1. Understanding the rapidity dependence of v 2 and HBT at RHIC M. Csanád (Eötvös University, Budapest) WPCF 2005 August 15-17, Kromeriz

2. M. Csanád @ WPCF’05 2/25 How analytic hydro works Scheme works also backwords * * For a certain time-interval Hydro equations + EoS Phase-space distribution Boltzmann-equation SourceS(x,p)SourceS(x,p) PRC67:034904,2003 hep-ph/0108067 Self-similar solution: PLB505:64-70,2001 hep-ph/0012127 Observables N 1 (p), C 2 (p 1,p 2 ), v 2 (p) Observables PRC54:1390-1403,1996 hep-ph/9509213

3. M. Csanád @ WPCF’05 3/25 Sensitivity to the EoS Different initial conditions, different equation of state but exactly the same hadronic final state possible. (!!) Different initial conditions, different equation of state but exactly the same hadronic final state possible. (!!) This is an exact, analytic result in hydro( !!) This is an exact, analytic result in hydro( !!) c s 2 = 2/3 c s 2 = 2/3 c s 2 = 1/3 c s 2 = 1/3

4. M. Csanád @ WPCF’05 4/25 Buda-Lund hydro 3D expansion, symmetry 3D expansion, symmetry Local thermal equilibrium Local thermal equilibrium Analytic expressions for the observables (no numerical simulations, but formulas) Analytic expressions for the observables (no numerical simulations, but formulas) Reproduces known exact hydro solutions (nonrelativistic, Hubble, Bjorken limit) Reproduces known exact hydro solutions (nonrelativistic, Hubble, Bjorken limit) Core-halo picture Core-halo picture

5. M. Csanád @ WPCF’05 5/25 Time dependence Blastwave or Cracow model type of cooling vs Buda- Lund cooling, c s 2 = 2/3, half freeze-out time Blastwave or Cracow model type of cooling vs Buda- Lund cooling, c s 2 = 2/3, half freeze-out time see: http://csanad.web.elte.hu/phys/3danim/

6. M. Csanád @ WPCF’05 6/25 A useful analogy Core  Sun Core  Sun Halo  Solar wind Halo  Solar wind T 0,RHIC  T 0,SUN  16 million K T 0,RHIC  T 0,SUN  16 million K T surface,RHIC  T surface,SUN  6000 K T surface,RHIC  T surface,SUN  6000 K R G  Geometrical size R G  Geometrical size  0  Radiation lifetime  0  Radiation lifetime  Radial flow of surface (~0)  Radial flow of surface (~0)  Longitudinal expansion (~0)  Longitudinal expansion (~0) Fireball at RHIC  Fireball Sun Fireball at RHIC  Fireball Sun

7. M. Csanád @ WPCF’05 7/25 Buda-Lund in spectra, HBT… J.Phys.G30: S1079-S1082, 2004 nucl-th/0403074

8. M. Csanád @ WPCF’05 8/25 Ellipsoidal generalization Axially symmetric case: R G, u t Axially symmetric case: R G, u t Main axes of expanding ellipsoid: Main axes of expanding ellipsoid: 3D expansion, 3 expansion rates: 3D expansion, 3 expansion rates: Introducing space-time eccentricity: Introducing space-time eccentricity: Hubble type of expansion: Hubble type of expansion: Additionally: Additionally: Aprroximation: Aprroximation:

9. M. Csanád @ WPCF’05 9/25 The ellipsoidal Buda-Lund model M.Cs., T.Csörgő, B. Lörstad: Nucl.Phys.A742:80-94,2004 ; nucl-th/0310040 The original model was developed for axial symmetry  central collisions The original model was developed for axial symmetry  central collisions In the most general hydrodynamical form In the most general hydrodynamical form (‘Inspired by’ nonrelativistic solutions): Four-velocity distribution: Hubble-flow Four-velocity distribution: Hubble-flow Proper-time distribution: Proper-time distribution: Shape of distributions: Shape of distributions: Generalized Cooper-Frye prefactor: Generalized Cooper-Frye prefactor: Temperature-distribution: Temperature-distribution: Fugacity: Fugacity:

10. M. Csanád @ WPCF’05 10/25 Observables from BL hydro Two-particle correlation function : Two-particle correlation function : Width of it are the HBT radiiWidth of it are the HBT radii One-particle spectrum with core-halo correction: One-particle spectrum with core-halo correction: Core-halo picture: Core-halo picture: Flow coefficients: Flow coefficients:

11. M. Csanád @ WPCF’05 11/25 HBT radii ‘Harmonic sum’ of geometrical and thermal radii Temperature gradient Expansion rate Scaling variable Space-time rapidity of the point of maximal emittivity

12. M. Csanád @ WPCF’05 12/25 HBT(m t, ,  ) Dramatic change at low m t No change at high m t Radii decreasing with increasing m t or y Dramatic change at low m t No change at high m t Radii decreasing with increasing m t or y R out R side R long

13. M. Csanád @ WPCF’05 13/25 Hydro scaling in HBT Radii depend on m t and y through m t Radii depend on m t and y through m t

14. M. Csanád @ WPCF’05 14/25 Hydro scaling in HBT Radii depend on m t and y through m t Radii depend on m t and y through m t

15. M. Csanád @ WPCF’05 15/25 Hydro scaling in HBT Radii depend on m t and y through m t Radii depend on m t and y through m t

