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 $T_{AA}(b)$.
   • In-medium parton energy loss (jet quenching) is incorporated following the BDMPS-Z formalism, parameterized by the transport coefficient $\hat q$.
   • 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 $^{129}$Xe nucleus is modeled with quadrupole deformation $\beta_2 = 0.18$, in a Woods–Saxon geometry:

$R(\theta) = R_0 [1 + \beta_2 Y_{20}(\theta)]$

with $R_0 = 5.36$ fm and diffuseness $a = 0.59$ 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 $\cos\theta_i > 0.8$ (tip–tip) or $|\cos\theta_i| < 0.2$ (body–body) per nucleus.

2.2 Event Generation and Centrality

• Impact parameters are sampled with probability $\propto b$, up to $b_{\rm max} \approx 15$ fm.
• Centrality percentiles are defined using final charged multiplicity at midrapidity, with typical bins: $0$–$5\%$, $5$–$10\%$, $10$–$20\%$, $20$–$30\%$, $30$–$50\%$, $50$–$70\%$.
• Glauber calculations give $\langle N_{\rm part}\rangle$ and $\langle N_{\rm coll}\rangle$ for each centrality class.

2.3 Tuned Parameters

Parameter	Value
Freeze-out temperature $T_{\rm fo}$	120 MeV
Max. transverse flow $\rho_{\rm max}$	$\tanh^{-1}(0.6)$
Baryochemical potential $\mu_B$	0 MeV
Minimum $p_T$ (hard scatterings) $p_T^{\rm min}$	2 GeV/$c$
Soft fraction (central)	90%
Transport coefficient $\hat q$	1.5 GeV$^2$/fm
PDF (pp baseline)	CTEQ6L
Nuclear shadowing	EKS98

3. Key Physics Observables: Spectra and Nuclear Modification

3.1 $p_T$-Spectra Construction

• The total $p_T$ spectrum is the sum of soft and hard contributions:

$$\frac{dN}{dy dp_T} = \frac{dN_{\rm soft}}{dy dp_T} + \frac{dN_{\rm hard}}{d^2p_T dy}$$

where the hard term for a given impact parameter $b$ is:

$$\frac{dN_{\rm hard}}{d^2p_T dy} = T_{AA}(b) \left[\frac{d\sigma_{pp}^{\rm hard}}{d^2p_T dy}\right]_{\rm PYQUEN} \otimes \text{Energy-loss kernel}(\hat q, L)$$

3.2 Nuclear Modification Factors

• $R_{AA}(p_T)$: Compares the observed yield to that expected from scaled $pp$ reference:

$$R_{AA}(p_T) = \frac{1}{\langle N_{\rm coll} \rangle} \frac{dN_{AA}/dp_T}{dN_{pp}/dp_T}$$

• $R_{CP}(p_T)$: Ratio of central to peripheral yields, both normalized by $\langle N_{\rm coll}\rangle$:

$$R_{CP}(p_T) = \frac{(1/\langle N_{\rm coll}^{\rm cent}\rangle)\ dN_{AA}^{\rm cent}/dp_T} {(1/\langle N_{\rm coll}^{\rm peri}\rangle)\ dN_{AA}^{\rm peri}/dp_T}$$

4. Model Performance and Empirical Validation

4.1 Agreement with ALICE Data

• $p_T$-Spectra: HYDJET++ reproduces the centrality-dependent $p_T$ spectrum at midrapidity up to $p_T \sim 15$ GeV/$c$ within 10–15%.
• $R_{AA}$: At $p_T \sim 10$ GeV/$c$ and $0$–$5$\% centrality, the model gives $R_{AA} \sim 0.18$, consistent with ALICE ($0.17 \pm 0.02$). The high-$p_T$ rise in $R_{AA}$ is reproduced.
• $R_{CP}$: Agreement in $3 < p_T < 8$ GeV/$c$; at $p_T < 2$ GeV/$c$ the model overpredicts $R_{CP}$ by $\sim20\%$.

4.2 Comparison With AMPT String Melting

• Both HYDJET++ and AMPT reproduce the $dN/dp_T$ shape below $p_T \sim 3$ GeV/$c$.
• HYDJET++ better reproduces the suppressed $R_{AA}$ at high $p_T$ (AMPT yields $R_{AA} \sim 0.25$ at 10 GeV/$c$ vs. data/model $\sim 0.18$).
• Global statistics: HYDJET++ $\chi^2/{\rm ndf} \approx 1.2$ vs. AMPT $\approx 2.5$ for $R_{AA}$.

5. Sensitivity to Collision Geometry and Limitations

• Observables ($\langle p_T \rangle$, $R_{AA}$, $R_{CP}$) 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–$p_T$ flow-to-high–$p_T$ 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 $p_T^{\rm min}$, $\hat q$, and $\rho_{\rm max}$.
• 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 $p_T$ 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 $\sqrt{s_{NN}}=5.44$ TeV, using $T_{\rm fo} = 120$ MeV, $\rho_{\rm max} = \tanh^{-1}(0.6)$, and $\hat q = 1.5$ GeV$^2$/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-$p_T$ 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.