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Nantes June 15, 2009 GDRE - 2009 Spectra and space-time scales in relativistic heavy ion collisions: the results based on HydroKinetic Model (HKM) Based on: Yu.S., I. Karpenko, A. Nazarenko J. Phys. G: Nucl. Part. Phys. 35 104071 (2008); S.V. Akkelin, Y. Hama, Iu. Karpenko, Yu.S., PR C 78, 034906 (2008 ) Yu. Sinyukov, BITP, Kiev Yu.S. Two lectures in Acta Phys Polon. 40 (2009)

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Nantes June 15, 2009 2 Heavy Ion Experiments E_lab/A (GeV)

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3 Thermodynamic QCD diagram of the matter states The thermodynamic arias occupied by different forms of the matter Theoretical expectations vs the experimental estimates

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4 UrQMD Simulation of a U+U collision at 23 AGeV

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Dnepropetrovsk, 3 May 2009 5 Soft Physics measurements x t A A KK p=(p 1 + p 2 )/2 q= p 1 - p 2 (QS) Correlation function Space-time structure of the matter evolution, e.g., T ch and ch soon after hadronization (chemical f.o.) Radial flow Landau/Cooper- Frye prescription

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NPQCD-2007 6 Collective transverse flows P T Initial spatial anisotropy different pressure gradients momentum anisotropy v 2

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7 Energy dependence of the interferometry radii Energy- and kt-dependence of the radii Rlong, Rside, and Rout for central Pb+Pb (Au+Au) collisions from AGS to RHIC experiments measured near midrapidity. S. Kniege et al. (The NA49 Collaboration), J. Phys. G30, S1073 (2004).

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8 Expecting Stages of Evolution in Ultrarelativistic A+A collisions Early thermalization at 0.5 fm/c 0.2?(LHC) Elliptic flows t Relatively small space-time scales (HBT puzzle) Early thermal freeze-out: T_th Tch 150 MeV 10-15 fm/c 7-8 fm/c 1-3 fm/c or strings

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9 HBT PUZZLE & FLOWS Possible increase of the interferometry volume with due to geometrical volume grows is mitigated by more intensive transverse flows at higher energies:, is inverse of temperature Why does the intensity of flow grow? More more initial energy density more (max) pressure p max BUT the initial acceleration is the same HBT puzzle Intensity of collective flows grow Time of system expansion grows Initial flows ( < 1-2 fm/c) develop

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10 Interferometry radii Borysova, Yu.S., Akkelin,Erazmus, Karpenko: PRC 73, 024903 (2006)

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Dnepropetrvsk May 2009 NPQCD-2009 11 Duration of particle emission is taken into account by means of enclosed freeze-out hypersurface: v i =0.35 fm/c volume emission surface emission

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Dnepropetrovsk May 3 2009 NPQCD-2009 12 Ro/Rs ratio and initial flows

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Dnepropetrovsk May 3 2009 NPQCD -2009 13 Hydrodynamic expansion: gradient pressure acts Free streaming: Gradient of density leads to non-zero collective velocities For nonrelativistic gas So, even if and : Yu.S. Acta Phys.Polon. B37 (2006) 3343; Gyulassy, Yu.S., Karpenko, Nazarenko Braz.J.Phys. 37 (2007) 1031. :at For thermal and non-thermal expansion In the case of thermalization at later stage it leads to spectra anisotropy Basic ideas for the early stage: developing of pre-thermal flows

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Dnepropetrovsk May 2009 NPQCD-2009 14 Comparision of flows at free streaming and hydro evolution

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15 Boost-invariant distribution function at initial hypersurface McLerran-Venugopalan CGC effective FT (for transversally homogeneous system) Transversally inhomogeneous system: of the gluon distribution proportional to the ellipsoidal Gaussian defined from the best fit to the density of number of participants in the collisions with the impact parameter b. A.Krasnitz, R.Venugopalan PRL 84 (2000) 4309; A. Krasnitz, Y. Nara, R. Venugopalan: Nucl. Phys. A 717 (2003) 268, A727 (2003) 427;T. Lappi: PRC 67 (2003) 054903, QM 2008 (J.Phys. G, 2008) is the variance of a Gaussian weight over the color charges of partons If one uses the prescription of smearing of the -function as, then. As the result the initial local boost-invariant phase-space density takes the form

