Continious emission vs sharp freeze-out:results of hydro-kinetic approach for A+A

collisions

Yu. Sinyukov, BITP, Kiev

Krakow 11-14 June WPCF-2008

Based on: Akkelin, Hama, Karpenko, and Yu.S - PRC 78

(2008) 034906

“Soft Physics” measurements

2

xt

A

A

ΔωK

p=(p1+ p2)/2

q= p1- p2

(QS) Correlation function

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

Cooper-Frye prescription (1974)

Landau, 1953

Continuous Emissiont

x

outt

F. Grassi, Y. Hama, T. Kodama (1995)

“The back reaction of the emission on the fluid dynamics is not reduced just to energy-momen-tum recoiling of emitted particles on the expan-ding thermal medium, but also leads to a re-arrangement of the medium, producing a devia-tion of its state from the local equilibrium, ac-companied by changing of the local temperature, densities, and collective velocity field. This complex effect is mainly a consequence of the fact that the evolution of the single finite system of hadrons cannot be split into the two compo-nents: expansion of the interacting locally equi-librated medium and a free stream of emitted particles, which the system consists of. Such a splitting, accounting only for the momentum-energy conservation law, contradicts the unde-rlying dynamical equations such as a Boltzmann one.”

Akkelin, Hama, Karpenko, Yu.SPRC 78 034906 (2008)

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

t

z

t

r

constrconstzt

at : 22

hadr 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

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

Hydro-kinetic approach

MODEL• is based on relaxation time approximation for relativistic finite expanding system;

• provides evaluation of escape probabilities and deviations (even strong) of distribution functions [DF] from local equilibrium;

3. accounts for conservation laws at the particle emission;

Complete algorithm includes: • solution of equations of ideal hydro;• calculation of non-equilibrium DF and emission function in first approximation;• solution of equations for ideal hydro with non-zero left-hand-side that accounts for conservation laws for non-equilibrium 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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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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Boltzmann eqs (integral form)

Spectra and Emission function

Index is omittedeverywhere

Spectrum

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

Relaxation time approximation ( increases with time!):

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is related to local rest system where collective veloc.

Representations of non-loc.eq. distribution function

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If at the initial (thermalization) time

Energy-momentum conservation:

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Iteration procedure:

I. Solution of perfect hydro equations with given initial conditions

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II. Decomposition of energy-momentum tensor

Kiev, June 18-22 WRNP-2007

12

13

where

III. Ideal hydro with “source” instead of non-ideal hydro

(known function)

IV Final distribution function:

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This approach accounts for conservation laws deviations from loc. eq. viscosity effects in hadron gas:

Saddle point approximation

Emission density

Spectrum

where

Normalization condition

Eqs for saddle point :

Physical conditions at

Cooper-Frye prescription

Spectrum in new variables

Emission density in saddle point representationTemporal width of emission

Generalized Cooper-Frye f-la

Generalized Cooper-Frye prescription:

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r

t

0

Escape probability

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

OPACITY

Toy model: one-component pion gas with initial T= 320 MeV at and different cross-sections.

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

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 homogeneity length of the distribution function: 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 . Such a generalized Cooper-Frye representation is related to freeze-out hypersurface of maximal emission 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)

Pt dependence of emission density

Transverse Spectra

Max at Pt = 0.3 GeV/c

Max at Pt = 1.2 GeV/c

Initial Conditions and Emission Function

Initial profile: Gaussian

Initial profile: Woods-Saxon

1st order ph. tr.

CrossoverCrossover

W/0 initial flow With initial flow

Conclusions

• The following factors reduces space-time scales of the emission

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)

• 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.

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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.