1. FASTMC A FAST HADRON FREEZE-OUT GENERATOR
a part of Universal Hydro Kinetic Model (UHKM)
, R. Lednicky, T.A. Pocheptsov: Joint Institute for Nuclear Research, Dubna, Russia
I.P. Lokhtin, A.M. Snigirev , L.V. Malinina: Moscow State University, Institute of Nuclear Physics, Russia
Iu.A. Karpenko, Yu.M. Sinyukov: Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine
N.S. Amelin
(PRC (2006))

2. Outline 1. Introduction- motivation.
Physical framework of the model .
Hadron generation procedure .
Model parameters.
Examples of calculations with generator and comparison with RHIC
experimental data
particle ratios
pseudorapidity
pt-spectra
correlation radii
Conclusions, perspectives, status

3.
The creation of the UHKM Universal Hydro Kinetic Model was initiated by Nikolai Amelin.

5. FASTMC: Introduction-Motivation
LHC very high hadron multiplicities fairly fast MC- generators for event simulation. (I. P. Lokhtin and A. M.Snigirev, Phys. Lett. B378, 247 (1996), Z. Phys. C76, 99 (1997).)
FASTMC- fast Monte Carlo procedure of hadron generation
Matter is thermally equilibrated. Particle multiplicities are determined by the temperature and chemical potentials. Statistical model. Chemical freeze-out.
Particles can be generated on the chemical or thermal freeze-out hypersurface represented by a parameterization or a numerical solution of relativistic hydrodynamics.
- Standard parameterizations of the hadron freeze-out hypersurface and flow velocity profile under the assumption of a common chemical and thermal freeze-out.
so-called Bjorken-like (BlastWave) and Hubble-like parametrizations.
(F. Retiere and M. Lisa, Phys.Rev. C70 (2004) ; THERMINATOR: Thermal heavy-Ion generator A. Kisiel, T. Taluc, W. Broniowski, W. Florkowski, nucl-th/ )

6. Introduction-Motivation
Decays of hadronic resonances is included (presently u,d and s quarks only).
Note that it is not pure “Blast-Wave” because the resonances are included
- Besides standard space-like sectors associated with the volume decay,
the hypersurface may also include non-space-like sectors related to the
emission from the surface of expanding system
(Kiev- Nantes model).
-The C++ generator code is written under the ROOT framework.

7. Physical framework of the model: Hadron multiplicities
1. We consider the hadronic matter created in heavy-ion collisions as a hydrodynamically expanding fireball with the equation of state of an ideal hadron gas.
2. “concept of effective volume” T=const and µ=const the total yield of particle species is: , total co-moving volume, ρ-particle number density
3. in central Au+Au or Pb+Pb collisions at GeV
parametrizations of the T and µB:
4. All macroscopic characteristics of particle system (T,µi) are determined via a set of equilibrium distribution functions in the fluid element rest frame:

8. Physical framework of the model: Hadron momentum distribution
We suppose that a hydrodynamic expansion of the fireball ends by a sudden system breakup
at given T and chemical potentials. Momentum distribution of produced hadrons keeps
the thermal character of the equilibrium distribution.
Cooper-Frye formula:
The invariant weight
It is convenient to transform the
four-vectors into
the fluid element rest frame,
and calculate the weight
in this frame
In the case when the normal four-vector coincides with the fluid element flow velocity
the weight is independent of x and isotropic in p. 100% efficiency
the four-momenta of the generated particles transformed back to the fireball rest frame using the velocity field v.

9. Physical framework of the model: Freeze-out surface parameterizations
At relativistic energies, due to dominant longitudinal motion, it is convenient to substitute the Cartesian coordinates t, z by the Bjorken ones , ,
Similarly, it is convenient to parameterize the fluid flow four-velocity
at a point x in terms of the longitudinal z and transverse r fluid flow rapidities: ,
Representing the freeze-out hypersurface by the equation
the hypersurface element:
for the azimuthaly symmetric hypersurface
we further assume the longitudinal boost invariance
The local quantities (such as particle density) are then functions of τ and r only
For the simplest freeze-out hypersurface

