UrQMD Transport Model Calculations

• UrQMD is a microscopic framework that simulates hadronic and nuclear collisions from intermediate to ultra‐relativistic energies using the relativistic Boltzmann equation.
• It incorporates key processes such as resonance formation/decay, string excitation/fragmentation, and mean‐field effects to model event-by-event dynamics.
• The model facilitates extraction of particle yields, transverse momentum spectra, and flow observables, aiding analysis across SPS, RHIC, FAIR, and NICA experiments.

The Ultra-relativistic Quantum Molecular Dynamics (UrQMD) transport model is a widely used microscopic framework for simulating hadronic and nuclear collisions from intermediate to ultra-relativistic energies. It provides a non-perturbative, event-by-event description of the time evolution of many-body strongly-interacting systems, with applications ranging from elementary $p+p$ reactions to heavy-ion collisions up to collider energies. UrQMD solves the covariant Boltzmann equation including an extensive set of hadronic states and implements essential dynamical processes such as resonance formation/decay, string excitation/fragmentation, and (optionally) mean-field effects.

1. Fundamental Equations and Model Structure

UrQMD simulates the phase-space evolution of particles using the relativistic Boltzmann equation,

$p^\mu\partial_\mu f_i(x,p) = C_i[f]$

where $f_i(x,p)$ is the single-particle distribution function for hadron species $i$, and $C_i[f]$ encodes the collision and decay integrals. The degrees of freedom include 55 baryon states and 32 meson states (including resonances up to 2.25 GeV), as well as their antiparticles, with particle–antiparticle, isospin, and SU(3) flavor symmetry enforced. Hadrons propagate on straight-line trajectories in "cascade mode," with dynamics extended by Skyrme-type density and momentum-dependent mean fields in potential versions or quantum molecular dynamics (QMD) mode (Ozvenchuk et al., 2017, Zhang et al., 2020, Kuttan et al., 2022).

Two-body collisions are implemented using the closest-approach criterion: a collision occurs if the covariant distance between two hadrons falls below $\sqrt{\sigma_{\text{tot}}/\pi}$, with total cross-section $\sigma_{\text{tot}}$ extracted from experimental data, additive quark model, or resonance parametrizations. Resonance excitation/decay follows the Breit-Wigner formalism; baryon-antibaryon annihilations are included.

At high energies, soft inelastic processes are modeled via Lund string excitation/fragmentation, and hard scatterings (transfer $Q>1.5$ GeV) handled by embedded PYTHIA routines (Ozvenchuk et al., 2017, Zhang et al., 2020, Yuan, 2024).

2. Implementation Details and Simulation Workflow

For elementary $p+p$ and nuclear collisions, UrQMD event generation proceeds as follows:

• Initialization: For $A+A$ collisions, nucleons are spatially positioned according to Woods–Saxon distributions (with deformation for heavy nuclei as needed) and assigned Fermi momenta. At $p+p$, only two nucleons are initialized at zero impact parameter.
• Event evolution: Time stepping typically uses $\Delta t \sim 0.2$ fm/$c$.
  • Free propagation of hadrons according to the Hamiltonian equations of motion.
  • At each step, check all particle pairs for collisions using the geometric criterion.
  • For colliding pairs, select outgoing channels (elastic, resonance, string) stochastically according to cross sections.
  • Propagate resonances and decay them according to lifetimes and branching ratios.
  • For potential or QMD mode, compute mean-field forces using local densities from Gaussian wave packets.
• Event analysis: Final-state particles are binned in momentum and rapidity for yield, spectra, and correlation observables. Coalescence afterburners are applied for light nuclei.
• Rapidity intervals and centrality classes are chosen according to the specific observable or experiment to be compared (e.g., $0Ozvenchuk et al., 2017, Yuan, 2024).

In UrQMD v3.4, as used for $p+p$ at CERN SPS energies, the final state hadrons ($\pi^\pm$, $K^\pm$, $p$, $\bar p$) in inelastic events are fully recorded for construction of double-differential yields $d^2N/(dp_T\,dy)$ (Ozvenchuk et al., 2017).

3. Key Applications and Observable Extraction

UrQMD is utilized for a diverse range of physics analyses:

a. Transverse Momentum Spectra and Inverse Slopes

The $p_T$ spectra of pions, kaons, protons, and antiprotons are extracted as $d^2N/(dp_T\,dy)$ over fine $(p_T, y)$ bins. Fits with the exponential or blast-wave-inspired function

$$\frac{d^2N}{dp_T\,dy} = \frac{S\,p_T}{T^2 + m\,T}\,\exp\left[ -(m_T - m)/T \right]$$

yield the normalization $S$ and inverse-slope parameter $T$ ("temperature"), with $m_T = \sqrt{m^2 + p_T^2}$ (Ozvenchuk et al., 2017). These parameters are mapped as a function of rapidity and collision energy for quantitative comparison with experiment (e.g., NA61/SHINE).

b. Particle Yields, Ratios, and Centrality Dependence

UrQMD provides total yields ($dN/dy$), average transverse momenta, and particle ratios ($\pi^-/\pi^+$, $K^-/K^+$, $\bar p/p$) across centrality bins, highlighting mechanisms such as pair production dominance or net-baryon number transport (Yuan, 2024).

