Transcript The centrality dependence of elliptic flow

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 arXiv:0704.3553
Outline
• A model for deviations from ideal hydro.
• Centrality and system-size dependence of
elliptic flow in ideal hydro: eccentricity
scaling.
• Eccentricity scaling+deviations from hydro:
explaining the centrality and system-size
dependence of elliptic flow at RHIC.
• Predictions for LHC (in progress).
Elliptic flow, hydro, and the Knudsen number
• Elliptic flow results from collisions among the produced particles
• The relevant dimensionless number is K=λ/R where λ is the mean
free path of a parton between two collisions, and R the system size.
• K»1: few collisions, little v2, proportional to 1/K.
• Ideal hydro is the limit K=0. 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
v2=v2hydro/(1+1.4 K)
The transport result smoothly converges to hydro as K goes to 0, as
expected
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.
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
λ is strongly time-dependent in 3D due to longitudinal expansion.
We estimate λ at the time when elliptic flow builds up.
Elliptic flow in ideal hydro
•
v2 in hydro scales like the initial eccentricity ε: requires a thorough
knowledge of initial conditions!
Recent breakthrough:
•
ε 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 ε 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)
Our model for the centrality and system-size
dependence of elliptic flow
We simply put together eccentricity scaling and deviations
from hydro:
v2/ε= h/(1+1.4 K)
Where
K-1= σ (1/S)(dN/dy)
(S = overlap area between the two nuclei)
ε and (1/S)(dN/dy) are computed using a model (Glauber or
CGC+fluctuations) as a function of system size and
centrality.
Both the hydro limit h and the partonic cross section σ are
free parameters, fit to Phobos Au-Au data for v2.
Results using Glauber model
(data from PHOBOS)
The « hydro limit » of v2/ε is 0.3, well above the value for
central Au-Au collisions.
Such a high value would require a very hard EOS (unlikely)
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 (Heinz&Kolb, Hirano)
LHC: deviations from hydro
• How does K evolve from RHIC to LHC ? Recall that
K-1 ~ σ (1/S)(dN/dy)
• dN/dy increases by a factor ~ 2
• Two scenarios for the partonic cross section σ:
• If σ 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 σ ~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.
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.
Summary
• The centrality and system-size dependence of elliptic
flow measured at RHIC are perfectly reproduced by a
simple model based on eccentricity scaling+deviations
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/ε 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/ε should be higher at LHC due to
the longer lifetime of the QGP.
Backup slides
v2 versus K in a 2D transport model
The lines are fits using v2=v2hydro/(1+K/K0), where K0 is a fit parameter
v4/v22 versus pt
Deviations from ideal hydro result in larger values,
closer to data (about 1.2) than hydro, but still too low
v2 versus pt: 2D transport versus hydro
3D transport versus hydro
Molnar and Huovinen, Phys. Rev. Lett. 94, 012302 (2005)
For small values of K, i.e., large values of σ, deviations from ideal hydro
should scale like 1/σ, which is clearly not the case here.