N.Amelin ,Project leader, author of the UKM R. Lednisky, Particle Correlations

Yu.Sinyukov, Hydrodynamic evolution

I. Lokhtin, author of the Hydro model

A. Snegirev, author of the Hydro model

L.Malinina, UKM-R coding and simulations T.Pocheptsov, UKM-R coding under ROOT

Development of the UKM -Universal-Kinetic-Model under the ROOT framework (UKM-R).

1. Motivation2. Structure 3. Kinetic part4. Initial states: -Thermal initial state, test the UKM algorithm -Hydro initial state5. First results: comparison with

RHIC data

“Such important effects as jet quenching, HBT correlations and elliptic flowhave rather poor implementation in available event generators (in the most of generators are not represented at all) and should be introduced in the future”CERN workshop on Monte-Carlo tools for LHC July 2003, Heavy ions working group

Attempts to describe large elliptic flow andsmall HBT radii observed in the RHIC data in the single model.Numerous hydrodynamic models elliptic flow, but disagree with HBT radii,Cascade models fail actually to describe the interferometry radii of the system (D.Ferenc, Nucl.Phys. A 610,523c,D.Teaney, j.Lauret, E.V.Shuryak, nucl-th/011037),Relativistic quantum molecular dynamics model (UrQMD) underestimates elliptic flow, and overestimates the HBT-radii, Hydro+UrQMD overestimates HBT-radii (S.Soff, S.A.Bass and A.Dumitru Ph.Lett.86 3981). Good description of the experimental data obtained in the simple rescattering model by T.Humanic (arXiv:nucl-th 0205053 ) early hadronization τ = 1 fm/c, very dense state of matter 8 GeV / fm3, T=300 MeV . rescatterings generate flow , no intial flow

Usually used for HBT simulations among ALICE generators: MeVSim, Hijing (no space information),UrQMDSome of the new ones: “simple rescattering model” of T.Humanic (nucl-th 0205053), THERMINATOR

(Thermal Heavy Ion Generator, W.Broniowski, W.Florkowski, A.Kisel, T.Taluc nucl-th 0504047) For HBT simulations we need of Monte Carlo generator which:-- includes flow; -- takes into account hadronic rescatterings,-- resonances decays;-- is rapid; -- is flexible;

MotivationUKM-R Universal-Kinetic-Model under ROOT framework

UKM-R is based on the UKM model created by N.S.Amelin which is a universal model, because there is a possibility to choose and add particle system

(hadrons, partons) and a particular numerical algorithm.

UKM-R with its cascade algorithm included is a solver of a system of Boltzmann equations. The implemented in UKM-R cascade algorithm was borrowed from the QGSM cascade algorithm

UKM-R realisation:The code is written in O.O. C++. The ROOT classes are used for particle properties description,

for operations with three and four vectors and for organization the containers and lists.UKM-R classesInitial state initialization class, here the initial space andmomentum distributions of the particles are described;

Particle densities class, here the mean particle densities are calculated

Particle interaction class to specify particle interactions;

Particle decay class to specify particle decays;

Boundary condition class to specify particle boundary conditions;

Kinetic algorithm class. Kinetic algorithm class to specify particle kinetic algorithm. In UKM-R is applied the algorithm function which is a solver of the Boltzmann equation system

UKM-R is a version of Universal-Kinetic-Model under the ROOT framework

Particle class: PDG-code, momentum components, space-time componentsParticle objects are grouped into particle lists.

UKM-R types of particle lists: UKM-R types of lists:

Primaries list, Collision list Secondary particles listNew secondaries list.

UKM-R cascade algorithm1. The List of Primaries is 3.The earliest collision is simulated,initialized and an initial value of the the two collided particles are removed fromcurrent time is set. all lists and the created particles are put in the list of new secondaries. The earliest decay occurs new secondaries.2. All possible pairs of colliding particles from the The particle coordinates are shifted inPrimaries list are searched for and the Collision list is filled accordance with the interaction (decay) time and particlein according to the increasing interaction time velocity vectors.

4. The collisions of the particles from new secondaries listThe binary particle collision is considered as possible if with all particles from primaries andd- the distance of closest approach of the two colliding secondaries lists are searched for and theparticles in their rest frame σ- total collision time ordered collision list is updated. cross-section, parameterized (using data for total elastic σ from PDG Phys. Rev. D 45, 83 (1992)). 5. The particles from the new secondaries list is moved to secondaries lists. Simulation continues starting from step 3 or stops if the current time > stop time or List of interactions is empty.