c. Flow and HBT Observables

Transverse flow is assessed via effective source-temperature fits and radial flow velocities, often using blast-wave models for extracted spectra. Two-pion HBT (Hanbury–Brown–Twiss) correlation radii ($R_O$, $R_S$, $R_L$) are calculated from final freeze-out distributions via three-dimensional Gaussian fits to the two-pion correlator in the LCMS, with the $R_O/R_S$ ratio revealing source explosivity (Li et al., 2012).

d. Higher-Order Observables and Fluctuations

In box simulations or event-by-event analyses, UrQMD calculates higher-order cumulants for baryon and proton numbers, cluster yields via coalescence afterburners, and the time-evolution of coordinate/momentum-space fluctuations, providing a probe for first-order phase transition signatures (Bumnedpan et al., 2024, Li et al., 2016).

4. Model Capabilities, Results, and Systematics

The predictive performance of UrQMD has been evaluated for a broad spectrum of phenomena:

• Hadron spectra and yields: UrQMD v3.4 accurately reproduces $\pi^+$ and $K^+$ $p_T$ spectra and inverse slopes at central rapidity and high SPS energies, but underestimates yields at lower beam momenta and overpredicts proton/antiproton production at high energies.
• Spectral shapes: Discrepancies in $p_T$-distribution tails (notably for forward-rapidity pions and kaons) arise due to incomplete modeling of diffraction, baryon stopping, and strangeness production mechanisms (Ozvenchuk et al., 2017).
• Central and peripheral collisions: Cascade mode suffices for low-$p_T$ spectra and peripheral events, but the inclusion of momentum-dependent soft mean fields (SM-EoS) is necessary to reproduce the spectral hardening at high $p_T$ in central U+U at top RHIC energies (Yuan, 2024).
• Coalescence and light nuclei: Yield and rapidity distributions of $^3$He and other clusters require a potential version ("UrQMD/M") with phase-space coalescence afterburners, including Lorentz (kinematic) corrections (Li et al., 2016).
• Systematic uncertainties: Model limitations include omitted three-body collisions, lack of a deconfined phase at high energy, and the absence of explicit mean fields unless run in QMD mode. The description of antiproton and multistrange hadron production at low energies is particularly sensitive to the implementation of strangeness-exchange and multi-step channels (Graef et al., 2014, Bumnedpan et al., 2024).

5. Model Limitations and Pathways for Refinement

Ongoing discrepancies with experiment point to specific physics ingredients that require further refinement:

• Underprediction of $K^+$ yields at low energies indicates missing strangeness production mechanisms.
• Excess central rapidity protons and incorrect rapidity transport imply the need for improved baryon stopping and diffractive excitation.
• Systematic overproduction of antiprotons is linked to incomplete modeling of annihilation/recreation channels.
• Spectral shape mismatches (e.g., for protons) show limitations of single-exponential fits and suggest missing physics in baryon transport and freeze-out (Ozvenchuk et al., 2017).

Suggested improvements include:

• Retuning binary inelastic and single-diffractive cross sections.
• Explicit implementation of low-mass diffraction channels ($NN \to \Delta N^*$ up to 2.25 GeV).
• Global best-fit tuning to both $p+p$ and $A+A$ datasets, and improved description of strange baryon production.

6. Comparative and Hybrid Approaches

UrQMD is frequently benchmarked against other transport models (PHSD, SMASH, PHQMD) and compared to ideal hydrodynamics as a means to assess both bulk and rare observables (Reichert et al., 2021):

Observable	UrQMD/SMASH (resonance)	PHSD/PHQMD (nonresonant)	Typical agreement
Bulk yields	$\lesssim$10% diff.	$\lesssim$10%	p, $\pi$, $\eta$, $K$
$\Xi^-$ yield	$3-4\times 10^{-5}$	$(2-3)\times 10^{-6}$	Order-of-magnitude diff.
$T_{\text{source}}$	80–95 MeV	80–95 MeV	All models similar
$\langle v_T \rangle$	0.22–0.30 c	0.25–0.28 c	All models similar

Hybrid models incorporate hydrodynamic evolution (for a deconfined fireball phase or stiffer EoS), Landau matching, and particlization for enhanced agreement with HBT, photon, and dilepton observables and flow data (Zhang et al., 2020, Li et al., 2012, Lang et al., 2012).

7. Impact and Future Directions

UrQMD calculations have been central to interpreting hadronic spectra, particle ratios, and fluctuation observables across a vast energy range. The model's flexibility in incorporating different EoS, collision cross-sections, mean fields, and clusterization schemes permits ongoing benchmarking and tuning against new high-precision data from RHIC Beam Energy Scan, SPS, and upcoming FAIR and NICA experiments.

Persistent deviations in specific channels continue to inform refinements in hadronic reaction mechanisms and the mapping of QCD matter properties, such as the nuclear EoS and the nature of the phase transition region. Targeted improvements—including consistent treatment of diffraction, explicit partonic degrees of freedom, and more sophisticated freeze-out and coalescence algorithms—represent the current frontier for UrQMD and related transport approaches (Ozvenchuk et al., 2017, Yuan, 2024, Li et al., 2016).