Monte Carlo event generator that integrates Fortran and C++ to simulate high-energy collisions with detailed parton and hadron cascades.
• It features advanced hadronization techniques using both Lund fragmentation and coalescence, accurately reproducing key observables like dNch/dη and pT spectra.
• The framework supports diverse collision types, offering flexible reaction channels, simulation of collective phenomena, and modeling of exotic states.

The PACIAE 4.0 model is a multipurpose Monte Carlo event generator designed to simulate high-energy collision processes involving leptons, hadrons, and nuclei. Developed as an extension of previous PACIAE versions and based on the PYTHIA generator, PACIAE 4.0 incorporates detailed physics of parton and hadron cascades, advanced hadronization mechanisms, and a modernized software framework utilizing both Fortran and C++. This allows for robust modeling of nuclear medium effects, collective phenomena, and critical observables across a wide range of experimental regimes.

1. Model Architecture and Transition to Modern Frameworks

PACIAE 4.0 represents a substantial advancement in both simulation methodology and computational architecture (Lei et al., 21 Nov 2024). The legacy fixed-format Fortran 77 codebase has been refactored to integrate modern Fortran and C++, enabling direct interfacing with PYTHIA 8.3—a major upgrade over previous versions relying solely on PYTHIA 6.4. The Fortran INTERFACE block manages seamless communication and coordination between Fortran and C++ routines, permitting dynamic instantiation of PYTHIA 8 objects. This modernization introduces enhanced modularity, increased maintainability, and access to improved physics models such as advanced color reconnection schemes, multiparton interactions, and the Angantyr model for heavy-ion collisions.

2. Core Simulation Stages and Dynamical Evolution

PACIAE 4.0 follows a staged simulation protocol for each event:

• Initial State Preparation: The initial partonic configuration (quarks, antiquarks, and gluons) is generated using PYTHIA 8.3, with string fragmentation temporarily disabled. Diquarks/antidiquarks are randomly broken up at this stage (Xie et al., 9 Mar 2025, Xie et al., 7 Aug 2025).
• Parton Cascade / Rescattering: Parton evolution proceeds via 2→2 leading-order pQCD scatterings. The differential cross section for partonic processes is

$$\frac{d\sigma}{dt}(ab\to cd; s, t) = K\frac{\pi\alpha_s^2}{s^2}\left|\overline{M}(ab\to cd)\right|^2,$$

with $\alpha_s$ the strong coupling, $s$ and $t$ Mandelstam invariants, $\overline{M}$ the matrix element, and $K$ an empirical factor accounting for higher-order and nonperturbative corrections.

• Hadronization: After parton rescattering, hadronization is implemented using either the Lund string fragmentation model or a coalescence scheme. The fragmentation function follows

$$f(z) \propto \frac{1}{z}(1-z)^a\exp\left(-\frac{b\,m_T^2}{z}\right),$$

with $z$ the energy fraction, $m_T$ hadron transverse mass, $a$ and $b$ tunable parameters (Xie et al., 9 Mar 2025, Xie et al., 7 Aug 2025). The coalescence approach uses phase-space constraints:

$$\frac{16\pi^2}{9}\,\Delta r^3\,\Delta p^3 = \frac{h^3}{d},$$

with $d$ a degeneracy factor (typically $d=4$).

• Hadronic Rescattering: Final hadrons undergo further two-body scatterings until kinetic freeze-out. Cross-section calculations use empirical parameterizations and, where necessary, the Additive Quark Model.

3. Physics Features and Model Innovations

Key advancements in PACIAE 4.0 include:

• Advanced Rescattering: Heavy-flavor processes receive explicit mass corrections in cross sections (e.g., $Qq \rightarrow Qq$ and $Qg \rightarrow Qg$ have heavy quark mass terms) (Lei et al., 21 Nov 2024).
• Multiple Hadronization Modes: Users may select between Lund fragmentation and a refined coalescence model with Altarelli–Parisi splitting for gluons and iterative deexcitation for energetic quarks.
• Flexible Simulation Environments: The model accommodates both low-energy hadronic and high-energy partonic–hadronic cascades. For heavy-ion collisions, the Angantyr model gives advanced handling of nucleon substructure.
• Expanded Reaction Channels: Additional inelastic channels for both soft and hard hadronic interactions are provided, with improved treatment for heavy hadrons and less-common species.

