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THERMINATOR 2: THERMal heavy IoN generATOR 2

Mikolaj Chojnacki, Adam Kisiel, Wojciech Florkowski, Wojciech Broniowski

arXiv:1102.0273v1nucl-thhep-phnucl-ex

TL;DR

THERMINATOR supports statistical hadronization studies in relativistic heavy-ion collisions. Its extended package adds femtoscopic analysis, and estimates indicate that post-freeze-out elastic rescattering is small for pions.

  • Problem

    The paper addresses statistical hadronization studies of particle production in relativistic heavy-ion collisions.

    Source: §1 · p. 3

  • Method

    The extended THERMINATOR package performs statistical hadronization, while FEMTO-THERMINATOR analyzes femtoscopic correlations with the two-particle method.

    Source: §1 · p. 3, §9 · p. 35

  • Results

    For pions, post-freeze-out elastic scattering averages 1.8 times in most-central collisions and fewer than 1 time in mid-central and peripheral collisions.

    Source: §8.2 · p. 34

  • Takeaways & Limitations

    Spectra, flow, and femtoscopy analyses are all implemented in THERMINATOR 2.

    Source: §9 · p. 35

    The simple HBT fitting examples do not address the problem that emission functions are often far from Gaussian despite the use of Gaussian fits.

    Source: §7.2 · p. 29

Abstract

from arXiv · show

We present an extended version of THERMINATOR, a Monte Carlo event generator dedicated to studies of the statistical production of particles in relativistic heavy-ion collisions. The increased functionality of the code contains the following features: The input of any shape of the freeze-out hypersurface and the expansion velocity field, including the 3+1 dimensional profiles, in particular those generated externally with various hydrodynamic codes. The hypersufraces may have variable thermal parameters, which allows for studies departing significantly from the mid-rapidity region, where the baryon chemical potential becomes large. We include a library of standard sets of hypersurfaces and velocity profiles describing the RHIC Au+Au data at sqrt(s_(NN)) = 200 GeV for various centralities, as well as those anticipated for the LHC Pb+Pb collisions at sqrt(s_(NN)) = 5.5 TeV. A separate code, FEMTO-THERMINATOR, is provided to carry out the analysis of femtoscopic correlations which are an important source of information concerning the size and expansion of the system. We also include several useful scripts that carry out auxiliary tasks, such as obtaining an estimate of the number of elastic collisions after the freeze-out, counting of particles flowing back into the fireball and violating causality (typically very few), or visualizing various results: the particle p_T-spectra, the elliptic flow coefficients, and the HBT correlation radii. We also investigate the problem of the back-flow of particles into the hydrodynamic region, as well as estimate the elastic rescattering in terms of trajectory crossings. The package is written in C++ and uses the CERN ROOT environment.

Program summary

THERMINATOR 2 is a C++ program using CERN ROOT for statistical hadronization simulations. Its distribution includes the code, configuration and hypersurface-library files, with documented runtime and storage requirements.

Program summary

  • The program is written in C++ with CERN ROOT libraries and BASH, and is distributed as a 2.7 MB tar.gz package.The listing reports 13,000 lines of code, 300 configuration lines, and 400,000 hypersurface-library input files.Source: Program summary · p. 2
  • Execution requires a C++ compiler and CERN ROOT 5.26 or later; the tested setup used Linux Ubuntu 10.10 x64.The program was tested on an Intel Core2 Duo E8400 at 3 GHz with 4 GB RAM.Source: Program summary · p. 2
  • Typical execution requires about 30 MB for therm2 events and 150 MB for femto.The listing reports 45 MB per 500-event data file and 35 MB per correlation file for one kT bin.Source: Program summary · p. 2
  • Typical runs take 8 minutes per 500 events, while one HBT kT bin takes 25 minutes with 5,000 events.The listing also reports 70 minutes for primordial multiplicities, calculated once per case, and 10 minutes to draw all figures.Source: Program summary · p. 2

1 Introduction

THERMINATOR 2 extends statistical hadronization simulations to general freeze-out conditions and adds femtoscopic analysis and auxiliary scripts. The introduction also reports small back-flow and post-freeze-out pion rescattering estimates for realistic cases.

