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

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, CG, Lappi, Ollitrault, arxiv: v1

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

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

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

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, (2007)

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

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

Partial solution of the HBT puzzle Piotr Bozek & al arXiv: v1 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

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:

AzHBT Observables y x P z

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

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

Backup slides

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)

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

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

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 (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

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

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)

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

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, (2004)

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

HBT vs data