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Yu. Sinyukov, BITP, Kiev

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Spectra and space-time scales in relativistic heavy ion collisions: the results based on HydroKinetic Model (HKM)

Yu. Sinyukov, BITP, Kiev

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.S. Two lectures in Acta Phys Polon. 40 (2009)

GDRE - 2009

LHC

FAIR

E_lab/A

(GeV)

- The thermodynamic arias
occupied by different forms of

the matter

Theoretical expectations vs the experimental estimates

A

x

t

ΔωK

Landau/Cooper-Frye prescription

A

p=(p1+ p2)/2

q= p1- p2

Tch and μchsoon after hadronization (chemical f.o.)

(QS) Correlation function

Space-time structure of the

matter evolution, e.g.,

Radial flow

P

T

Initial spatial anisotropydifferent pressure gradients

momentum anisotropy v2

NPQCD-2007

Energy- and kt-dependence of the radii Rlong, Rside, and Rout forcentral Pb+Pb(Au+Au) collisions from AGS to RHICexperiments measurednear midrapidity. S. Kniege et al. (The NA49 Collaboration), J. Phys. G30, S1073 (2004).

t

Relatively small space-time scales

(HBT puzzle)

10-15 fm/c

Early thermal freeze-out: T_th Tch 150 MeV

7-8 fm/c

Elliptic flows

1-3 fm/c

Early thermalization at 0.5 fm/c

0.2?(LHC)

or strings

8

- 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

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

volume

emission

surface

emission

vi=0.35 fm/c

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Basic ideas for the early stage: developing of pre-thermal flows

Yu.S. Acta Phys.Polon. B37 (2006) 3343; Gyulassy, Yu.S., Karpenko, Nazarenko Braz.J.Phys. 37 (2007) 1031.

For nonrelativistic gas

For thermal and non-thermal expansion

at

:

Hydrodynamic expansion: gradient pressure acts

So, even if

:

and

Free streaming:

Gradient of density leads to non-zero collective velocities

In the case of thermalization at later stage it leads to spectra anisotropy

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McLerran-Venugopalan

CGC effective FT

(for transversally homogeneous system)

is the variance of a Gaussian weight over the color charges of partons

A.Krasnitz, R.Venugopalan PRL84 (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)

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.

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

15

- Equation for partonic free streaming in hyperbolic coordinates between

- Solution

where

16

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

freestreaming and free expansion of classical field. The results are compared

with thehydrodynamic evolution of perfect fluid with hard equation of state

p = 1/3 epsilon startedat 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 freestreaming. The results are compared with thehydrodynamic evolution of perfect fluid with hard equation of state p = 1/3 epsilon startedat tau_0 .Impact parameter b=6.3 fm.

Yu.S., Nazarenko, Karpenko: Acta Phys.Polon. B40 1109 (2009)

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:

is a fitting parameter

where

For central Au+Au (Pb+Pb) collisions

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 gasconsistiong 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

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:

MeV

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 decaychannel.

Cooper-Frye prescription (CFp)

t

t

z

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.

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

t

t

t

UrQMD

HYDRO

z

r

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

- 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.

20 May 2009

23

Yu.S., Akkelin, Hama: PRL. 89, 052301 (2002); + Karpenko: PRC 78 034906 (2008).

Hydro-kinetic approach

Continuous emission

t

- 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)

x

F. Grassi,Y. Hama, T. Kodama

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.

Is related to local

*

Boltzmann equations and Escape probabilities

Boltzmann eqs (differential form)

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

Escape probability

(for each component )

26

Spectra and Emission function

Boltzmann eqs

(integral form)

Index

is omitted

everywhere

Spectrum

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

27

Saddle point approximation

Spectrum

Emission density

where

Normalization condition

Eqs for saddle point :

Physical conditions at

Cooper-Frye prescription

Spectrum in new variables

Emission density in saddle point representation

Temporal width of emission

Generalized Cooper-Frye f-la

t

0

Escape probability

r

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

30

Momentum dependence of freeze-out

Here and further for Pb+Pb collisions we use:

initial energy density

Pt-integrated

EoS from Lattice QCD when T< 160 MeV, and EoS of chemically frozen hadron gas with 359 particle species at T< 160 MeV.

RANP08

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 of135 MeV (bottom) is superimposed.

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

Conditions for the utilization of the generalized Cooper-Frye prescription

- 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 relaxationtime ( 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 saddlepoint methodis violated does not affect essentially theparticle momentum spectrum.

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)

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 necessarilyencloses the initially

dense matter.

- 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.

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.

If the extreme internal fluid element has initialtransverse velocity and undergo mean acceleration, then

.

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

49

- 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.

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 functionat fixed momentum p in an appropriate regionof r.

- 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

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

t

t

t

UrQMD

HYDRO

z

r

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

- 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.

NPQCD-2009

53

Nov Dnepropetrovsk May 3 2009