Development of the UKM-R: Universal Kinetic Model under the ROOT framework

1. Development of the UKM -Universal-Kinetic-Model under the ROOT framework (UKM-R). 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 1. Motivation 2. Structure 3. Kinetic part 4. Initial states: -Thermal initial state, test the UKM algorithm -Hydro initial state 5. First results: comparison with RHIC data

2. Motivation UKM-R Universal-Kinetic-Modelunder ROOT framework “Such important effects as jet quenching, HBT correlations and elliptic flow have 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 and small 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),UrQMD Some 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;

3. 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 aparticular numerical algorithm. UKM-Rwith 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 classes Initial state initialization class, here the initial space and momentum distributions of the particles are described; Particle densities class, here the mean particle densities are calculated Particle interaction classto specify particle interactions; Particle decay classto specify particle decays; Boundary condition classto 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-Ris a version of Universal-Kinetic-Model under the ROOT framework

4. UKM-R Universal-Kinetic-Model under ROOT framework • Particle class:PDG-code, momentum components, space-time components • Particle objects are grouped into particle lists. • UKM-R types of particle lists:UKM-R types of lists: • Primaries list, Collision list • Secondary particles list • New secondaries list. • UKM-R cascade algorithm • The List of Primaries is 3.The earliest collision is simulated, • initialized and an initial value of the the two collided particles are removed from • current time is set. all lists and the created particles are put in the list ofnew • secondaries. The earliest decay occurs new secondaries. • 2. All possible pairs of colliding particles from the The particle coordinates are shifted in • Primaries list are searched for and the Collision list is filled accordance with the interaction (decay) time and particle • in according to the increasing interaction time velocity vectors. • 4. The collisions of the particles from • new secondaries list • The binary particle collision is considered as possible if with all particles from primaries and • d- the distanceof closest approach of the two collidingsecondaries lists are searched for and the • particles in their rest frame σ- total collisiontime 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 timeor • List of interactions is empty.

5. UKMstructure UniversalKineticModel ----------------------------------- #Projectiles #Targets #Primaries #Secondaries Scatterer Initializer #LastSecondaries ----------------------------------- ----------------------------------- #ListOfInteractions ----------------------------------- +ParticleEvolution() ----------------------------------- +AfterInteractionTest()=0 +Camera()= 0 +CollisionTime()=0 -+InitialState()=0---------------------- +Scatter()=0 +InitializeProjectileTarget()=0 BoundaryCondition Decayer ----------------------------------- ----------------------------------- ----------------------------------- ----------------------------------- +TimeToBoundary()=0 +DecayTime()=0 +FulfilCondition()=0 +Decay()=0

6. Initial state class It 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 distributionof a hadrons of mass misotropic Bolzmann distribution • Non-relativistic • Relativistic • Test of the correctness of any cascade algorithm was applied to UKM-R • Resonances decay

7. - UKM-R test results From: “EVOLUTION OF OBSERVABLES IN A NUMERICAL KINETIC MODEL” in preparation N.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). 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). The initial and after rescattering CFs are the same: The initial and after rescattering momentum distributions are the same:

8. From: “EVOLUTION OF OBSERVABLES IN A NUMERICAL KINETIC MODEL” in preparation N.S. Amelin, R. Lednicky, L. I. Malinina, T. A. Pocheptsov and Yu.M. Sinyukov It should be noted that the above algorithm violates the Lorenz covariance (non-causality) introduces action at distance and breaks local character of the BE. One can diminish nonlocality of the algorithm simultaneously increasing the number of particles and decreasing interaction cross section by the same factor k. (Invariance of BE with respect of this transformation). To test covariance of the algorithm we have repeated simulations with pion mass particles for 40000 particles in one event with the scattering cross section 4~mb and obtained the same results. “Our study shows that for nonrelativistic particles with proton masses, which are initially thermal and distributed symmetrically in configuration space, their momentum distributions at the initial time and after the last rescattering are almost the same for the values of the cross sections till 1000~ mb; The initial and final CFs coincide rather well at any choice of the cross sections: the difference between the interferometry radii 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 Universal Kinetic Model. Summarizing our studies we conclude that numerical analysis of solutions of the Boltzmann equation demonstrate that the approximate conservation of the momentum spectra and the interferometry volume could take place also in more general and realistic cases when the initial hadronic system is asymmetric and a relativistic one.”