4. Validation Against Experimental Observables

PACIAE 4.0 has been systematically benchmarked against key experimental data (Xie et al., 9 Mar 2025, Xie et al., 7 Aug 2025). Noteworthy findings include:

• Charged Particle Pseudorapidity ($dN_{ch}/d\eta$) and Transverse Momentum Spectra ($p_T$): Simulated results across $\sqrt{s}=200$ GeV to 13 TeV accurately reproduce STAR, ALICE, and CMS data for both minimum bias and non-single diffractive (NSD) pp collisions, requiring only a single parameter set for fragmentations ($a=0.68$, $b=0.8$ at NSD energies), and $K=0.8$ for hard scatterings.
• Energy Dependence and Extrapolation: The $b$ parameter in the fragmentation function shows a monotonic increase with collision energy, permitting empirical fits and reliable extrapolation to energies where experimental data are unavailable (Xie et al., 9 Mar 2025).
• Consistency Across Pseudorapidity Bins: Systematic comparison in narrow $\eta$ bins verifies the reliability of both central and forward production predictions.

The robust agreement, achieved without retuning parameters for each energy or collision mode, supports the model’s suitability for baseline calculations in heavy-ion physics, where pp collisions serve as a benchmark for medium effects.

5. Specialized Applications: Collective Phenomena and Exotic States

PACIAE 4.0 is equipped to simulate nuclear medium effects and collective phenomena, including:

• Elliptic Flow in Small Systems: Simulations of pp collisions demonstrate nontrivial elliptic flow ($v_2$) originating from initial geometry fluctuations and multi-stage rescattering, with the flow coefficient $v_2/\epsilon$ saturating above $\sqrt{s}\sim7$ TeV (Zhou et al., 2010).
• Vorticity and Polarization: Detailed studies extract local non-relativistic, kinematic, temperature and thermal vorticities, revealing non-monotonic energy dependence, quadrupole spatial patterns, and quantitative agreement with global $\Lambda$ polarization data (Lei et al., 2021).
• Production of Vector Bosons and Exotic Hadrons: The model incorporates bias-corrected hard scattering channels to paper $J/\psi$, $W^\pm$, and $Z^0$ production (Ye et al., 2023, Zhou et al., 2020). DCPC coalescence is integrated to paper the formation of glueball-like ($X$(2370)) and tetraquark states in pp and $e^+e^-$ collisions (Cao et al., 7 Aug 2024, She et al., 10 Jul 2024, Wang et al., 25 Jun 2025).

This versatility makes PACIAE 4.0 applicable across a spectrum of high-energy nuclear physics topics.

6. Computational Aspects and Parameterization

Simulation efficiency is increased through modern fortran and C++ integration, with advanced sampling of the Woods–Saxon nuclear density, impact parameter definition, and event-by-event Glauber modeling (Yan et al., 2022). Key computational prescriptions include:

• Impact Parameter and Centrality Mappings: Centrality percentile $c$ is mapped to $b$ via $b = \sqrt{c}\, b_{max}$, where $b_{max}$ is set according to experimental guidance (typically up to 20 fm in Pb–Pb collisions).
• Minimum Distance for Collisions: $d_{min} \leq \sqrt{\sigma_{tot}/\pi}$, where $\sigma_{tot}$ may follow the Donnachie–Landshoff parameterization.
• Debye Mass Regulation: Divergences in the partonic matrix elements are regulated using a Debye mass ($\mu$) in $t$-channel singularities.
• Event Sampling and Yield Estimation: For exotic state formation, yields are calculated from integrals over phase space with coalescence constraints (see equations above).

7. Model Limitations and Directions for Future Development

While PACIAE 4.0 achieves strong empirical validation, ongoing development aims to:

• Implement medium-induced showering using a time-like scheme with improved Sudakov factors and non-collinear splitting.
• Integrate hybrid hadronization schemes combining fragmentation and coalescence, particularly for heavy-flavor observables.
• Transition hadron decay mechanics to PYTHIA 8 for consistency and up-to-date particle tables.
• Expand the catalog of rescattering reactions in response to new empirical data and theoretical insight.
• Refine the treatment of initial-state fluctuations, viscous effects, and centrality definition for better theoretical alignment with experimental methodologies (Lei et al., 21 Nov 2024, Yan et al., 2022, Lei et al., 2023).

This suggests that PACIAE 4.0 is positioned as a high-fidelity, extensible platform for simulating a broad range of high-energy collisions, with an architecture that enables ongoing incorporation of new physics and methodological improvements.