Perfect Fluid: flow measurements are described by ideal hydro

Problem: all fluids have some viscosity -- can we measure it?

I. Radial flow fluctuations: Dissipated I. Radial flow fluctuations: Dissipated by shear viscosityby shear viscosity

II. Contribution to transverse momentum II. Contribution to transverse momentum correlationscorrelations

III. Implications of current fluctuation III. Implications of current fluctuation data data

Phys. Rev. Lett. 97162302 (2006)

Measuring Viscosity at RHIC Measuring Viscosity at RHIC

Sean Gavin Sean Gavin Wayne State University Wayne State University

Mohamed Abdel-AzizMohamed Abdel-Aziz Institut für Theoretische Physik, Institut für Theoretische Physik, FrankfurtFrankfurt

Measuring Viscosity at RHIC Measuring Viscosity at RHIC

Sean Gavin Sean Gavin Wayne State University Wayne State University

Mohamed Abdel-AzizMohamed Abdel-Aziz Institut für Theoretische Physik, Institut für Theoretische Physik, FrankfurtFrankfurt

Measuring Shear ViscosityMeasuring Shear Viscosity

flow vx(z)

shear viscosity

€

⇒ Tzx = −η∂vx

∂z

Elliptic and radial flow suggest small shear viscosityElliptic and radial flow suggest small shear viscosity• Teaney; Kolb & Heinz; Huovinen & Ruuskanen

Problem: initial conditions & EOS unknown • CGC more flow? larger viscosity? Hirano et al. Kraznitz et al.; Lappi & Venugopalan; Adil et al.

Additional viscosity probes?

What does shear viscosity do? It resists shear flow.

Small viscosity or strong flow?

Radial Flow FluctuationsRadial Flow Fluctuations

€

vr

neighboring fluid elements flow past one another viscous friction

small variations in radial flow in each event

shear viscosity shear viscosity drives velocity drives velocity toward the averagetoward the average

zvT rzr ∂∂−=

damping of radial flow fluctuations viscosity

Evolution of Evolution of FluctuationsFluctuations

€

∂∂t

−ν∇ 2 ⎛

⎝ ⎜

⎞

⎠ ⎟gt = 0

diffusion equationdiffusion equation for momentum current

kinematic viscosity

€

ν = /Ts shear viscosity entropy density s, temperature T

r

z

€

gt ≡ T0r − T0r ≈ Ts u

€

Tzr ≈ −η∂u

∂z≈ −

η

Ts

∂gt

∂z

€

∂∂t

T0r +∂

∂zTzr = 0

momentum current momentum current for small fluctuations

momentum conservation

u

u(z,t) ≈ vr vr

shear stress

viscous diffusion viscous diffusion drives fluctuation gt

V = 2ν/0

0 2 4 6

/0

€

∂gt

∂τ=

ν

τ 2

∂2gt

∂y 2

€

V = 2ν1

τ 0

−1

τ

⎛

⎝ ⎜

⎞

⎠ ⎟ V

random walk in random walk in rapidityrapidity y vs. proper time

kinematic viscosity ν Ts

formation at 0

Viscosity Broadens Rapidity Viscosity Broadens Rapidity DistributionDistribution

€

y€

gt

€

( )width2 ≡V rapidity broadeningrapidity broadening

increase

Hydrodynamic Momentum CorrelationsHydrodynamic Momentum Correlations

€

rg = gt (x1)gt (x2) − gt (x1) gt (x2)

momentum flux density correlation functionmomentum flux density correlation function

rg rg r g,eq satisfies diffusion equation satisfies diffusion equation

Gardiner, Handbook of Stochastic Methods, (Springer, 2002)

fluctuations diffuse through volume, driving rg rg,eq

€

σ 2 = σ 02 + 2ΔV (τ f )

width in relative rapidity grows from initial value σ

How Much Viscosity?How Much Viscosity?viscosity in collisions -- Hirano & Gyulassy

supersymmetric Yang-Mills: s pQCD and hadron gas: s ~ 1

viscous liquid νpQGP ~ ν HRG ~ 1 fmnearly perfect νsQGP ~ (4 Tc)-1 ~ 0.1 fm

Abdel-Aziz & S.G

€

ν = Ts

kinematic viscosity

effect on momentum diffusion:

limiting cases:

wanted:wanted: rapidity dependence of momentum correlation rapidity dependence of momentum correlation functionfunction

observable:observable:

€

C = pt1pt 2 − pt

2

Transverse Momentum CovarianceTransverse Momentum Covariance

€

pt1pt 2 ≡1

N2 pti ptj

pairs i≠ j

∑

€

pt ≡1

Npti∑

propose:propose: measure C() to extract width σ2 of rg

measures momentum-density correlation function

€

C = 1

N2 Δrg dx1dx2∫

C depends on rapidity interval

C()

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4€

ν /τ 0 =1

€

ν /τ 0 = 0.1

s ~ 0.08 ~ 1/4

Current Data?Current Data?

