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)

Nantes June 15, 2009

2

Heavy Ion Experiments

E_lab/A (GeV)

3

Thermodynamic QCD diagram of the matter states

The thermodynamic arias

occupied by different forms of

the matter

Theoretical expectations vs the experimental estimates

4

UrQMD Simulation of a U+U collision at 23 AGeV

Dnepropetrovsk, 3 May 2009

5

Soft Physics” measurements

x

t

A

A

ΔωK

p=(p1+ p2)/2

q= p1- p2

(QS) Correlation function

Space-time structure of the matter evolution, e.g.,

Tch and μch soon after hadronization (chemical f.o.)

Radial flow

Landau/Cooper-Frye prescription

NPQCD-20076

Collective transverse flows

P

T

Initial spatial anisotropy different pressure gradients momentum anisotropy v2

1 2~ 1 2 cos( ) 2 cos(2 )dN

v vd

1 2~ 1 2 cos( ) 2 cos(2 )dN

v vd

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).

8

Expecting Stages of Evolution in Ultrarelativistic A+A collisions

Early thermalization at 0.5 fm/c

0.2?(LHC)

Elliptic flows

tRelatively 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

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 pmax

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

10

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

Dnepropetrvsk May 2009

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Duration of particle emission is taken into account by means of enclosed freeze-out hypersurface:

vi =0.35 fm/c

volumeemission

surfaceemissio

n

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Ro/Rs ratio and initial flows

Dnepropetrovsk May 3 2009 NPQCD -200913

Hydrodynamic expansion: gradient pressure acts

Free streaming:

Gradient of density leads to non-zero

collective velocities

For nonrelativistic gas

So, even if an

d

:

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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Comparision of flows at free streaming and hydro evolution

15

Boost-invariant distribution function at initial hypersurface

McLerran-Venugopalan CGC effective FT (for transversally homogeneous system)

Transversally inhomogeneous system: <transverse profile> 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

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

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)

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)

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

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

Hydro-evolution

MeVChemically 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-thparticles coming from arbitrary decay of j-th resonance, is branching ratio, is a number of i-th particles produced in decay channel.

Cooper-Frye prescription (CFp)

t

z

t

rconstrconstzt at : 22

f.o. 0zat )(:f.o. 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.

)(r

f.o.

Hybrid models: HYDRO + UrQMD (Bass, Dumitru (2000))

t

z

t

r

constr

constzt

at

: 22hadr 0zat )(:hadr 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.

)(r

t

HYDRO

UrQMD

UrQMD

hadr

hadrhadr

The initial conditions for hadronic cascade models should be based on non-local equilibrium distributions

20 May 2009 23

24

t

x

outt

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

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.

2626

and are G(ain), L(oss) terms for p. species

Boltzmann eqs (differential form)

Escape probability(for each component )

Boltzmann equations and Escape probabilities

2727

Boltzmann eqs (integral form)

Spectra and Emission function

Index

is omitted

everywhere

Spectrum

Method of solution (Yu.S. et al, PRL, 2002)

28

Saddle point approximation

Emission density

Spectrum

where

Normalization condition

Eqs for saddle point :

Physical conditions at

29

Cooper-Frye prescription

Spectrum in new variables

Emission density in saddle point representation

Temporal width of emission

Generalized Cooper-Frye f-la

30

Generalized Cooper-Frye prescription:

30

r

t

0

Escape probability

Yu.S. (1987)-particle flow conservation; K.A. Bugaev (1996) (current form)

Nov 3-6

RANP0831

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

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

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

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

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)

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/fm3 = 4.7 GeV/fm3), for the top RHIC energy GeV/fm3 = 8.0 GeV/fm3 ) We also demonstrate results at = 40 GeV/fm3 and = 50

GeV/fm3

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/fm3 . The chosen values of approximately correspond to the logarithmic dependence of the energy density on the collision energy.

Energy dependence of pion spectra

Energy dependence of the interferometry scales, R_long

Energy dependence of the interferometry scales, R_side

Energy dependence of the interferometry scales, R_out

Energy dependence of ratio

42

Pion emission density at different energies in HKM

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.

Pion and kaon interferometry Long- radii (preliminary)

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Pion and kaon interferometry Side- radii (preliminary)

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Pion and kaon interferometry Out- radii (preliminary)

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

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

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

50

Conclusions-2

The CFp might be applied only in a generalized form, accounting for thedirect 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.

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.

CONCLUSION FINAL

Nov Dnepropetrovsk May 3 2009

NPQCD-200953

Hybrid models: HYDRO + UrQMD (Bass, Dumitru (2000))

t

z

t

r

constr

constzt

at

: 22hadr 0zat )(:hadr 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.

)(r

t

HYDRO

UrQMD

UrQMD

hadr

hadrhadr

The initial conditions for hadronic cascade models should be based on non-local equilibrium distributions