Does HBT interferometry probe thermalization

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Does HBT interferometry probe thermalization?

• Jean-Yves Ollitrault, Saclay
• Collaborators Clément Gombeaud, Tuomas Lappi
• Tamura Symposium on heavy-ion physics
• UT Austin, November 20, 2008

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Introduction hydrodynamics, viscosity,

• RHIC experiments have taught us
• Hydrodynamics gives the best qualitative
  description of soft particle (lt2 GeV/c)
  production in ultrarelativistic heavy-ion
  collisions at RHIC
• Ideal hydrodynamics is not so good, viscous
  corrections needed.
• Reminder (textbook) dimensional analysis in
  fluid mechanics
• ? viscosity, usually scaled by the
  mass/energy density
• ?/e ? vthermal, where ? mean free path of
  a particle
• R typical (transverse) size of the system
• vfluid fluid velocity vthermal because
  expansion into the vacuum
• The Reynolds number characterizes viscous
  effects
• Re R vfluid /(?/e) R/?
• Viscous corrections scale like the
  viscosity Re-1 ?/R.
• If they are not 1, the hydrodynamic picture
  breaks down!

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Motivations of our work

• Study soft observables for arbitrary values of
  the Knudsen number K?/R beyond the validity of
  hydrodynamics.
• The theoretical framework Boltzmann transport
  theory, which means particles undergoing 2 ? 2
  elastic collisions, easily solved numerically by
  Monte-Carlo simulations.
• One recovers ideal hydro for K?0 (we check this
  explicitly).
• One should recover viscous hydro to first order
  in K (not checked).
• The price to pay dilute system (?
  interparticle distance), which implies, ideal gas
  equation of state no phase transition
  connection with real world (data) not
  straightforward
• Additional simplifications 2 dimensional
  system (transverse only), massless particles.
  Extensions to 3 d and massive particles under
  study.

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Outline

• Elliptic flow
• Gombeaud, JYO, nucl-th/0702075
• The centrality dependence of elliptic flow at
  RHIC
• Drescher, Dumitru, Gombeaud, JYO, arXiv0704.3553
• HBT for central collisions, and the HBT puzzle.
• HBT for noncentral collisions
• Gombeaud, Lappi, JYO, in preparation

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Elliptic flow
f
Non-central collision seen in the transverse
plane the overlap area, where particles are
produced, is not a circle.
A particle moving at fp/2 from the x-axis is
more likely to be deflected than a particle
moving at f0, which escapes more easily.
Initially, particle momenta are distributed
isotropically in f. Collisions results in
positive v2.
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The initial eccentricity
?y

• v2 scales like the eccentricity e of the initial
  density profile, defined as

?x
This eccentricity depends on the collision
centrality, which is well known experimentally.
Ideal hydrodynamics is scale invariant v2 a e,
where a constant for all systems (Au-Au and
Cu-Cu) and all centralities at a given energy
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Elliptic flow versus time
Convergence to ideal hydro clearly seen!
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Elliptic flow versus K
v2a e/(11.4 K)
Elliptic flow increases with number of collisions
(1/K) Smooth convergence to ideal hydro as K?0
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The centrality dependence of v2 explained

• Phobos data for v2
• e obtained using Glauber or CGC initial
  conditions fluctuations
• Fit with
• v2a e/(11.4 K)
• assuming
• 1/K(s/S)(dN/dy)
• with the fit parameters s and a.

K0.3 for central Au-Au collisions v2 30 below
ideal hydro!
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HBT radii
Rs
pt
Ro
For particles with a given momentum pt along x Ro
measures the dispersion of xlast-v tlast Rs
measures the dispersion of ylast Where
(tlast,xlast,ylast) are the space-time
coordinates of the particle at the last scattering
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Boltzmann versus hydro
R/?0.1
R/?0.5
Ro
R/? 3
R/? 10
hydro
pt
Boltzmann again converges to ideal hydro for
small Kn However, Ro is not very sensitive to
thermalization pt dependence is already present
in (CGC-inspired) initial conditions
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Ro and Rs versus K
Ro
Radii
Rs
Au-Au, b0
R/?1/K
As the number of collisions increases, Ro
increases and Rs decreases, but this is a very
slow process The hydro limit requires a
huge number of collisions !
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The HBT puzzle revisited
Ro/Rs
pt
A simulation with R/?3 (inferred from elliptic
flow) gives a value compatible with data, and
significantly lower than hydro.
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Radii too small
Ro
pt
due to the hard equation of state and 2
dimensional geometry
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Azimuthally-sensitive HBT
y
x
z
J. Adams al, nucl-ex/0411036v2
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Azimuthal oscillations versus K
?HBT/?
?s/?
?o/?
R/?1/K
The eccentricity seen in Ro decreases as 1/K
increases sensitive to thermalization.
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Comparison with STAR data
The initial anisotropy is washed out in the data,
not in our calculation.
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Conclusions

• v2 smoothly converges to hydro as ? decreases.
• Ro/Rs also converges to hydro but much more
  slowly the experimental value Ro/Rs1 is
  consistent with estimates of ? inferred from v2.
• The pt dependence of Ro and Rs is not a signature
  of hydro behaviour. It is likely to be present
  already in initial conditions.
• The azimuthal oscillations of HBT radii seen by
  STAR is smaller than we expect based on the
  initial eccentricity.

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Backup slides
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Dimensionless numbers

• Parameters
• Transverse size R
• Cross section s (length in 2d!)
• Number of particles N
• Other physical quantities
• Particle density nN/R2
• Mean free path ?1/sn
• Distance between particles dn-1/2
• Relevant dimensionless numbers
• Dilution parameter Dd/?(s/R)N-1/2
• Knudsen number K?/R(R/s)N-1

The hydrodynamic regime requires both D1 and
K1. Since ND-2K-2, a huge number of particles
must be simulated. (even worse in 3d)
The Boltzmann equation requires D1 This is
achieved by increasing N (parton subdivision)
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Test of the Monte-Carlo algorithm
thermalization in a box
Initial conditions monoenergetic particles.
Kolmogorov test Number of particles with energy
ltE in the system Versus Number of particles with
energy ltE in thermal equilibrium
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Elliptic flow versus time
Convergence to ideal hydro clearly seen!
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Elliptic flow versus pt
Convergence to ideal hydro clearly seen!
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v4 data
(Bai Yuting, STAR)
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Higher harmonics v4
Recall RHIC data above ideal hydro
preferred value from v2 fits
Boltzmann also above ideal hydro but still below
data
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Particle densities per unit volume at RHIC(MC
Glauber calculation)
The density is estimated at the time tR/cs
(i.e., when v2 appears), assuming 1/t dependence.
H-J Drescher (unpublished)
The effective density that we see through
elliptic flow depends little on colliding system
centrality !
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Effects of thermalization
?Ro/?Rs
R/?
The ratio increases with the degree of
thermalisation, as the ratio Ro/Rs
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Work in progress

• Extension to massive particles
• Small fraction of massive particles embedded in a
  massless gas
• Study how the mass-ordering of v2 appears as the
  mean free path is decreased
• Extension to 3 dimensions with boost-invariant
  longitudinal cooling
• Repeat the calculation of Molnar and Huovinen
• Different method boost-invariance allows
  dimensional reduction Monte-Carlo is 3d in
  momentum space but 2d in coordinate space, which
  is much faster numerically.