UKM-R Universal-Kinetic-Model under ROOT framework

)( s

d

UniversalKineticModel----------------------------------- #Projectiles#Targets #Primaries

#Secondaries #LastSecondaries#ListOfInteractions -----------------------------------+ParticleEvolution()+AfterInteractionTest()=0+Camera()= 0

Scatterer-----------------------------------

-----------------------------------+CollisionTime()=0+Scatter()=0

Initializer-----------------------------------

-+InitialState()=0----------------------

+InitializeProjectileTarget()=0

Decayer-----------------------------------

-----------------------------------+DecayTime()=0+Decay()=0

BoundaryCondition-----------------------------------

-----------------------------------+TimeToBoundary()=0+FulfilCondition()=0

UKM structure

Initial state classIt allows one to create different initial space and momentum particle distributions.

1) Initial states to perform cascade algorithm tests:

Static spherically symmetric fireball consisting of the particles of one type;

The initial 4-momentum distribution of a hadrons of mass m isotropic Bolzmann

distribution

Non-relativistic

Relativistic

1) Test of the correctness of any cascade algorithm was applied to UKM-R

2) Resonances decay

mTpeppf 2/2 2

~)(

TEemEEEf /22~)(

From: “EVOLUTION OF OBSERVABLES IN A NUMERICAL KINETIC MODEL” in preparationN.S. Amelin, R. Lednicky, L. I. Malinina, T. A. Pocheptsov and Yu.M. Sinyukov

The test of the correcteness of cascade algorithm was applied to the created code: For some specific initial conditions we have obtained analytically known solution of the non-relativistic Boltzmann equation, then the evolution of interacting gas is similar to a free streaming.Suggested in (Yu. M. Sinyukov, S.V. Akkelin and Y. Hama,Phys. Rev. Lett. 89 (2002) 052301).

The initial and after rescattering momentumdistributions are the same:

The initial and after rescattering CFs are the same:

- UKM-R test results

Initial conditions:

Spherically symmetric fireball initially at rest with Gaussian radius 7Fm and temperature 0.13 GeV, containing 400 particles with mass m=1GeV , which elastically scatter with each other with constant cross sections 40 mb (400 mb).

“Our study shows that for nonrelativistic particles with proton masses,which are initially thermal and distributed symmetrically inconfiguration space, their momentum distributions at the initial timeand after the last rescattering are almost the same for the values ofthe cross sections till 1000~ mb; The initial and final CFs coincide rather well at anychoice of the cross sections: the difference between the interferometryradii of the initial and final CFs is about 2 %.

These results are in correspondence with the results of Yu.M. Sinyukov, S.V. Akkelin and Y. Hama,Phys. Rev. Lett. 89 (2002) 052301.and demonstrate the positive testing of the UniversalKinetic Model.

Summarizing our studies we conclude that numerical analysis ofsolutions of the Boltzmann equation demonstrate that the approximateconservation of the momentum spectra and the interferometryvolume could take place also in more general and realistic caseswhen the initial hadronic system is asymmetric and a relativistic one.”

It should be noted that the above algorithm violatesthe Lorenz covariance (non-causality) introduces action at distance and breaks local character ofthe BE. One can diminish nonlocality of the algorithmsimultaneously increasing the number of particles and decreasing interaction crosssection by the same factor k. (Invariance of BE with respect of this transformation).To test covariance of the algorithm we have repeated simulations withpion mass particles for 40000 particles in one event with thescattering cross section 4~mb and obtained the same results.

From: “EVOLUTION OF OBSERVABLES IN A NUMERICAL KINETIC MODEL” in preparationN.S. Amelin, R. Lednicky, L. I. Malinina, T. A. Pocheptsov and Yu.M. Sinyukov

Study of resonance decays effects on (π+π+) CF with UKM-R

Initial conditions:a. Spherically symmetric fireball initially at rest with Gaussian radius 7 Fm and temperature 0.13 GeV, containing

400 particles , which decay. Thermal-like energy distribution.The decay products scatter elastically. The realistic parametrised cross sections were taken.

Resonance source (π+π+) Qinv CF :

b. Short lived resonances. For example ρ Mean lifetime 1.3 fm/c. The shape of CF ~ initial one

c. Moderately lived resonanceFor example ω. Mean lifetime 23.4 fm/c. Modifies CF essentially the shape of CF.

d. Long-lived resonances For example η’Mean lifetime 1000 fm/c.