1 Introduction

  • THERMINATOR 2 uses C++ and CERN ROOT and is intended to be adaptable for experimental-data analysis, detector modeling, and heavy-ion studies.The introduction also points to related packages for particle abundances, fragmented-fireball production, and hydrodynamics with jets.Source: §1 · pp. 3–4
  • The code accepts freeze-out hypersurfaces and expansion velocity fields of arbitrary shape, including externally generated 2+1- and 3+1-dimensional hydrodynamic profiles.This extends the original code's simple boost-invariant hypersurfaces.Source: §1 · p. 3
  • Space-time-dependent thermal parameters support non-boost-invariant studies, including fragmentation regions where the baryon chemical potential becomes large.Source: §1 · p. 3
  • The package includes standard profiles for RHIC Au+Au collisions at 200 GeV across centralities and anticipated LHC Pb+Pb collisions at 5.5 TeV.The anticipated LHC profiles can be used to model detectors before real data are available.Source: §1 · p. 3
  • FEMTO-THERMINATOR analyzes femtoscopic correlations, which provide complementary information about the system's size and expansion.Source: §1 · p. 3
  • Auxiliary scripts count back-flowing particles and estimate elastic collisions, and visualize pT spectra, elliptic flow v2, and HBT radii.For realistic profiles, the back-flow count is negligible, below 1%.Source: §1 · pp. 3–4
  • For realistic cases, back-flow is below 1%, while pion-pion elastic rescattering averages about 1.5 collisions, supporting the single-freeze-out approximation.The rescattering estimate is described as small; the scripts estimate it through trajectory crossings.Source: §1 · p. 4

2 Statistical hadronization

The statistical approach uses a small set of physical inputs to describe a broad range of heavy-ion observables, with hadronic resonances contributing significantly at freeze-out temperatures of about 140–165 MeV.

2 Statistical hadronization

  • At freeze-out temperatures of ∼140−165 MeV, hadronic resonances contribute significantly to observables, and THERMINATOR 2 incorporates four-**** and three-*** resonances.The statistical approach has been used to describe particle abundances, transverse-momentum spectra, balance functions, elliptic flow, and femtoscopic observables.Source: §2 · p. 4

2.1 Cooper-Frye formalism

  • The Cooper-Frye formula gives hadron production on the freeze-out hypersurface; resonance decays in two- and three-body cascades produce stable particles.THERMINATOR provides the space-time positions and velocities of produced particles.Source: §2 · pp. 4–5, §2.1 · p. 5

2.2 Freeze-out in 3+1-dimensional perfect-fluid hydrodynamics

  • For 3+1-dimensional perfect-fluid hydrodynamics without assumed symmetries, the freeze-out surface is parametrized using a numerically obtained distance function d(φ, ζ, θ).The coordinates are defined through the hypersurface parameters, with τ = τ_i + d(φ, ζ, θ) sinθ sinζ.Source: Figure 2 · p. 7, §2.2 · p. 8

2.3 Freeze-out in 2+1-dimensional boost-invariant hydrodynamics

  • For boost-invariant systems, the fluid rapidity equals the space-time rapidity, Y_f = Y_s, and the surface function d depends only on φ and ζ.This is the stated 2+1-dimensional reduction of the freeze-out description.Source: §2.3 · p. 9, Figure 3 · p. 10

2.4 Blast-wave models

  • The modified Blast-Wave model uses τ(ρ) = τ_f + Aρ: A > 0 makes outer regions freeze out later, A < 0 earlier, and A = 0 recovers the standard parametrization.The standard model assumes constant proper time; the generalized condition gives a sloped freeze-out curve.Source: §2.4 · p. 10

2.5 Cracow model

  • The Cracow model defines freeze-out by τ² − ρ² = τ_f² and sets the fluid four-velocity proportional to the space-time position.Its hypersurface volume element is proportional to u^μ, as in the Siemens and Rasmussen model.Source: §2.5 · p. 11

2.6 Resonance decays

  • Two- and three-body resonance decays are implemented as in the original THERMINATOR code.The SHARE table format is presented in Appendix A.Source: §2.6 · p. 12

3 Structure of THERMINATOR 2

THERMINATOR 2 is organized as a sequence of modules that read particle data, generate and decay events, and write output. Its files and configuration separate the event generator, freeze-out models, and analysis tools.