9. Study of resonance decays effects on (π+π+) CF with UKM-R • Initial conditions: • 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 resonance For 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 experimentalresolution is introduced the long-lived resonances can’t be resolved. They only decrease the intersept of the correlation function withouit changing the CF shape. The realistic resonances contributions in pion spectra simulated with HYDRO+UKM-R

10. 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 fireball at 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 rescatterings occur 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 equilibrium Chemical 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 demonstrated That 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.

11. Grand canonical class, here the mean primordial densities are calculated 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 N.S.Amelin, Proceedings of the Forth International Workshop “Very High Multiplicity physics (2004), p.83 In 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

12. The fit of the experimental particle ratios allows to determine and fix: Tchem, µi. All particles species made up of the light quarks (u, d, s) from PDG table are included in 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 approach T, MeV 168 μb, MeV 41 μs, MeV 10 μI, MeV -1 UKM-R decays experiment nucl-th/0212052 π- / π+ 1.001.00 ± 0.02 1.02 ˉp / π- 0.06 0.08 ± 0.02 0.09 K- / K+ 0.890.92 ± 0.03 0.92 K- / π- 0.190.15 ± 0.02 0.16 ˉp / p 0.61 0.61 ± 0.07 0.65 ˉΛ / Λ0.650.71 ± 0.04 0.69 ˉΞ / Ξ0.83 0.83 ± 0.06 0.77 φ / K- 0.160.13 ± 0.03 0.15 Λ / p 0.47 0.49 ± 0.03 0.47 Ξ- / π- 0.0100.0088 ± 0.0020 0.0072 Ω+ / Ω- 0.91 0.95 ± 0.16 0.86

13. 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 isgiven by the superposition of thermal initial distribution and collective flow.The formation of the cylindrically symmetric hot matter expanding preferable along the cylinder axis is expected. Variables: 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-out are calculated in Grand Canonical class: Poisson multiplicity distribution. Τch-hadronization proper time System is considered at the moment of hadronization then τ = τch=const Rf-radius of the system at this moment 1. Thermal distribution of the produced hadrons in the rest frame of the fluid element. HYDRO UKM-R

14. maximal longitudinal transverse collective rapidities From I.P. Lokhtin and A.M. Snegirev hep-ph/0312204: of liquid element 4. Particlespace-time information The longitudinal and transverse particle hadronisation positions and time are: 5. Anisotropic flow  spatial ellipticity of freeze-out region  ellipticity of system formed in the region of the initial overlap of nuclei

15. In fitting procedure we use RICH data: 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 Charged hadron rapidity density dN/dy~740±50; at different centralities: pt – spectra, mt-spectra, v2 from particle ratios and mt spectra Now are fixed as: Tchem, MeV 168 μb, MeV 41 μs, MeV 10 μI, MeV -1 The first tests of the simulations with UKM-R have been done. The parameters are not yet optimized, just the “reasonable” description of the yields, pt-spectra, mt-spectra and v2 were required.

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

17. Comparison of the simulated Mt-spectra with Phenix data at 200 AGeV (central collisions) Mt-Spectra simulated with Hydro+UKM-R are fitted with <Vt> average radial transverse flow Tth –temperature at kinetic freeze-out 0-5 % centrality Inverse 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

18. 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 plane • anisotropic pressure gradients more rapid expansion into the reaction plane • than in perpendicular to it anisotropy of the final transverse momentum distribution • Elliptic flow, which is expressed via v2 second harmonic coefficient of a Fourier expansion of: V2 at centrality 15-30 % all charged The comparison with RICH data at different centralities is underway.

19. Conclusions • The work is underway on creation of the Monte Carlo • generator of heavy ion collisions performed at SPS, RHIC, or LHC • mean 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. • 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