The «ideal» correlation measurements were considered. If the real experimental resolution is introduced the long-lived resonances can’t be resolved. They only decrease the intersept of the correlation function withouit changing theCF shape.

The realistic resonances contributions in pion spectrasimulated with HYDRO+UKM-R

Let’s try to construct more realistic initial state model for the UKM-R: Expanding fireball.

We used the hydrodynamic model to describe the expanding fireballat the chemical freeze-out. This model produces the momentum and space-time distributions of hadrons at the post hadronization stage. This is the initial state of the evolution, described by the UKM-R cascade algorithm. System is considered at the moment of hadronization then τ = τh =const. Decays of resonances and elastic rescatteringsoccur till kinetic freeze-out.

Justification of this approach:From: S.V.Akkelin, P.Braun-Muzinger, and Yu.M.Sinyukov, nucl-th/0111050

Chemical freeze-out:

1. Initial hadronic state is in chemical equilibriumChemical concentrations of hadronic gase do not change during evolution

2. It gives the possibility to use the approach of local thermal equilibrium and hydrodynamic expansion at approximately frozen chemical composition.

3. Uniform temperature, baryon and strangeness chemical potentials, unique hadronization hypersurface for all particles are assumed.

4. In boost-invariant scenario as observed at RHIC, chemical freeze-out conditions are the same for any individual rapidity slices (enough large)

5. It was demonstratedThat if chemical freeze-out is incorporated into the hydrodynamics then the final spectra and fireball lifetimes are insensitive to the temperature at which the switch from hydrodynamics to cascade is made.

In the statistical model of an ideal particle gas we can determine all macroscopic characteristics of the considered particle system at given Tchem, µi

ParticleNumberDensities,

ParticleEnergyDensity,(ParticleBaryonNumberDensity,ParticleStrangenessDensity ets)

Spin degeneracy factor

Set of the distributions functions

Chemical potential

Series expansions

K2 modified Hankel function of 2-nd order

)(2)exp()1

(2

)exp()1(

)1/)(exp(

1

)2(

1)(

12

),(2

),(2

1

12

2

1

1

3

22

22

T

mnK

Tn

n

Tmgn

T

wnf

ISB

Twpf

Jg

mpfwdppg

w

mpfdppg

n

ii

n

niii

ii

n

ni

iI

iS

iB

i

iii

ii

iiii

i

iii

i

Grand canonical class, here the mean primordialdensities are calculated

N.S.Amelin, Proceedings of the Forth International Workshop “Very High Multiplicity physics (2004), p.83In UKM-R we rewrite these classes with ROOT-PDG table and created additional Particle Properties Table needed for GrandCanonical functions calculations. Now one can calculate these thermo dynamical characteristics using 85 stable particles and resonances

The fit of the experimental particle ratios allowsto determine and fix: Tchem, µi.

All particles species made up of the light quarks (u, d, s) from PDG table are includedin the calculations.

Parameters of the model: for this test the parameters were not optimized but were taken from W.Florkowski and W.Broniowski nucl-th/0212052 which successfully describes RHIC particle ratios in the statistical model approachT, MeV 168μb, MeV 41μs, MeV 10μI, MeV -1 UKM-R decays experiment nucl-th/0212052 π- / π+ 1.00 1.00 ± 0.02 1.02

ˉp / π- 0.06 0.08 ± 0.02 0.09

K- / K+ 0.89 0.92 ± 0.03 0.92

K- / π- 0.19 0.15 ± 0.02 0.16

ˉp / p 0.61 0.61 ± 0.07 0.65

ˉ Λ / Λ 0.65 0.71 ± 0.04 0.69

ˉ Ξ / Ξ 0.83 0.83 ± 0.06 0.77

φ / K- 0.16 0.13 ± 0.03 0.15

Λ / p 0.47 0.49 ± 0.03 0.47

Ξ- / π- 0.010 0.0088 ± 0.0020 0.0072 Ω+ / Ω- 0.91 0.95 ± 0.16 0.86

For the space-momentum description of the initial state HYDRO was used.

Hydrodynamic model of I.P. Lokhtin and A.M. Snegirev (hydro.f generator for fast simulation of flow effects (elliptic and transverse flow) in central and semi-central heavy ion collisions at LHC http://cern.ch/lokhtin/hydro, Phys.Lett. B 378 (1996)

In this model the final hadron spectrum is given by the superposition of thermal initialdistribution and collective flow. The formation of the cylindrically symmetric hot matter expanding preferable along the cylinder axis is expected. Variables:

iI

iS

iB

i

TEi

ISB

emEEEf i

1/)0(22 ]1[00~)0(TEemEEEf /02200~)0(

HYDRO UKM-R

1. Thermal distribution of the produced hadrons in the rest frame of the fluid element.

22

22

zt

yxr

0. Mean charge particle multiplicity per unit rapidity is a parameter in HYDRO ; In UKM-R multiplicities are calculated:

Mean primordial densities <ni> of stable hadrons and resonances at chemical freeze-outare calculated in Grand Canonical class:

Poisson multiplicity distribution.

