how perfect is the rhic liquid? jean-yves ollitrault, ipht saclay bnl colloquium, feb. 24, 2009...
TRANSCRIPT
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How perfect is the RHIC liquid?
Jean-Yves Ollitrault, IPhT Saclay
BNL colloquium, Feb. 24, 2009based on
Drescher Dumitru Gombeaud & JYO, arXiv:0704.3553Gombeaud, Lappi, JYO, arXiv:0901.4908
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Two Lorentz-contracted nuclei collide
A nucleus-nucleus collision at RHIC
« hard » processes, accessible to perturbative calculations
High-density, strongly-interacting hadronic matter (quark-gluon plasma?) is created and expands, and eventually reaches the detectors as hadrons.
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Hot QCD…
If interactions are strong enough, the matter produced in a
nucleus-nucleus collision at RHIC reaches local thermal equilibrium
The thermodynamics of strongly coupled, hot QCD can be computed on the lattice:• equation of state• correlation functions
However, some quantities are very hard to compute on the lattice, such as the viscosity
Karsch, hep-lat/0106019
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…and string theory…
Using the AdS/CFT correspondence, one can compute exactly the viscosity to entropy ratio in strongly coupled supersymmetric N=4 gauge theories, and it has been postulated that the result is a universal lower bound.
η/s=ħ/4πkB
Kovtun Son Starinets hep-th/0405231
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…and RHIC ?
• In 2005, a press release claimed that nucleus-nucleus collisions at RHIC had created a perfect liquid, with essentially no viscosity. Since then, many works on viscous hydrodynamics
Martinez, Strickland arXiv:0902.3834Luzum Romatschke arXiv:0901.4588
Bouras Molnar Niemi Xu El Fochler Greiner Rischke arXiv:0902.1927Song Heinz arXiv:0812.4274
Denicol Kodama Koide Mota arXiv:0807.3120Molnar Huovinen arXiv:0806.1367
Chaudhuri arXiv:0801.3180Dusling Teaney arXiv:0710.5932
• For a given substance, the minimum of η/s occurs at the liquid-gas critical point : are we seing the QCD critical point?
Csernai et al nucl-th/0604032
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Outline
• Ideal hydrodynamics, what it predicts• Viscous corrections & dimensional analysis• The system size dependence of v2: a natural
probe of viscous effects• Comparison with data: how large are viscous
corrections? • Other observables• Conclusions
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Elliptic flow
xx
Non-central collision seen in the transverse plane: the overlap area, where particles are produced, is
not a circle.
A particle moving at φ=π/2 from the x-axis is more likely to be deflected than a particle moving at φ=0, which escapes more easily.
φ
Initially, particle momenta are distributed isotropically in φ.Collisions results in positive v2.
...)2cos2cos21(2
121
vv
d
dN
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Eccentricity scaling
v2 scales like the eccentricity ε of the initial
density profile, defined as :
22
22
xy
xy
y
x
Ideal (nonviscous) hydrodynamics (∂μTμν= 0) is scale invariant:v2= α ε, where α constant for all colliding systems (Au-Au and Cu-Cu) at all impact parameters at a given energy
This eccentricity depends on the collision centrality, which is well known experimentally.
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Caveat: hydro predictions for v2 are model dependent
The initial eccentricity ε is model dependent
The color-glass condensate prediction is 30% larger than the Glauber-model prediction
Drescher Dumitru Hayashigaki Nara, nucl-th/0605012
Eccentricity fluctuations are importantPHOBOS collaboration, nucl-ex/0510031
The ratio v2/ε depends on the equation of state
A harder equation of state gives more elliptic flow for a given eccentricity.
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No scale invariance in data
v2/ε is not constant at a given energy: non-viscous hydro fails !But viscosity breaks scale invariance
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How viscosity breaks scale invariance: dimensional analysis in fluid dynamics
η = viscosity, usually scaled by the mass/energy density: η/ε ~ λ vthermal, where λ 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 /(η/ε) ~ R/λ Viscous corrections scale like the viscosity: ~ Re-1 ~ λ/R.
If viscous effects are not « 1, the hydrodynamic picture breaks down!
In this talk, I use instead the Knudsen number K= λ/R1/K ~ number of collisions per particle
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A simplified approach to viscous effects
• Our motivation : study arbitrary values of the Knudsen number K≡λ/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 (λ» 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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Elliptic flow versus time
Convergence to ideal hydro clearly seen!
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Elliptic flow versus K
v2=α ε/(1+1.4 K)
Elliptic flow increases with number of collisions (~1/K)
Smooth convergence to ideal hydro as K→0
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How is K related to RHIC data?
