HYDJET++ Heavy-Ion Event Generator

• HYDJET++ is a hybrid Monte Carlo generator that simulates relativistic heavy-ion collisions by coupling parameterized hydrodynamic freeze-out with QCD-inspired hard scattering and jet-quenching processes.
• The model employs detailed methodologies including the Cooper–Frye prescription for soft emissions and BDMPS-Z based energy loss for hard parton scatterings, accurately reproducing charged-hadron pT spectra and nuclear modification factors.
• Despite its strengths in modular event generation and geometry sensitivity, HYDJET++ is limited by the absence of post-hadronic rescattering and simplified energy-loss fluctuations, suggesting clear paths for future improvements.

HYDJET++ is a hybrid Monte Carlo event generator designed for relativistic heavy-ion collisions, systematically combining parameterized hydrodynamic modeling of soft processes with a QCD-inspired, jet-quenching-modified simulation of hard partonic scatterings. The model has been extensively applied to describe Xe–Xe collisions at LHC energies, notably at $\sqrt{s_{NN}} = 5.44$ TeV, and has been benchmarked against ALICE charged-hadron $p_T$ spectra and nuclear modification observables, as well as the AMPT String Melting model (Pandey et al., 2022).

1. Theoretical Structure: Soft and Hard Components

HYDJET++ models each nucleus-nucleus (AA) event as the incoherent sum of two physically distinct processes:

1. Soft (Hydrodynamic) Component:
   • Implements bulk hadron emission from a freeze-out hypersurface using parameterized relativistic hydrodynamics.
   • Utilizes the FAST MC generator to sample the Cooper–Frye prescription at a fixed kinetic freeze-out temperature $T_{\rm fo}$ and transverse flow rapidity profile $\rho(r)$.
   • The local distribution for each hadron species follows:

   $$\frac{dN_{\rm soft}}{dy dp_T} \propto p_T m_T \int_0^{R} r dr \; I_0\left(\frac{p_T \sinh \rho(r)}{T_{\rm fo}}\right) K_1\left(\frac{m_T \cosh \rho(r)}{T_{\rm fo}}\right)$$

   with $m_T = \sqrt{p_T^2 + m^2}$ and $\rho(r) = \rho_{\rm max}(r/R)$. - Key physical assumptions: instantaneous kinetic freeze-out at $T_{\rm fo}$ (no post-hadronic rescattering), with chemical composition either fixed earlier (usual) or taken coincident with $T_{\rm fo}$ here.

2. Hard (Jet/Partonic) Component:

   • Simulates initial high-$p_T$ partonic scatterings with a standard pQCD nucleon-nucleon cross-section, sampled via the nuclear overlap function $p_T$0.
   • In-medium parton energy loss (jet quenching) is incorporated following the BDMPS-Z formalism, parameterized by the transport coefficient $p_T$1.
   • Fragmentation after energy loss is performed with the Lund string model (PYTHIA-derived).
   • Nuclear PDF shadowing is treated with EKS98 corrections.

This dual construction enables independent and composable control over soft (bulk flow-dominated) and hard (jet quenching-dominated) observables.

2. Implementation for Deformed Xe–Xe Collisions

2.1 Nuclear Geometry and Event Classification

• The $p_T$2Xe nucleus is modeled with quadrupole deformation $p_T$3, in a Woods–Saxon geometry:

$p_T$4

with $p_T$5 fm and diffuseness $p_T$6 fm.

• Two limiting geometrical configurations are constructed:
  • Tip–Tip: Both nuclei aligned with major axes parallel to the beam direction.
  • Body–Body: Major axes lie transverse to the beam.
• In simulation, events are generated with random orientations and binned post-facto by requiring $p_T$7 (tip–tip) or $p_T$8 (body–body) per nucleus.

2.2 Event Generation and Centrality

• Impact parameters are sampled with probability $p_T$9, up to $T_{\rm fo}$0 fm.
• Centrality percentiles are defined using final charged multiplicity at midrapidity, with typical bins: $T_{\rm fo}$1–$T_{\rm fo}$2, $T_{\rm fo}$3–$T_{\rm fo}$4, $T_{\rm fo}$5–$T_{\rm fo}$6, $T_{\rm fo}$7–$T_{\rm fo}$8, $T_{\rm fo}$9–$\rho(r)$0, $\rho(r)$1–$\rho(r)$2.
• Glauber calculations give $\rho(r)$3 and $\rho(r)$4 for each centrality class.