10. Physical framework of the model: Freeze-out surface parameterizations
The Bjorken model with hypersurface
Assuming and the linear transverse flow rapidity profile:
here R is the fireball transverse radius;
The total effective volume for particle production at
We refer this choice of the freeze-out hyper-surface and the flow 4-velocity profile as the Bjorken-like parametrization
We also consider so called Cracow model scenario corresponding to the Hubble-like freeze-out hypersurface
and spherically-symmetric Hubble flow with four-velocity

11. Hadron generation procedure
Initialization of the chosen model parameters
The particle three-momenta in the fluid element rest frames
according to the probability
by sampling uniformly distributed cos(θ*p) and φ*p
Calculation of Veff and particle number densities
von Neumann rejection/acceptance procedure to account
for diff. between the true prob. and prob. corresponding
to 3-5. Residual weight
simulated x, p are accepted W> ξ- a test variable randomly
simulated in [0,W_max], otherwise-> 3
The mean multiplicities
Multiplicities by Poisson distr.
Simulation of particle freeze-out 4-coordinates
in the fireball rest frame :
on each hypersurface segment accord. to the element
by sampling uniformly distributed r,η, φ
the hadron four-momentum is
boosted to the fireball rest frame
Calculation of
the corresponding collective flow four-velocities
The two-body, three-body and many-body
decays are simulated with the branching ratios
calculated via ROOT utilities;--
Boltzmann equation solver

12. Hadron generation procedure
It should be stressed that a high generation speed is achieved due to 100 % generation efficiency of the freeze-out four-coordinates and four-momenta in steps 3-5 as well as due to a weak non-uniformity of the residual weight W
in the cases of practical interest.
For example, in the Bjorken-like model, the increase of the maximal transverse flow rapidity from zero to 0.65 leads only to a few percent decrease of the generation speed. Compared, e.g., to THERMINATOR our generator appears more than two order of magnitude faster.

13. Validation of the fast MC procedure
In the Boltzmann approximation for the equilibrium distribution function,
the transverse momentum spectrum in the Bjorken-like model takes
the form:

14. Validation of the MC procedure:
Exact formulae (solid line), the corresponding MC results (black triangles),
MC results obtained with a constant residual weight (black points).
transverse flow rapidity

15. Model parameters: 1. Number of events to generate.
2. Thermodynamic parameters at chemical freeze-out:
T temperature
{µB, µS, µQ} chemical potentials per a unit charge
3. As an option, γS < 1 taking into account the strangeness suppression
4. Volume parameters:
τ -the freeze-out proper time
R- firebal transverse radius
maximal transverse flow rapidity for Bjorken-like parametrization
ηmax -maximal space-time longitudinal rapidity which determines the rapidity interval [- ηmax, ηmax] in the collision center-of-mass system.
7. To account for the violation of the boost invariance, an option corresponding to the substitution of the uniform distribution of the space-time longitudinal rapidity by a Gaussian distribution.
8. Option to calculate T, µB using phenomenological dependence of

16. The parameters used to model central Au+Au collisions at = 200 GeV
parameter Bjorken-like Hubble-like
T, GeV
µB, GeV
µS, GeV
µQ, GeV
γS (0.8) (0.8)
τ, fm/c
R, fm
ηmax (3,5) (3,5)
ρu max
Particle number ratios near mid-rapidity in central Au + Au collisions
= GeV
FASTMC PHENIX
π -/π ± 0.004
K-/K ±0.008
K-/ π ± 0.001
p-/p ± 0.011

17. Examples of calculations and comparison with RHIC experimental data:
Particle ratios near mid-rapidity in central Au + Au collisions at = 130 GeV.
T = GeV, µB = GeV, µS = GeV and µQ = GeV.

18. Pseudo-rapidity spectrum of charged hadrons
PHOBOS data on pseudo-rapidity spectrum of charged hadrons in central Au+Au collisions at =200 GeV (black points)
our MC results obtained within the Bjorken-like and Hubble-like
models for different values of (lines).
Single freeze-out scenario allows
to fix the effective volume

19. p_t spectra
The mid-rapidity PHENIX data on π, K and proton p_t spectra in Au+Au collisions at 200 GeV with our MC results obtained within the Bjorken-like and Hubble-like models. .
For kaons, the discrepancy can be diminished with the help of the strangeness suppression parameter γs of 0.8
The overestimated slope of the kaon and proton p_t spectra
can also be related with the oversimplified assumption of a common
thermal and chemical freeze-out or insufficient number of the
accounted heavy resonance states.

