The centrality dependence of elliptic flow

1
The centrality dependence of elliptic flow
Workshop on heavy ion collisions at the LHC Last
call for predictions, May 30, 2007

• Jean-Yves Ollitrault, Clément Gombeaud (Saclay),
  Hans-Joachim Drescher, Adrian Dumitru (Frankfurt)
• nucl-th/0702075 and arXiv0704.3553

2
Outline

• A model for deviations from ideal hydro.
• Centrality and system-size dependence of elliptic
  flow in ideal hydro eccentricity scaling.
• Eccentricity scalingdeviations from hydro
  explaining the centrality and system-size
  dependence of elliptic flow at RHIC.
• Predictions for LHC (in progress).

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Elliptic flow, hydro, and the Knudsen number

• Elliptic flow results from collisions among the
  produced particles
• The relevant dimensionless number is K?/R where
  ? is the mean free path of a parton between two
  collisions, and R the system size.
• K1 few collisions, little v2, proportional to
  1/K.
• Ideal hydro is the limit K0. Does not reproduce
  all RHIC results.
• Viscous hydro is the first-order correction
  (linear in K)
• The Boltzmann transport equation can be used for
  all values of K. We have solved numerically a
  2-dimensional Boltzmann equation (no longitudinal
  expansion, transverse only) and we find
• 
  v2v2hydro/(11.4 K)
• The transport result smoothly converges to
  hydro as K goes to 0, as expected

4
Why a 2-dimensional transport calculation?

• Technical reason numerical, finite-size
  computer.
• The Boltzmann equation (2 to 2 elastic collisions
  only) only applies to a dilute gas (particle size
  distance between particles). This requires
  parton subdivision.
• To check convergence of Boltzmann to hydro, we
  need both a dilute system and a small mean free
  path, i.e., a huge number of particles.
• In the 2-dimensional case, we were able to
  reproduce hydro within 1 using 106 particles. A
  similar achievement in 3 dimensions would require
  109 particles.

5
Does v2 care about the longitudinal expansion?
Time-dependence of elliptic flow in transport and
hydro
Little difference between 2D and 3D ideal hydro.
Deviations from hydro should also be similar,
but the mean free path ? is strongly
time-dependent in 3D due to longitudinal
expansion. We estimate ? at the time when
elliptic flow builds up.
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Elliptic flow in ideal hydro

• v2 in hydro scales like the initial eccentricity
  e requires a thorough knowledge of initial
  conditions!
• Recent breakthrough
• e was underestimated in early hydro calculations
  it is increased by fluctuations in the positions
  of nucleons within the nucleus, which are large
  for small systems and/or central collisions
• Miller Snellings nucl-ex/0312008,
• PHOBOS nucl-ex/0610037
• The CGC predicts a larger e than Glauber (binary
  collisions participants) scaling.
• Hirano Heinz Kharzeev Lacey Nara, Phys. Lett.
  B636, 299 (2006)
• Adil Drescher Dumitru Hayashigaki Nara, Phys.
  Rev. C74, 044905 (2006)

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Our model for the centrality and system-size
dependence of elliptic flow

• We simply put together eccentricity scaling and
  deviations from hydro
• v2/e h/(11.4 K)
• Where
• K-1 s (1/S)(dN/dy)
• (S overlap area between the two nuclei)
• e and (1/S)(dN/dy) are computed using a model
  (Glauber or CGCfluctuations) as a function of
  system size and centrality.
• Both the hydro limit h and the partonic cross
  section s are free parameters, fit to Phobos
  Au-Au data for v2.

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Results using Glauber model(data from PHOBOS)
The hydro limit of v2/e is 0.3, well above
the value for central Au-Au collisions. Such a
high value would require a very hard EOS
(unlikely)
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Results using CGC
The fit is exactly as good, but the hydro limit
is significantly lower 0.22 instead of 0.3,
close to the values obtained by various groups
(HeinzKolb, Hirano)
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LHC deviations from hydro

• How does K evolve from RHIC to LHC ? Recall
  that
• K-1 s
  (1/S)(dN/dy)
• dN/dy increases by a factor 2
• Two scenarios for the partonic cross section s
• If s is the same, deviations from ideal hydro
  are smaller by a factor 2 at LHC than at RHIC
  (12 for central Pb-Pb collisions for CGC initial
  conditions)
• Dimensional analysis suggests s T-2
  (dN/dy)-2/3. Then, K decreases only by 20
  between RHIC and LHC, and the centrality and
  system-size dependence are similar at RHIC and
  LHC.

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LHC the hydro limit

• Lattice QCD predicts that the density falls by a
  factor 10 between the QGP and the hadronic
  phase
• If deviations from ideal hydro are large in the
  QGP, this means that the hadronic phase
  contributes little to v2.
• The density decreases like 1/t the lifetime of
  the QGP scales like (dN/dy) roughly 2x larger
  at LHC than at RHIC. There is room for
  significant increase of v2.
• Hydro predictions should be done with a smooth
  crossover, rather than with a first-order phase
  transition.

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Summary

• The centrality and system-size dependence of
  elliptic flow measured at RHIC are perfectly
  reproduced by a simple model based on
  eccentricity scalingdeviations from hydro
• Elliptic flow is at least 25 below the hydro
  limit , even for the most central Au-Au
  collisions
• Glauber initial conditions probably underestimate
  the initial eccentricity.
• v2/e will still increase as a function of system
  size and/or centrality at LHC, and 12 to 20
  below the hydro limit for the most central
  Pb-Pb collisions.
• The hydro limit of v2/e should be higher at LHC
  due to the longer lifetime of the QGP.

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Backup slides
14
v2 versus K in a 2D transport model
The lines are fits using v2v2hydro/(1K/K0),
where K0 is a fit parameter
15
v4/v22 versus pt
Deviations from ideal hydro result in larger
values, closer to data (about 1.2) than hydro,
but still too low
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v2 versus pt 2D transport versus hydro
17
3D transport versus hydro
Molnar and Huovinen, Phys. Rev. Lett. 94, 012302
(2005) For small values of K, i.e., large values
of s, deviations from ideal hydro should scale
like 1/s, which is clearly not the case here.