2.3 Tuned Parameters

Parameter	Value
Freeze-out temperature $\rho(r)$5	120 MeV
Max. transverse flow $\rho(r)$6	$\rho(r)$7
Baryochemical potential $\rho(r)$8	0 MeV
Minimum $\rho(r)$9 (hard scatterings) $$\frac{dN_{\rm soft}}{dy dp_T} \propto p_T m_T \int_0^{R} r dr \; I_0\left(\frac{p_T \sinh \rho(r)}{T_{\rm fo}}\right) K_1\left(\frac{m_T \cosh \rho(r)}{T_{\rm fo}}\right)$$0	2 GeV/$$\frac{dN_{\rm soft}}{dy dp_T} \propto p_T m_T \int_0^{R} r dr \; I_0\left(\frac{p_T \sinh \rho(r)}{T_{\rm fo}}\right) K_1\left(\frac{m_T \cosh \rho(r)}{T_{\rm fo}}\right)$$1
Soft fraction (central)	90%
Transport coefficient $$\frac{dN_{\rm soft}}{dy dp_T} \propto p_T m_T \int_0^{R} r dr \; I_0\left(\frac{p_T \sinh \rho(r)}{T_{\rm fo}}\right) K_1\left(\frac{m_T \cosh \rho(r)}{T_{\rm fo}}\right)$$2	1.5 GeV$$\frac{dN_{\rm soft}}{dy dp_T} \propto p_T m_T \int_0^{R} r dr \; I_0\left(\frac{p_T \sinh \rho(r)}{T_{\rm fo}}\right) K_1\left(\frac{m_T \cosh \rho(r)}{T_{\rm fo}}\right)$$3/fm
PDF (pp baseline)	CTEQ6L
Nuclear shadowing	EKS98

3. Key Physics Observables: Spectra and Nuclear Modification

3.1 $$\frac{dN_{\rm soft}}{dy dp_T} \propto p_T m_T \int_0^{R} r dr \; I_0\left(\frac{p_T \sinh \rho(r)}{T_{\rm fo}}\right) K_1\left(\frac{m_T \cosh \rho(r)}{T_{\rm fo}}\right)$$4-Spectra Construction

• The total $$\frac{dN_{\rm soft}}{dy dp_T} \propto p_T m_T \int_0^{R} r dr \; I_0\left(\frac{p_T \sinh \rho(r)}{T_{\rm fo}}\right) K_1\left(\frac{m_T \cosh \rho(r)}{T_{\rm fo}}\right)$$5 spectrum is the sum of soft and hard contributions:

$$\frac{dN_{\rm soft}}{dy dp_T} \propto p_T m_T \int_0^{R} r dr \; I_0\left(\frac{p_T \sinh \rho(r)}{T_{\rm fo}}\right) K_1\left(\frac{m_T \cosh \rho(r)}{T_{\rm fo}}\right)$$6

where the hard term for a given impact parameter $$\frac{dN_{\rm soft}}{dy dp_T} \propto p_T m_T \int_0^{R} r dr \; I_0\left(\frac{p_T \sinh \rho(r)}{T_{\rm fo}}\right) K_1\left(\frac{m_T \cosh \rho(r)}{T_{\rm fo}}\right)$$7 is:

$$\frac{dN_{\rm soft}}{dy dp_T} \propto p_T m_T \int_0^{R} r dr \; I_0\left(\frac{p_T \sinh \rho(r)}{T_{\rm fo}}\right) K_1\left(\frac{m_T \cosh \rho(r)}{T_{\rm fo}}\right)$$8

3.2 Nuclear Modification Factors

• $$\frac{dN_{\rm soft}}{dy dp_T} \propto p_T m_T \int_0^{R} r dr \; I_0\left(\frac{p_T \sinh \rho(r)}{T_{\rm fo}}\right) K_1\left(\frac{m_T \cosh \rho(r)}{T_{\rm fo}}\right)$$9: Compares the observed yield to that expected from scaled $m_T = \sqrt{p_T^2 + m^2}$0 reference:

$m_T = \sqrt{p_T^2 + m^2}$1

• $m_T = \sqrt{p_T^2 + m^2}$2: Ratio of central to peripheral yields, both normalized by $m_T = \sqrt{p_T^2 + m^2}$3:

