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

1. Continious emission vs sharp freeze-out:results of hydro-kinetic approach for A+A collisions Yu. Sinyukov, BITP, Kiev Based on: Akkelin, Hama, Karpenko, and Yu.S - PRC 78 (2008) 034906 WPCF-2008

2. “Soft Physics” measurements A x Landau, 1953 t ΔωK Cooper-Frye prescription (1974) A p=(p1+ p2)/2 q= p1- p2 (QS) Correlation function Space-time structure of the matter evolution, e.g.,

3. Continuous Emission “The back reactionof the emission on the fluid dynamics is not reduced justto energy-momen-tum recoiling of emitted particles on theexpan-ding thermal medium, but also leads to a re-arrangementof the medium, producing a devia-tion of its state from thelocal equilibrium, ac-companied by changing of the localtemperature, densities, and collective velocity field. Thiscomplex effect is mainly a consequence of the fact that theevolution of the single finite system of hadrons cannot be splitinto the two compo-nents: expansion of the interacting locallyequi-librated medium and a free stream of emitted particles,which the system consists of. Such a splitting, accountingonly for the momentum-energy conservation law, contradictsthe unde-rlying dynamical equations such as a Boltzmannone.” t x F. Grassi, Y. Hama, T. Kodama (1995) Akkelin, Hama, Karpenko, Yu.S PRC 78 034906 (2008)

4. 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.

5. Yu.S. , Akkelin, Hama: PRL89 , 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.

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

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

8. Relaxation time approximation ( increases with time!): is related to local rest system where collective veloc.

9. Representations of non-loc.eq. distribution function If at the initial (thermalization) time

10. Energy-momentum conservation:

11. Iteration procedure: I. Solution of perfect hydro equations with given initial conditions

12. II. Decomposition of energy-momentum tensor WRNP-2007

13. III. Ideal hydro with “source” instead of non-ideal hydro where (known function)

14. IV Final distribution function: • This approach accounts for • conservation laws • deviations from loc. eq. • viscosity effects in hadron gas:

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

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

17. Generalized Cooper-Frye prescription: t 0 Escape probability r Yu.S. (1987)-particle flow conservation; K.A. Bugaev (1996) (current form) 17

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

19. 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.

20. 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: 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 . Such a generalized Cooper-Frye representation is related to freeze-out hypersurface of maximal emission that depends on momentum p and does not necessarilyencloses the initially dense matter.

21. Pt dependence of emission density

22. Transverse Spectra Max at Pt = 0.3 GeV/c Max at Pt = 1.2 GeV/c

23. Initial Conditions and Emission Function Crossover Crossover 1st order ph. tr. Initial profile: Woods-Saxon W/0 initial flow With initial flow Initial profile: Gaussian

24. Conclusions 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. • 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.