3.1 Modules and their functions

  • Event generation reads particle and decay data, computes primordial multiplicities, generates particles with Poisson-distributed multiplicities, decays resonances, and writes each event.The event-generation flow is divided among Parser.cxx, ParticleDB.cxx, Integrator.cxx, Event.cxx, and ParticleDecay.cxx.Source: §3.1 · p. 12

3.2 File structure

  • Freeze-out model files store configuration and hypersurface data, while ROOT macros, SHARE data, and auxiliary programs support analysis and related tasks.The supplied model configurations include Blast-Wave, Krakow Single Freeze-out, 2+1D boost-invariant hydrodynamics, and 3+1D hydrodynamics; the distribution does not include hypersurfaces for the 3+1D case.Source: §3.2 · pp. 16–18
  • The package includes freeze-out hypersurface files for RHIC Au+Au at 200 GeV and LHC Pb+Pb at 5.5 TeV across multiple centrality classes.File names encode collision system, energy, centrality, initial temperature, hydrodynamic starting time, and freeze-out temperature.Source: §3.2 · p. 17

3.3 *.ini files

  • The main events.ini file selects the model and event-generation settings, while a separate model configuration file supplies model-specific parameters.IntegrateSamples defaults to 5000000; increasing NumberOfEvents above 20000 is needed for stable HBT results, while about 1000 events suffices for pT spectra.Source: §3.3 · p. 18

4 Storage of events

THERMINATOR 2 stores generated events as either plain text or ROOT files. The text format is human-readable and complete, while ROOT organizes event, particle, and parameter data in separate trees.

4 Storage of events

  • Each event is written after generation and decay, in either plain ASCII text or ROOT format.Particles for one event are held in memory until the event is complete, then written and erased from memory.Source: §4 · p. 18

4.1 Plain text output file

  • Plain-text output records event headers and one entry per particle, with particle identities, decay ancestry, four-momentum, and creation coordinates.The eid, fathereid, pid, and rootpid fields can trace a resonance cascade back to its primordial particle.Source: §4.1 · p. 19

4.2 ROOT file

  • ROOT output separates event, parameter, and particle information into three TTree objects, with event and particle data stored in leaves.By default, each file contains 500 event entries in the events tree and particle data from 500 events in the particles tree.Source: §4.2 · p. 20

5 Installation and sample runs

The installation workflow downloads and compiles THERMINATOR 2, then provides documentation and scripts for test runs, event generation, and analysis. The sample run uses RHIC Au+Au collisions at 200 GeV and takes several hours on the reported processor.

5.1 Downloading the latest version

  • THERMINATOR 2 is distributed as a tar.gz archive containing source code, scripts, configuration files, and data files.Compilation and execution require a C++ compiler and ROOT version 5.26 or later.Source: §5.1 · pp. 21–22

5.2 Compilation

  • After compilation, the executable binaries are available in the main directory.The package is compiled with the provided make configuration.Source: §5.2 · p. 22

5.3 Doxygen documentation

  • Doxygen generates HTML and PDF reference manuals documenting the classes, variables, and files.The generated main-directory files are manual.html and manual.pdf.Source: §5.3 · p. 22

5.4 A test run

  • 5000 events in the RHIC Au+Au 200 GeV, 20–30% centrality test run take 4 h 20 min on a 3 GHz processor.The run uses 2D boost-invariant hydrodynamics and produces event files and figures.Source: §5.4 · p. 23

5.5 Running

  • The run scripts generate events and figures from the selected model settings, with ROOT macros also available for individual plots.The default event-generation configuration uses 2D boost-invariant hydrodynamics for RHIC top energy at 20–30% centrality.Source: §5.5 · pp. 23–24

5.6 Auxiliary files

  • Auxiliary temporary files let scripts track event directories and file counts, supporting multiple THERMINATOR 2 processes at once.This includes parallel runs on a computer cluster.Source: §5.6 · p. 24

6 Scripts to generate physical results

ROOT scripts process THERMINATOR 2 event files to produce transverse-momentum spectra and flow coefficients. The scripts expect the event files to remain in sequence, starting from event000.root.