20. Momentum correlations
Due to the effects of QS and FSI,
the momentum correlations of two or more particles at small relative momenta in their center-of-mass system are sensitive to the space-time characteristics of the production
process so serving as a correlation femtoscopy tool.
q = p1- p2 , x = x1- x2
w=1+cos qx .
The corresponding correlation widths are usually parameterized in terms of the Gaussian correlation radii R_i:
side
out  transverse
pair velocity vt
long  beam
We choose as the reference frame the longitudinal co-moving system (LCMS)

21. Momentum correlations
STAR collaboration data (open points)
FASTMC Tth=Tch=165 MeV, R=10 fm, τ=6.1 fm/c (solid line)
FASTMC Tth=Tch=165 MeV, R=8 fm, τ=9 fm/c (dashed line)
with negative r-t correlation coefficient (red dotted line)
T=165 MeV
T=100 MeV
T=130 MeV

22. Momentum correlations: other approaches
The concept of a later thermal freeze-out occurring at
Tth<Tch and with no multiplicity constraint on the thermal effective volume
can help to resolve this problem:
More realistic models as compared with the simple
Bjorken-like and Hubble-like ones (particularly, consider a more complex form
of the freeze-out hypersurface taking into account particle emission
from the surface of expanding system M.S.Borysova, Yu.M.Sinyukov, S.V.Akkelin,
B.Erazmus and Iu.A.Karpenko, Phys. Rev. C 73, (2006)
and study the problem of particle rescattering and resonance excitation after the
chemical and/or thermal freeze-out (only minor effect of elastic rescatterings on particle spectra and correlations is expected N.S.Amelin, R.Lednicky, L.V.Malinina, T.A.Pocheptsov and Y.M.Sinyukov, Phys. Rev. C 73, (2006)
For the latter, our earlier developed C++ kinetic code can be coupled to FASTMC.
F. Retiere and M. Lisa, Phys.Rev. C70 (2004)

23. Conclusions
We have developed a MC simulation procedure and the corresponding C++ code allowing for a fast but realistic description of multiple hadron production in central relativistic heavy ion collisions.
A high generation speed and an easy control through input parameters make our MC generator code particularly useful for detector studies.
-As options, we have implemented two freeze-out scenarios with coinciding and with different chemical and thermal freeze-outs.
Also implemented are various options of the freeze-out hypersurface parameterizations
-The generator code is quite flexible and allows the user to add other scenarios and freeze-out surface parameterizations as well as additional hadron species in a simple manner.
-We have compared the RHIC experimental data with our MC generation results obtained within the single freeze-out scenario and Bjorken-like and Hubble-like freeze-out surface parameterizations.

24. Status and Perspectives http://uhkm.jinr.ru
The fast MC procedure is extended
to describe non-central collisions.
Scenarios with single f.o. and thermal
f.o. are implemented in the model
Generation from the external file
containing SPHES output is tested and very soon will be accessible
in our site.
FASTMChydro + high-pt part related to the partonic states:
We are studying influence of the mini-jet production on CFs at RHIC energies. Jet-quenching model PYTHIA/PYthiaQUENched is implemented.
FASTMS is implemented in AliRoot
We started to study the possibility to add the QGSM inelastic cascade part to UKM elastic cascade.
SPHES is tested and very soon
will be accessible for users in our site.

25. Additional slides

26. Preliminary:

31. Jet-quenching model PYQUEN/PYTHIA is implemented.
High-p_t part related to the partonic states created in ultra-relativistic heavy
PYTHYA+PYQUEN (I.P.Lokhtin and A.M.Snigirev, Eur. Phys. J. C 45, 211 (2006).)

32. Momentum correlations
The fitted correlation radii and strength parameter are compared with those measured by STAR collaboration
The Bjorken-like model describes the decrease of the correlation radii with increasing k_t but overestimates their values.The situation is even worth with the Hubble-like model which is
more constraint than the Bjorken-like one and yields the longitudinal radius by a factor two larger.