16. M. Csanád @ WPCF’05 16/25 The elliptic flow Pseudorapidity dependence mostly not understood (except see Hama/SPHERIO) Pseudorapidity dependence mostly not understood (except see Hama/SPHERIO) Depends on pseudorapidity and transverse momentum Depends on pseudorapidity and transverse momentum The m-th Fourier component is the m-th flow The m-th Fourier component is the m-th flow One-particle spectrum: One-particle spectrum:

17. M. Csanád @ WPCF’05 17/25 At large pseudorapidities… and If the point of maximal emittivity (saddlepoints) is near the longitudinal axis: If the point of maximal emittivity (saddlepoints) is near the longitudinal axis: Here  s is the space-time rapidipy of the saddlepoint Here  s is the space-time rapidipy of the saddlepoint, and so, and so Rapidity grows  the asymmetry vanishes (saddlepoint goes to the z axis)  elliptic flow vanishes Rapidity grows  the asymmetry vanishes (saddlepoint goes to the z axis)  elliptic flow vanishes, introducing, introducing

18. M. Csanád @ WPCF’05 18/25 Hydro predicts scaling Scaling variable Scaling variable For every type of measurement: For every type of measurement: Elliptic flow depends on every physical parameter only through w Elliptic flow depends on every physical parameter only through w Scaling curve I 1 / I 0 ? Scaling curve I 1 / I 0 ?

19. M. Csanád @ WPCF’05 19/25 Fits to PHOBOS data

20. M. Csanád @ WPCF’05 20/25 Fit parameters Fitted parameters: eccentricity &  Fitted parameters: eccentricity &  Fixed (non-essential) parameters (from spectra and HBT fits): Fixed (non-essential) parameters (from spectra and HBT fits): T0T0T0T0 RsRsRsRs utututut RgRgRgRg 175 MeV 12.38 fm 1.5 13.5 fm  200 GeV 0.394  0.006 2.56  0.04 130 GeV 0.376  0.005 2.46  0.04 62.4 GeV 0.349  0.008 2.16  0.05 19.6 GeV 0.294  0.029 1.70  0.25

21. M. Csanád @ WPCF’05 21/25 Error contours

22. M. Csanád @ WPCF’05 22/25 Universal scaling Scale parameter w Scale parameter w The perfect fluid extends from very small to very large rapidities at RHIC

23. M. Csanád @ WPCF’05 23/25 Conclusions I. Buda-Lund model describes HBT data @RHIC Buda-Lund model describes HBT data @RHIC Predictedion of the rapidity dependence of HBT radii Predictedion of the rapidity dependence of HBT radii Hydro scaling present in HBT radii? Straightforward to check! Hydro scaling present in HBT radii? Straightforward to check!

24. M. Csanád @ WPCF’05 24/25 Conclusions II. Buda-Lund model describes v 2 (  ) data @RHIC Buda-Lund model describes v 2 (  ) data @RHIC The vanishing elliptic flow at large  : The vanishing elliptic flow at large  : Hubble flow + finite longitudinal size v 2 (  ) data (2005) collapse to the theoretically predicted (2003) scaling function of v 2 (  ) data (2005) collapse to the theoretically predicted (2003) scaling function of The perfect fluid is present in AuAu in the whole  space The perfect fluid is present in AuAu in the whole  space

25. M. Csanád @ WPCF’05 25/25 Thanks for your attention Spare slides coming …

26. M. Csanád @ WPCF’05 26/25 Nonrelativistic hydrodynamics Equations of nonrelativistic hydro: Equations of nonrelativistic hydro: Not closed, EoS needed: Not closed, EoS needed: We use the following scaling variable: We use the following scaling variable: X, Y and Z are characteristic scales, depend on (proper-) time X, Y and Z are characteristic scales, depend on (proper-) time

27. M. Csanád @ WPCF’05 27/25 A nonrelativistic solution A general group of scale-invariant solutions (hep- ph/0111139): A general group of scale-invariant solutions (hep- ph/0111139): This is a solution, if the scales fulfill: This is a solution, if the scales fulfill:  (s) is arbitrary, e.g.  constant   gaussian, or:  (s) is arbitrary, e.g.  constant   gaussian, or: Buda-LundBondorf-Zimanyi-Garpman

28. M. Csanád @ WPCF’05 28/25 Some numeric results from hydro Propagate the hydro solution in time numerically: Propagate the hydro solution in time numerically:

29. M. Csanád @ WPCF’05 29/25 A relativistic solution Relativistic hydro: Relativistic hydro: with with A general group of solutions (nucl-th/0306004): A general group of solutions (nucl-th/0306004): Overcomes two shortcomings of Bjorken’s solution: Overcomes two shortcomings of Bjorken’s solution: Rapidity distribution Rapidity distribution Transverse flow Transverse flow Hubble flow  lack of acceleration Hubble flow  lack of acceleration

30. M. Csanád @ WPCF’05 30/25 The emission function The phase-space distribution looks like The phase-space distribution looks like Maxwell-Boltzman, for sake of simplicity with the constant: Consider the collisionless Boltzmann-equation Consider the collisionless Boltzmann-equation Calculates the source of a given phase-space distribution: Emission function in the simplest case (instant. source, at t=t 0 ): Emission function in the simplest case (instant. source, at t=t 0 ):