Scripts to generate physical results

  • Analysis scripts read chains of event files beginning with event000.root, so files should not be deleted from the middle of the sequence.The scripts produce figures and ROOT figure macros.Source: §6 · p. 24
  • The figure_distpt.C macro produces transverse-momentum spectra from a requested number of event files.The number of event files to process is supplied as an argument.Source: §6 · p. 25
  • The figure_v2pt.C script computes elliptic flow v2 as a function of transverse momentum in the hydrodynamics reaction plane; figure_v4pt.C runs analogously.The scripts process the requested number of event files.Source: §6 · p. 25

7 HBT analysis

THERMINATOR 2 adds two-particle femtoscopic analysis to calculate HBT correlation radii. The section describes the correlation calculation, fitting workflow, and conditions that qualify interpretation of the fitted radii.

7 HBT analysis

7.1 Method

  • The analysis uses the LCMS and Bertsch-Pratt momentum axes, while the pair wave function is represented in the PRF.The long axis follows the beam, the out axis follows the pair’s average transverse momentum, and the side axis is perpendicular to both.Source: §7.1 · p. 27
  • Monte Carlo sums over generated particles or pairs to calculate binned correlation functions, with mixed events removing non-femtoscopic correlations and increasing statistics.For identical charged particles, event mixing is described as innocuous; calculation without mixing is not implemented.Source: §7.1 · pp. 27–28

7.2 Calculation of correlation functions extraction of the HBT radii

  • Charged pions are paired within each event, and their wave-function contributions populate the correlation numerator while pair counts populate the denominator.The resulting ratio is the correlation function; pairs can be selected by kT for femtoscopic results in a chosen momentum range.Source: §7.2 · pp. 28–29
  • The fitting procedure extracts λ and Rout, Rside, and Rlong from a Gaussian form, but the examples do not address non-Gaussian emission functions.Statistically stable HBT radii may require 20,000 or even 50,000 events, with correspondingly long execution times.Source: §7.2 · pp. 29–31

8 Yielding auxiliary results

The section presents analyses of back-flow and post-freeze-out elastic rescattering as probes of freeze-out assumptions. In the realistic calculations, back-flow is very small, and estimated pion rescattering is limited.

8 Yielding auxiliary results

  • These analyses address physical aspects of the freeze-out model, including the back-flow problem and the assumption that chemical and thermal freeze-outs occur simultaneously.The paper motivates trajectory crossings as a simple estimate of how well that simultaneous-freeze-out approximation holds.Source: §8 · p. 31, §8.2 · pp. 33–34

8.1 Backflow; 8.2 Elastic rescattering

  • The back-flow estimate compares particles with pμdΣμ < 0 against the case without that constraint, while the collision estimate uses trajectory crossings.A geometric collision is counted when πd² is less than the elastic cross section at the pair’s Mandelstam variable s.Source: §8.1 · pp. 32–33, §8.2 · p. 34

8.1 Backflow

  • Back-flow is below 0.5% for pions and even smaller for more massive particles in the realistic calculation.The authors attribute this smallness to large transverse fluid velocity inhibiting opposite-direction thermal motion.Source: §8.1 · p. 33

8.2 Elastic rescattering

  • Pions scatter elastically off other pions an average of 1.8 times in most-central collisions, and fewer than once in mid-central and peripheral collisions.The authors conclude that post-freeze-out elastic rescattering is small for pions and that the approximation is appropriate.Source: §8.2 · p. 34

9 Conclusion

THERMINATOR 2 combines general freeze-out conditions, particle spectra and flow, and femtoscopic analysis, with applications to hydrodynamic predictions and detector modeling. The conclusion also outlines extensions to viscosity, femtoscopic fitting, flow and event modeling.

