Thermalization and elliptic flow at RHIC and LHC

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Thermalization and elliptic flow at RHIC and
LHC

• J-Y Ollitrault (SPhT, Saclay)
• CERN, Heavy Ion Forum
• Dec. 13, 2006

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Outline

• Elliptic flow in ideal-fluid dynamics
• The perfect liquid scenario, and what it means in
  terms of final-state interactions.
• Modeling deviations from the perfect liquid
• Has RHIC created a perfect liquid? What do we
  expect at LHC?

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Elliptic flow v2
Interactions among the produced particles
Pressure gradients generate positive elliptic
flow v2
Early observation at RHIC v2 as large as
predicted by perfect fluid dynamics!
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v2 in ideal hydro

• Universal features
• Linear increase with pT for pions
• Lower for heavier particles at same pT (mass
  ordering)
• System-dependent features
• The time at which v2 appears scales with the
  transverse size of the system R
• v2 scales with the initial eccentricity e
• It increases with the stiffness of the equation
  of state, i.e., with the velocity of sound cs

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1. When is v2 created?
At a time R/cs, where R(1/ltx2gt1/lty2gt)-1/2
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1. When is v2 created (continued)?
The smaller the system, the earlier v2 appears!
(for dimensional reasons)
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2. v2 scales like the initial eccentricity e

• We used to think that we know how to calculate e,
  but
• The analysis of Cu-Cu data by PHOBOS has shown
  that fluctuations in the positions of
  participant nucleons, first pointed out by Miller
  and Snellings, are large, may even dominate!
• The Color Glass Condensate predicts
  significantly larger e than the standard Glauber
  model.
• Drescher Dumitru Hayashigaki Nara nucl-th/0605012

e lty2-x2gt/lty2x2gt

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3. v2 depends on the eq. of state
The relevant density here is not the initial
density but the density at time R/cs when v2 is
created, and the good news is, this density does
not vary much at a given energy Cu-Cu and Au-Au
collisions at RHIC are probing essentially the
same density! (flow and soft pT spectra)
Note density of ideal 2-flavor QGP n34 ?(3)p -2
T 32.8 fm-3 at T174 MeV elliptic flow at RHIC
is created at the transition to QGP
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What is a perfect liquid ?

• Liquid, as opposed to gas, means that the
  distance between particles is not much larger
  than the size of a particle (as defined, say,
  from its interaction cross section).
• But the RHIC liquid is expanding into the
  vacuum, i.e., it is compressible, unlike water.
• Perfect means low viscosity ?
• At the microscopic (particle) scale, low
  viscosity means large cross sections perfect
  liquid is the opposite of ideal gas (this is
  counter-intuitive to many people)

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Black-hole physics and RHIC

• The viscosity/entropy ratio can be computed for
  strongly coupled, N4 supersymmetric gauge
  theories using the Maldacena conjecture (AdS/CFT
  correspondence)
• ?/sh/4pkB, Son et al hep-th/0104066
• It has been postulated that this value is a
  universal lower bound (ex H2O, 25x higher) Son
  et al hep-th/0405231
• In the real world (QCD) the value is still
  unknown. Many RHIC scientists think the
  string-theory value is not incompatible with
  observed data.
• For a given substance, the minimum of ?/s occurs
  at the liquid-gas critical point Csernai et al
  nucl-th/0604032 are we seing the QCD critical
  point?

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From viscosity ? to mean free path ? and
partonic cross-sections s

• Viscosity describes momentum transport, which is
  achieved by collisions among the produced
  particles. For a relativistic fluid, transport
  theory shows that
• ?/s?T/c
• (remember, more collisions means lower
  viscosity)
• QCD plasma kBT200 MeVhc/fm thus
• ?/s(h/kB)
  ?fm
• The string theory prediction translates into ?
  0.1 fm
• Since ?1/sn and n2.5 fm-3 this in turn gives
• s40 mb .
  huge!

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The Knudsen number Kn

• The relevant dimensionless number for string
  theory is ?/s
• The relevant dimensionless number for heavy-ion
  collisions is Kn?/R
• Kn-1R/? is the typical number of collisions per
  particle
• Perfect fluid is the limit of a large number of
  collisions, i.e., Kn0.
• The string theory prediction translates into
  Kn0.03, a very small value indeed!

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Reading Kn from the multiplicity
Kn is related to the total (chargedneutral)
multiplicity through
Kn-1R/?s/S (dN/dy) where S overlap area
4pvltx2gtlty2gt
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Modelling deviations from the perfect fluid

• Need a theory that goes to ideal hydro in some
  limit.
• First method viscous hydrodynamics (papers by
  Teaney, Muronga, Baier Romatschke Wiedemann,
  Heinz Chaudhuri, Pratt) this is a general
  approach to small deviations from ideal hydro,
  but quantitative results are not yet available
• Second method Boltzmann equation. Limitation
  applies only to a dilute system (not to the
  liquid produced at RHIC). Advantage directly
  involves microscopic physics through collisional
  cross-sections

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Previous transport (Boltzmann) calculations
Boltzmann ?hydro although Kn1??
Molnar, Huovinen, nucl-th/0404065, Phys. Rev.
Lett.
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A new transport calculation
(C. Gombeaud JYO, in preparation)

• Two-dimensions
• Massless particles
• Billiard-ball calculation, but with Lorentz
  contraction taken into account this ensures
  Lorentz invariance of the number of collisions
  (?Molnar)
• N particles of size r in a box of size R dilute
  system if rR/vN

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pT dependence of v2
v2/e
pT
The transport calculation coincides with the
hydro calculation in the limit of small Kn, as it
should!
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Time dependence of v2
Boltzmann again coincides with hydro for small Kn
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Variation of v2 with Kn-1Nb collisions/particle
Best fit v2v2hydro/(11.76 Kn) goes to hydro
for Kn?0 Takes 2-3 collisions per particle to
reach 50 of hydro With Kn0.03 from string
theory, v20.95 v2hydro.
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Which data for model comparisons?

• Caveats v2 from hydro depends (rather strongly)
  not only on the initial eccentricity, but also on
  the equation of state, which is not an ideal gas
  (?Boltzmann)
• We must identify robust observables, which are
  insensitive to such model dependences.
• Bhalerao Blaizot Borghini JYO nucl-th/0508009

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v2/e Data from SPS and RHIC
Continuous increase with Kn-1, no saturation seen
in data
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Hexadecupole flow v4
Ideal hydro universal prediction v4/ (v2)20.5
at large pt Borghini JYO nucl-th/0506045 Confirme
d by numerical calculations
v4/(v2)2
pT
Data 1.2 suggest Kn1No thermalisation at RHIC!
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Qualitative predictions for LHC
Multiplicity higher by a factor 2. This has
several consequences

• Kn smaller by a factor 2 closer to hydro.
• Density larger by a factor 2 all v2 develops
  in the QGP phase.
• THEREFORE
• There is room for significant increase of v2
• v4/(v2)2 somewhat smaller than at RHIC

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Conclusions

• 18 months after the claim for the discovery of
  the perfect liquid at RHIC, a lot of effort is
  devoted to better understanding viscous effects
  in heavy-ion collisions.
• It is likely to result in a much improved
  quantitative understanding of the fully
  non-perturbative (i.e. soft) observables at LHC.