$m_T = \sqrt{p_T^2 + m^2}$4

4. Model Performance and Empirical Validation

4.1 Agreement with ALICE Data

• $m_T = \sqrt{p_T^2 + m^2}$5-Spectra: HYDJET++ reproduces the centrality-dependent $m_T = \sqrt{p_T^2 + m^2}$6 spectrum at midrapidity up to $m_T = \sqrt{p_T^2 + m^2}$7 GeV/$m_T = \sqrt{p_T^2 + m^2}$8 within 10–15%.
• $m_T = \sqrt{p_T^2 + m^2}$9: At $\rho(r) = \rho_{\rm max}(r/R)$0 GeV/$\rho(r) = \rho_{\rm max}(r/R)$1 and $\rho(r) = \rho_{\rm max}(r/R)$2–$\rho(r) = \rho_{\rm max}(r/R)$3\% centrality, the model gives $\rho(r) = \rho_{\rm max}(r/R)$4, consistent with ALICE ($\rho(r) = \rho_{\rm max}(r/R)$5). The high-$\rho(r) = \rho_{\rm max}(r/R)$6 rise in $\rho(r) = \rho_{\rm max}(r/R)$7 is reproduced.
• $\rho(r) = \rho_{\rm max}(r/R)$8: Agreement in $\rho(r) = \rho_{\rm max}(r/R)$9 GeV/$T_{\rm fo}$0; at $T_{\rm fo}$1 GeV/$T_{\rm fo}$2 the model overpredicts $T_{\rm fo}$3 by $T_{\rm fo}$4.

4.2 Comparison With AMPT String Melting

• Both HYDJET++ and AMPT reproduce the $T_{\rm fo}$5 shape below $T_{\rm fo}$6 GeV/$T_{\rm fo}$7.
• HYDJET++ better reproduces the suppressed $T_{\rm fo}$8 at high $T_{\rm fo}$9 (AMPT yields $T_{\rm fo}$0 at 10 GeV/$T_{\rm fo}$1 vs. data/model $T_{\rm fo}$2).
• Global statistics: HYDJET++ $T_{\rm fo}$3 vs. AMPT $T_{\rm fo}$4 for $T_{\rm fo}$5.

5. Sensitivity to Collision Geometry and Limitations

• Observables ($T_{\rm fo}$6, $T_{\rm fo}$7, $T_{\rm fo}$8) depend sensitively on the collision geometry (body-body vs. tip-tip), reflecting the underlying eccentricity and path length variations.
• The model allows flexible assignment of nuclear deformation and orientation, capturing realistic initial state effects for deformed ions.

Principal strengths:

• Modular, fast event generation with decoupled soft/hard production.
• Accurate low–$T_{\rm fo}$9 flow-to-high–$p_T$0 suppression transition.
• Flexible geometry implementation for systematic studies of deformation effects.

Principal limitations:

• No hadronic afterburner: yields of short-lived resonances are underestimated.
• The freeze-out temperature is fixed: inability to capture potential centrality-dependent kinetic decoupling.
• The energy-loss kernel lacks fluctuations beyond mean BDMPS-Z average: non-Gaussian path-length fluctuations are not described.

6. Scaling, Diagnostics, and Applicability

• The separation between soft (hydrodynamic, flow-dominated) and hard (suppression-dominated) regimes is controlled via $p_T$1, $p_T$2, and $p_T$3.
• Computationally, event-by-event independence between modules enables clear diagnostics of hydrodynamic vs. quenching contributions, facilitating parameter scans and geometry studies.
• The model is readily extendable to other deformed systems (see U+U, Pb+Pb), with geometry parameterization following the same Woods–Saxon deformation framework.
• Within the cited implementation and parameter set, the model provides a robust, predictive framework for high-precision $p_T$4 spectral and nuclear modification observables in midmass, deformed collision systems at LHC energies.

7. Summary and Outlook

The application of HYDJET++ to deformed Xe–Xe at $p_T$5 TeV, using $p_T$6 MeV, $p_T$7, and $p_T$8 GeV$p_T$9/fm, yields a quantitative description of charged-hadron spectra and suppression observables over all centralities (Pandey et al., 2022). The model outperforms AMPT (string melting) in matching high-$p_T$00 suppression, accurately captures the centrality and geometry dependence of key observables, and establishes a flexible methodology for incorporating complex nuclear shapes and configurations in event generator frameworks. Its remaining deficiencies, particularly in detailed resonance yields and fluctuating energy-loss dynamics, suggest directions for future development, such as the inclusion of post-hadronic transport or event-by-event fluctuating energy-loss modules.