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

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.

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

…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

…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

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

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

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.

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.

No scale invariance in data

v2/ε is not constant at a given energy: non-viscous hydro fails !But viscosity breaks scale invariance

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

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.

Elliptic flow versus time

Convergence to ideal hydro clearly seen!

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

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

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!

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.

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

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

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.

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.

Backup slides

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.

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.

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)

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

Elliptic flow versus pt

Convergence to ideal hydro clearly seen!

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)

v4 data

(Bai Yuting, STAR)

Higher harmonics : v4

Recall : RHIC data above ideal hydro

Boltzmann also above ideal hydro but still below data

preferred

value from

v2 fits

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.

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

Radii too small

Ro

pt

due to the hard equation of state and 2 dimensional geometry