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16 Developing of collective velocities in partonic matter at pre-thermal stage ( Yu.S. Acta Phys. Polon. B37, 2006 ) Equation for partonic free streaming in hyperbolic coordinates between Solution where

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Collective velocity developed at pre-thermal stage from proper time tau_0 =0.3 fm/c by supposed thermalization time tau_i = 1 fm/c for scenarios of partonic free streaming and free expansion of classical field. The results are compared with the hydrodynamic evolution of perfect fluid with hard equation of state p = 1/3 epsilon started at tau_0. Impact parameter b=0. Yu.S., Nazarenko, Karpenko: Acta Phys.Polon. B40 1109 (2009)

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Collective velocity developed at pre-thermal stage from proper time tau_0 =0.3 fm/c by supposed thermalization time tau_i = 1 fm/c for scenarios of partonic free streaming. The results are compared with the hydrodynamic evolution of perfect fluid with hard equation of state p = 1/3 epsilon started at tau_0. Impact parameter b=6.3 fm. Yu.S., Nazarenko, Karpenko: Acta Phys.Polon. B40 1109 (2009)

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is a fitting parameter Initial parameters becomes to be isotropic at =1 fm/c: The "effective" initial distribution - the one which being used in the capacity of initial condition bring the average hydrodynamical results for fluctuating initial conditions: where is a fitting parameter For central Au+Au (Pb+Pb) collisions

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20 Hydro-Evolution: Equation of State EoS from LattQCD (in form proposed by Laine & Schroder, Phys. Rev. D73, 2006) MeV The EoS accounts for gradual decays of the resonances into expanding hadronic gas consistiong of 359 particle species with masses below 2.6 GeV. The EoS in this non chemically equilibrated system depends now on particle number densities of all the 359 particle species. Since the energy densities in expanding system do not directly correlate with resonance decays, all the variables in the EoS depends on space-time points and so an evaluation of the EoS is incorporated in the hydrody- namic code. We calculate the EoS below in the approximation of ideal multi- component hadronic gas. MeV

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Hydro-evolution MeV Chemically equilibrated evolution of the quark-gluon and hadron phases. For : MeV The system evolves as non chemically equilibra- ted hadronic gas. The conception of the chemical freeze-out imply that mainly only elastic collisi- ons and the resonance decays among non-elastic reactions takes place because of relatively small densities allied with a fast rate of expansion at the last stage. Thus, in addition to (1) the equa- tions accounting for the particle number conser- vation and resonance decays are added: where denote the average number of i-th particles coming from arbitrary decay of j-th resonance, is branching ratio, is a number of i-th particles produced in decay channel.

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Cooper-Frye prescription (CFp) t z t r CFp gets serious problems: Freeze-out hypersurface contains non-space-like sectors artificial discontinuities appears across Sinyukov (1989), Bugaev (1996), Andrelik et al (1999); cascade models show that particles escape from the system about whole time of its evolution. Hybrid models (hydro+cascade) and the hydro method of continuous emission starts to develop.

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Hybrid models: HYDRO + UrQMD ( Bass, Dumitru (2000)) t z t r The problems: the system just after hadronization is not so dilute to apply hadronic cascade models; hadronization hypersurface contains non-space-like sectors (causality problem: Bugaev, PRL 90, 252301, 2003); hadronization happens in fairly wide 4D-region, not just at hypersurface, especially in crossover scenario. t HYDRO UrQMD The initial conditions for hadronic cascade models should be based on non-local equilibrium distributions 20 May 2009 23