Τch-hadronization proper timeSystem is considered at the moment of hadronization then τ = τch =constRf-radius of the system at this moment

xytgzt

zt

/)(

ln2

1

),(2

2

22

max2

iii

i

Lfeff

mpfdppg

n

YRniniVNi

4. Particle space-time information

The longitudinal and transverse particle hadronisation positions and time are:

)(

)(

)(

)(

SinRy

CosRx

Cosht

Sinhz

f

f

From I.P. Lokhtin and A.M. Snegirev hep-ph/0312204:

of liquid element

maximal longitudinal

transverse collective rapidities max

maxmax

T

L

Y

Y

5. Anisotropic flow spatial ellipticity of freeze-out region ellipticity of system formed in the region of the initial overlap of nuclei

2222 / xyxy

)1,1min()0()(

2/0

022

0022

0

CosSinCosSinbRbR

Rb A

Parameters of UKM-R with HYDRO initial state:

Rf (b=0) radius of the system at hadronization

b impact parameter (fix and distributed)

τ, Fm/c hadronisation time maximal transverse YTmax,

and longitudinal rapidity of collective motion

YLmax = η,

Tchem, μS, μB, μI – chemical potentials

The first tests of the simulations with UKM-R have been done.The parameters are not yet optimized, just the “reasonable” descriptionof the yields, pt-spectra, mt-spectra and v2 were required.

Charged hadron rapidity density dN/dy~740±50;

at different centralities: pt – spectra, mt-spectra, v2

from particle ratios and mt spectraNow are fixed as:

Tchem, MeV 168μb, MeV 41μs, MeV 10 μI, MeV -1

In fitting procedure we useRICH data:

2

2

)2( R

bc

:

Comparison of the simulated with HYDRO+UKM-R pt-spectra with Phenix data (central collisions) 0-5%

2

0

0

2

)exp()(2

1

2

Tth

T

TT

vmTT

T

mmA

mTTdydmm

Nd

Mt-Spectra simulated with Hydro+UKM-Rare fitted with

0-5 % centralityInverse slope parameters PHENIX:Tth = 177 MeV ; < ut >=0.48

Π+ 210.2±0.MeV

K+ 290.2±2.2

P 414.8±7.5

Simulated central collisions with UKM-R Π+ 243±5 MeV

K+ 296±10 MeV

P 432±26 MeV

Comparison of the simulated Mt-spectra with Phenix data at 200 AGeV (central collisions)

<Vt> average radial transverse flow

Tth –temperature at kinetic freeze-out

Simulated with HYDRO+UKM-R elliptic flow :

Thermal pressure causes a rapid collective expansion –flow of the reaction zone.In non-central collisions the initial overlap region is elliptically deformed in the transverse planeanisotropic pressure gradients more rapid expansion into the reaction planethan in perpendicular to it anisotropy of the final transverse momentum distributionElliptic flow, which is expressed via v2 second harmonic coefficient of a Fourier expansion of:

0

22

]221[2

02

react

Cosv

CosvN

d

dN

V2 at centrality 15-30 % all charged

The comparison with RICHdata at different centralitiesis underway.

Conclusions

1. The work is underway on creation of the Monte Carlo generator of heavy ion collisions performed at SPS, RHIC, or LHCmean for simulation of the flow effects and HBT radii.

UKM-R allows: 1) to determine all macroscopic characteristics of the considered particle system at

given Tchem, µi; 2) to generate all stable particles and resonances consisting of u, d, s quarks from PDG table at the chosen by user hypersurface; 3) to perform the subsequent space-time evolution and resonances decay 4) to calculate transverse momentum and transverse mass spectra, elliptic flow, CFs. The static or expanding fireball in the initial state can be chosen. Now the hypersurface is determined in HYDRO model.The code is written in O.O. C++ language and complies to the ROOT environment.

2. The kinetic part of the generator can be used separately with different initial states chosen by user.

3. The cascade algorithm was successfully tested: - Lorenz covariance - exact solutions of Boltzmann equations