9 Conclusion

  • General freeze-out conditions read from external files support detailed studies of hydrodynamic predictions with statistical hadronization, an approach reported to reproduce RHIC and LHC data.The authors describe it as a practical tool for other physical studies and detector modeling.Source: §9 · p. 35
  • FEMTO-THERMINATOR adds two-particle femtoscopic analysis, completing the spectra, flow, and femtoscopy studies used in heavy-ion data analyses.The package also includes ROOT macros for calculating and presenting physical results.Source: §9 · p. 35
  • Viscosity effects may be included through a straightforward modification of the approach, and fugacities can represent incomplete chemical equilibrium.The authors identify these as ways to use the code in viscous-hydrodynamics studies and to model incomplete chemical equilibrium.Source: §9 · p. 35
  • The code is documented for adaptation to other applications and supplied with ROOT macros for calculating and presenting physical results.The authors describe the program structure as clear and straightforward to adapt.Source: §9 · p. 35
  • Planned extensions include viscous-hydrodynamics freeze-out, advanced femtoscopic fitting such as imaging, and analysis of triangular and multipole flow.The current FEMTO-THERMINATOR output already contains three-dimensional correlation functions for advanced analyses.Source: §9 · p. 35
  • Other proposed extensions cover correlations beyond Bose-Einstein effects and more general multiplicity distributions for realistic event-by-event fluctuations.The authors also mention approximate conservation of global charges and transverse momentum, and file-format interfaces developed according to user demand.Source: §9 · pp. 35–36

A Particle data files (particles.data and decays.data)

The appendix describes the particle and decay data files that THERMINATOR 2 reads, including their fields and how branching ratios control decay feed-down.

A Particle data files (particles.data and decays.data)

  • The particle file records labels, masses, widths, spin, isospin, quark content, and Monte Carlo identifiers in the listed field order.Where available, the Monte Carlo identifiers follow the standard convention.Source: A Particle data files (particles.data… · p. 37
  • The decay file specifies parent and daughter particles, branching ratio, and whether to apply a Clebsch–Gordan coefficient.A third daughter entry is used only for three-body decays; the text says two-body decays normally use the coefficient, while three-body branching ratios are supplied in full.Source: A Particle data files (particles.data… · p. 37
  • Setting a decay channel’s branching ratio to zero switches off that channel, allowing control of feed-down from weak decays.Source: A Particle data files (particles.data… · p. 37

B General definitions

The appendix lists definitions and parameter fields for the supported freeze-out models, from hydro-inspired profiles to 2+1- and 3+1-dimensional hydrodynamic inputs.

B General definitions

  • General definitions cover particle kinematics, angles, quark counts, particle identifiers, Cooper-Frye integrand quantities, and decay-channel counts.The listed variables include projected four-momentum, energy in the fluid local rest frame, and transverse-momentum and transverse-velocity angles.Source: B General definitions · pp. 38–39

C Hydro-inspired models

  • Hydro-inspired model parameters include the firecylinder transverse size and baryon, isospin, strange, and charm chemical potentials.Source: C Hydro-inspired models · p. 39

C.1 Cracow Single Freeze-out Model

  • The Cracow single-freeze-out model has model-specific parameters, including a definition of proper time.Source: §C.1 · p. 40

D.2 Lhyquid 2D boost-invariant

  • The listed Lhyquid 2D boost-invariant and Blast-Wave model entries identify model-specific parameters without describing their values in these passages.Source: §C.2 · p. 40, §D.2 · p. 42

C.3 Blast-Wave A-Class

  • Blast-Wave A-Class options include linear transverse velocity profiles, exponential creation-time shifts in delayed models, and a formation-time option that shifts particle position.The transverse velocity and the A parameter are also identified among the model-specific quantities.Source: §C.3 · p. 40

D Freeze-out models based on 2+1 and 3+1 hydrodynamic calculations

  • Hydrodynamic-model inputs include chemical potentials, accelerator and nuclei labels, central initial temperature, phase-space integration ranges, and distance to the hypersurface element.Source: D Freeze-out models based on 2+1 and… · p. 41

D.1 Lhyquid 3D

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