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24 t x F. Grassi,Y. Hama, T. Kodama Continuous emission Hydro-kinetic approach is based on combination of Boltsmann equation and hydro for finite expanding system; provides evaluation of escape probabili- ties and deviations (even strong) of distri- bution functions from local equilibrium; accounts for conservation laws at the particle emission; PROVIDE earlier (as compare to CF-prescription) emission of hadrons, because escape probability accounts for whole particle trajectory in rapidly expanding surrounding (no mean-free pass criterion for freeze-out) Yu.S., Akkelin, Hama: PRL. 89, 052301 (2002); + Karpenko: PRC 78 034906 (2008). Basic ideas for the late stage

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25 * Is related to local Hydro-kinetic approach MODEL provides evaluation of escape probabilities and deviations (even strong) of distribution functions [DF] from local equilibrium; is based on relaxation time approximation for relativistic finite expanding system; (Shear viscosity ). accounts for conservation laws at the particle emission; Complete algorithm includes: solution of equations of ideal hydro [THANKS to T. Hirano for possibility to use code in 2006] ; calculation of non-equilibrium DF and emission function in first approximation; [Corresponding hydro-kinetic code: Tytarenko, Karpenko,Yu.S.(to be publ.)] Solution of equations for ideal hydro with non-zero left-hand-side that accounts for conservation laws for non-equlibrated process of the system which radiated free particles during expansion; Calculation of exact DF and emission function; Evaluation of spectra and correlations.

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26 and are G(ain), L(oss) terms for p. species Boltzmann eqs (differential form) Escape probability (for each component ) Boltzmann equations and Escape probabilities

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27 Boltzmann eqs (integral form) Spectra and Emission function Index is omitted everywhere Spectrum Method of solution (Yu.S. et al, PRL, 2002)

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28 Saddle point approximation Emission density Spectrum where Normalization condition Eqs for saddle point : Physical conditions at

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29 Cooper-Frye prescription Spectrum in new variables Emission density in saddle point representation Temporal width of emission Generalized Cooper-Frye f-la

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30 Generalized Cooper-Frye prescription: 30 r t 0 Escape probability Yu.S. (1987)-particle flow conservation; K.A. Bugaev (1996) (current form)

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Nov 3-6 RANP08 31 Momentum dependence of freeze-out Here and further for Pb+Pb collisions we use: initial energy density EoS from Lattice QCD when T< 160 MeV, and EoS of chemically frozen hadron gas with 359 particle species at T< 160 MeV. Pt-integrated

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The pion emission function for different pT in hydro-kinetic model (HKM) The isotherms of 80 MeV is superimposed.

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The pion emission function for different pT in hydro-kinetic model (HKM). The isotherms of 135 MeV (bottom) is superimposed.

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Transverse momentum spectrum of pi in HKM, compared with the sudden freeze-out ones at temperatures of 80 and 160 MeV with arbitrary normalizations.

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35 Conditions for the utilization of the generalized Cooper-Frye prescription i)For each momentum p, there is a region of r where the emission function has a sharp maximum with temporal width. ii) The width of the maximum, which is just the relaxation time ( inverse of collision rate), should be smaller than the corresponding temporal homogeneitylength of the distribution function: 1% accuracy!!! iii) The contribution to the spectra from the residual region of r where the saddle point method is violated does not affect essentially the particle momentum spectrum. Then the momentum spectra can be presented in Cooper-Frye form despite it is, in fact, not sadden freeze-out and the decaying region has a finite temporal width. Also, what is very important, such a generalized Cooper-Frye representation is related to freeze-out hypersurface that depends on momentum p and does not necessarily encloses the initially dense matter. iiii) The escape probabilities for particles to be liberated just from the initial hyper-surface t0 are small almost in the whole spacial region (except peripheral points)