The mean free path of a particle in medium is λ=1/σn
1/K= σnR ~ (σ/S)(dN/dy), where• σ is a (partonic) cross section• S is the overlap area between
nuclei• (dN/dy) is the particle multiplicity per
unit rapidity.
K can be tuned by varying the system size and centrality
S
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The centrality dependence of v2 explained
1. Phobos data for v2
2. ε obtained using Glauber or CGC initial conditions +fluctuations
3. Fit with
v2=α ε/(1+1.4 K)
assuming
1/K=(σ/S)(dN/dy)
with the fit parameters σ and α.
K~0.3 for central Au-Au collisions
v2 : 30% below ideal hydro!
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From cross-section σ to viscosity η
• Viscosity describes momentum transport, which is achieved by collisions among the produced particles. For a gas of massless particles with isotropic cross sections, transport theory gives
η=1.264 kBT/σc (remember, more collisions means lower viscosity)• The entropy is proportional to the number of particles
S=4NkB• This yields our estimates
η/s~0.16-0.19 (ħ/kB) depending on which initial conditions we use.
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Other observables : HBT radii
pt
Rs
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
HBT puzzle : Ro/Rs~1.5 in hydro, 1 in RHIC data
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Ro and Rs versus K
Au-Au, b=0
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 !
Radii
Ro
Rs
HBT puzzle
R/λ=1/K
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The HBT puzzle revisited
Ro/Rs
pt
A simulation with R/λ=3 (inferred from elliptic flow) gives a value
compatible with data, and significantly lower than hydro.
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Conclusions
• The centrality and system size dependence of elliptic flow is a specific probe of viscous effects in heavy ion collisions at RHIC.
• Viscosity is important : elliptic flow is 25 % below the «hydro limit», even for central Au-Au collisions !
• Quantitative understanding of RHIC results in the soft momentum sector requires viscosity, probably larger than the lower bound from string theory.
• Viscous effects are larger on HBT radii than on elliptic flow: the experimental value Ro/Rs~1 is consistent with estimates of viscous effects inferred from v2.
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Backup slides
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Estimating the initial eccentricity
Nucleus 1
Nucleus 2
Participant Region
x
y
b
Until 2005, this was thought to be the easy part. But puzzling results came:1. v2 was larger than predicted by hydro in central Au-Au collisions.2. v2 was much larger than expected in Cu-Cu collisions. This was interpreted by the PHOBOS collaboration as an effect of fluctuations in initial conditions [Miller & Snellings nucl-ex/0312008]
In 2005, it was also shown that the eccentricity depends significantly on the model chosen for initial particle production. We compare two such models, Glauber and Color Glass Condensate.
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Dimensionless numbers in fluid dynamics
They involve intrinsic properties of the fluid (mean free path λ, thermal/sound velocity cs, shear viscosity η, mass density ρ) as well as quantities specific to the flow pattern under study (characteristic size R, flow velocity v)
Knudsen number K= λ/R K «1 : local equilibrium (fluid dynamics applies)
Mach number Ma= v/cs
Ma«1 : incompressible flow
Reynolds number R= Rv/(η/ρ) R»1 : non-viscous flow (ideal fluid)
They are related ! Transport theory: η/ρ~λcs implies R * K ~ Ma
Remember: compressible+viscous = departures from local eq.
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Dimensionless numbers in the transport calculation
• Parameters:– Transverse size R– Cross section σ (~length in 2d!)– Number of particles N
• Other physical quantities– Particle density n=N/R2
– 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 K=λ/R=(R/σ)N-1
The hydrodynamic regime requires both D«1 and K«1.
Since N=D-2K-2, a huge number of particles must be simulated.
(even worse in 3d)
The Boltzmann equation requires D«1This 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 <E
in the systemVersusNumber of particles with energy <E in thermal equilibrium
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Elliptic flow versus pt
Convergence to ideal hydro clearly seen!
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Particle densities per unit volume at RHIC(MC Glauber calculation)
The density is estimated at the time t=R/cs
(i.e., when v2 appears), assuming 1/t dependence.
The effective density that we see through elliptic flow depends little on colliding system & centrality !
H-J Drescher
(unpublished)
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v4 data
(Bai Yuting, STAR)
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Higher harmonics : v4
Recall : RHIC data above ideal hydro
Boltzmann also above ideal hydro but still below data
preferred
value from
v2 fits
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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.
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Boltzmann versus hydro
Ro
pt
hydro
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
R/λ~0.1
R/λ~0.5
R/λ~ 3
R/λ~ 10
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Radii too small
Ro
pt
due to the hard equation of state and 2 dimensional geometry