1. Partial thermalization, a key ingredient of the HBT Puzzle Clément Gombeaud CEA/Saclay-CNRS Quark-Matter 09, April 09

2. Outline Introduction- Femtoscopy Puzzle at RHIC Motivation Transport numerical tool –Boltzmann solution –Dimensionless numbers HBT for central HIC –Boltzmann versus hydro –Partial resolution of the HBT-Puzzle –Effect of the EOS Azimuthally sensitive HBT (AzHBT) Conclusion CG, J. Y. Ollitrault, Phys. Rev C 77, 054904 CG, Lappi, Ollitrault, arxiv:0901.4908v1

3. P y x z Introduction HBT, the femtoscopic observables HBT puzzle: Experiment R o /R s =1 Ideal hydro R o /R s =1.5

4. Motivation Ideal hydrodynamics gives a good qualitative description of soft observables in ultrarelativistic heavy-ion collisions at RHIC But it is unable to quantitatively reproduce data: Full thermalization not achieved Using a transport simulation, we study the sensitivity of the HBT radii to the degree of thermalization, and if this effect can explain, even partially, the HBT puzzle

5. Numerical solution of the 2+1 dimensional Boltzmann equation. The Boltzmann equation (v·∂)f=C[f] describes the dynamics of a dilute gas statistically, through its 1- particle phase-space distribution f(x,t,p) The Monte-Carlo method solves this equation by –drawing randomly the initial positions and momenta of particles according to the phase-space distribution –following their trajectories through 2→2 elastic collisions –averaging over several realizations. Monte-Carlo simulation method

6. Dimensionless quantities Average distance between particles d Mean free path We define 2 dimensionless quantities Dilution D=d/ Knudsen K= /R~1/N coll_part characteristic size of the system R Boltzmann requires D<<1 Ideal hydro requires K<<1 Previous study of v 2 for Au-Au At RHIC gives Central collisions  K=0.3 Drescher & al, Phys. Rev. C76, 024905 (2007)

7. Boltzmann versus hydro The same behaviour is seen in both partially thermalized Boltzmann and short lived ideal hydro simulation Small sensitivity of the Pt dependence to the thermalization

8. Evolution vs K -1 Solid lines are fit with F(K)=F0+F1/(1+F2*K) V 2 goes to hydro three times faster than HBT K -1 =3 b=0 Au-Au At RHIC v 2 hydro Hydro limit of the HBT radii Regarding the values of F2

9. Partial solution of the HBT puzzle Piotr Bozek & al arXiv:0902.4121v1 Similar results for K=0.3 (extracted from v2 study) and for the short lived ideal hydro Partial thermalization (=few collisions per particles) explains most of the HBT Puzzle

10. Effect of the EOS Realistic EOS Viscosity  Partial thermalization Our Boltzmann equation implies Ideal gas EOS (  =3P) Pratt find that EOS is more important than viscosity Our K=0.3 (~viscous) simulation solves most of the Puzzle Pratt arxiv:0811.3363

11. AzHBT Observables y x P z

12. Evolution vs K -1  R o 2 /  R s 2 evolve qualitatively as Ro/Rs  s misses the data even in the hydro limit EOS effects

13. Conclusion The Pt dependence of the HBT radii is not a signature of the hydro evolution Hydro prediction R o /R s =1.5 requires unrealistically large number of collisions. Our K=0.3 (extracted from v2) explains most of the HBT Puzzle. 3+1d simulation using boost invariance

14. Backup slides

15. Dimensionless numbers Parameters: –Transverse size R –Cross section σ (~length in 2d!) –Number of particles N Other physical quantities –Particle density n=N/R 2 –Mean free path λ=1/σn –Distance between particles d=n -1/2 Relevant dimensionless numbers: –Dilution parameter D=d/λ=(σ/R)N -1/2 –Knudsen number Kn=λ/R=(R/σ)N -1 The hydrodynamic regime requires both D «1 and Kn«1. Since N=D -2 Kn -2, a huge number of particles must be simulated. (even worse in 3d) The Boltzmann equation requires D «1 This is achieved by increasing N (parton subdivision)

16. Impact of dilution on transport results Transport usually implies instantaneous collisions Problem of causality r  KD 2  D<<1 solves this problem when K fixes the physics

17. Viscosity and partial thermalization Non relativistic case Israel-Stewart corresponds to an expansion in power of Knudsen number

18. Implementation Initial conditions: Monte-Carlo sampling –Gaussian density profile (~ Glauber) –2 models for momentum distribution: Thermal Boltzmann (with T=n 1/2 ) CGC (A. Krasnitz & al, Phys. Rev. Lett. 87 19 (2001)) (T. Lappi Phys. Rev. C. 67 (2003) ) With a1=0.131, a2=0.087, b=0.465 and Qs=n 1/2 Ideal gas EOS

19. Elliptic flow versus Kn v 2 =v 2 hydro /(1+1.4 Kn) Smooth convergence to ideal hydro as Kn→0

20. The centrality dependence of v 2 explained 1.Phobos data for v 2 2.ε obtained using Glauber or CGC initial conditions +fluctuations 3.Fit with v 2 =v 2 hydro /(1+1.4 Kn) assuming 1/Kn=(σ/S)(dN/dy) with the fit parameters σ and v 2 hydro /ε. Kn~0.3 for central Au-Au collisions v 2 : 30% below ideal hydro! (Density in the azimuthal plane)

21. AzHBT radii evolution vs K -1 Better convergence to hydro in the direction of the flow

22. EOS effects Ideal gas R o R s R l product is conserved In nature, there is a phase transition Realistic EOS s deacrese, but S constant at the transition (constant T) –Increase of the volume V at constant T Phase transition implies an increase of the radii values S. V. Akkelin and Y. M. Sinyukov, Phys. Rev. C70, 064901 (2004)

23. AzHBT vs data Pt in [0.15,0.25] GeV 20-30% Pt in [0.35,0.45] GeV 10-20%

24. HBT vs data