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36 Inititial conditions in Pb+Pb and Au+Au colliisions for SPS, RHIC and LHC energies We use 0.2 (the average transverse flow = 0.17) for SPS energies and 0.35 ( = 0.26) for all the RHIC and LHC energies. Note that this parameter include also a correction of underestimated transverse flow since we did not account for the viscosity effects. For the top SPS energy GeV/fm 3 = 4.7 GeV/fm 3 ), for the top RHIC energy GeV/fm 3 = 8.0 GeV/fm 3 ) We also demonstrate results at = 40 GeV/fm 3 and = 50 GeV/fm 3 that probably can correspond to the LHC energies 4 and 5.6 A TeV. As for latter, according to Lappy, at the top LHC energy recalculated to the time 1 fm/c is 0.07*700=49 GeV/fm 3. The chosen values of approximately correspond to the logarithmic dependence of the energy density on the collision energy.

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Energy dependence of pion spectra

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Energy dependence of the interferometry scales, R_long

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Energy dependence of the interferometry scales, R_side

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Energy dependence of the interferometry scales, R_out

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Energy dependence of ratio

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42 Pion emission density at different energies in HKM

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The ratio as function on in-flow and energy If the extreme internal fluid element has initial transverse velocity and undergo mean acceleration, then. Suppose, that the fluid elements, that finally form almost homogeneous in time emission with duration time are initially situated within radial interval shifted to the periphery of the system.

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Pion and kaon interferometry Long- radii (preliminary) Dnepropetrovsk May 2009 NPQCD-2009 44

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Pion and kaon interferometry Side- radii (preliminary) Dnepropetrovsk May 3 2009 NPQCD-2009 45

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Pion and kaon interferometry Out- radii (preliminary) Dnepropetrovsk May 3 2009 NPQCD-2009 46

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Hydrodynamics with initial tube-like fluctuations (formation of ridges, initially 10 small tubes)

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Hydrodynamics with initial tube-like fluctuations (formation of ridges, initially 1 small tube)

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49 Conclusions-1 The following factors reduces space-time scales of the emission and Rout/Rside ratio. developing of initial flows at early pre-thermal stage; more hard transition EoS, corresponding to cross-over; non-flat initial (energy) density distributions, similar to Gaussian; early (as compare to standard CF-prescription) emission of hadrons, because escape probability account for whole particle trajectory in rapidly expanding surrounding (no mean-free pass criterion for freeze-out) Viscosity [Heinz, Pratt] The hydrokinetic approach to A+A collisions is proposed. It allows one to describe the continuous particle emission from a hot and dense finite system, expanding hydrodynamically into vacuum, in the way which is consistent with Boltzmann equations and conservation laws, and accounts also for the opacity effects. 49

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50 Conclusions-2 The CFp might be applied only in a generalized form, accounting for the direct momentum dependence of the freeze-out hypersurface corresponding to the maximum of the emission function at fixed momentum p in an appropriate region of r.

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51 Conclusions-3 A reasonable description of the pion spectra and HBT (except some an overestimate for ) in cental Au+Au collisions at the RHIC energies is reached with the value of the fitting parameter or the average energy density at the initial time The initial time fm/c and transverse width 5.3 fm (in the Gaussian approximation) of the energy density distribution are obtained from the CGC estimates. The EoS at the temperatures corresponds to the lattice QCD calculations at The used temperature of the chemical freeze-out MeV is taken from the latest results of particle number ratios analysis (F. Becattini, J.Phys. G, 2008). The anisotropy of pre-thermal transverse flows in non-central collisions, bring us a hope for a successful description of the elliptic flows with thermalization reached at a relatively late time:1-2 fm/c.

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CONCLUSION FINAL

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Nov Dnepropetrovsk May 3 2009 NPQCD-2009 53 Hybrid models: HYDRO + UrQMD ( Bass, Dumitru (2000) ) t z t r The problems: the system just after hadronization is not so dilute to apply hadronic cascade models; hadronization hypersurface contains non-space-like sectors (causality problem: Bugaev, PRL 90, 252301, 2003); hadronization happens in fairly wide 4D-region, not just at hypersurface, especially in crossover scenario. t HYDRO UrQMD The initial conditions for hadronic cascade models should be based on non-local equilibrium distributions