STAR measures rapidity width of pt fluctuationsSTAR measures rapidity width of pt fluctuations

• most peripheral σ* ~ 0.45 €

σ p t :n =1

Npti − pt( ) ptj − pt( )

i≠ j

∑

find width σ* increases in central collisions

• central σ* ~ 0.75

€

σ central2 −σ peripheral

2 = 4ν1

τ f ,p

−1

τ f ,c

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

naively identifynaively identify σ* with σ

freezeout f,p ~ 1 fm, f,c ~ 20 fm ν ~ 0.09 fm

at Tc ~ 170 MeV

J.Phys. G32 (2006) L37

we want: 2

2

1t

jitjti ppp

NC −= ∑

≠

Uncertainty RangeUncertainty Range

STAR measures:

maybe σn 2σ* STAR, PRC 66, 044904 (2006)

uncertainty range σ* σ 2σ* 0.08 s 0.3

€

N Δσ p t :n = pti − pt( ) ptj − pt( )i≠ j

∑

€

= N2C − pt

2(density correlations)

density correlation functiondensity correlation function may differ from rg

Current Data Current Data Viscosity Viscosity Hints!Hints!

MeasureMeasure p ptt Covariance Covariance for Clearer Picturefor Clearer Picture

current fluctuation data compatible with current fluctuation data compatible with

compare to other observables with different compare to other observables with different uncertaintiesuncertainties

Summary: small viscosity or strong Summary: small viscosity or strong flow?flow?

€

0.08 < η /s < 0.3

test the perfect liquid test the perfect liquid viscosity info viscosity info

• viscosity broadens momentum correlations in rapidity viscosity broadens momentum correlations in rapidity

• pptt covariance covariance measures these correlations measures these correlations

0

1

2

3

4

0 5 10 15 20

sQGP + Hadronic CoronasQGP + Hadronic Coronaviscosity in collisions -- Hirano & Gyulassy

supersymmetric Yang-Mills: s pQCD and hadron gas: s ~ 1

€

ν =T−1(η /s)

Broadening from viscosity

€

/τ 0

€

σ 2 −σ 02

€

d

dτσ 2 ≈

4ν (τ )

τ 2,

H&G

Abdel-Aziz & S.G, in progress

QGP + mixed phase + hadrons T()

Centrality DependenceCentrality Dependence

freezeout time modelTeaney, Lauret & Shuryak

rapidity width

€

centrality, 2nbin /npart

€

σ 2 −σ 02

( " )max

00.10.20.30.40.50.60.70.80.9

1

1 2 3 4 5 6

STAR, J.Phys. G32 (2006) L37

€

F ~ r2

part

centrality: mean parton path length

€

path length ~ 2nbin /npart

€

ν =const.

€

ν ∝ τ

€

d

dτσ 2 ≈

4ν (τ )

τ 2,

Abdel-Aziz & S.G, in progress

covariance

€

C =1

N2 pti ptj

pairs i≠ j

∑ − pt

2

Covariance Covariance Momentum FluxMomentum Flux

unrestricted sum:

€

rgdx1dx2 =∫ pti ptj∑ − N2

pt

2= pti

2∑ + N2C

€

rg = gt (x1)gt (x2) − gt (x1) gt (x2)

€

C = 1

N2 (rg − rg, eq )dx1dx2∫

∫∑ = 2121,all

dndnpppp ttji

tjti

€

= dx1dx2 dp1 pt1 f1∫( )∫ dp2 pt 2 f2∫( )

€

gt (x) = dp pt Δf x, p( )∫

€

dn = f x, p( )dpdx

correlation function:

C =0 in equilibrium

∫→ 2121 )()( dxdxxgxg

pptt Fluctuations Energy IndependentFluctuations Energy Independent

Au+Au, 5% most central collisions

STAR, Phys.Rev. C72 (2005) 044902

€

σ p t :n =1

Npti − pt( ) ptj − pt( )

i≠ j

∑ =N(N −1)

Nδpt1δpt 2

€

⇒ σ p t :n ≈ N δpt1δpt 2

€

Fp t

obs ∝N

obs

N=1− e− p t ,max /Teff 1+ pt,max /Teff( )

Flow ContributionFlow Contribution

0

1

2

3

4

0 0.5 1 1.5 2 2.5€

Fp t≈ N δpt1δpt 2 /2σ 2

σ 2 = pt2 pt 2

€

Fp t(%)

€

pt,max (GeV)

PHENIX

2/1

1

1⎟⎟⎠

⎞⎜⎜⎝

⎛

−+

≈r

reff v

vTT

explained by blueshift?explained by blueshift?

thermalization

flow added

dependence on dependence on maximummaximum pt, max

needed:needed: more realistic hydro + flow fluctuations more realistic hydro + flow fluctuations

want: ∫∑ −=−=≠

21,121 )(22 dxdxrrpppC eqggNt

jitjtiN

Covariance Covariance Momentum FluxMomentum Flux

STAR measures:

)()()()( 2121 xgxgxgxgr ttttg −=

two correlation functions:

€

σ p t :n =1

Npti − pt( ) ptj − pt( )

i≠ j

∑

€

=1

Npti ptj

i≠ j

∑ −N(N −1)

Npt

2

€

= N C − pt

2R( )

∫ −=−−

= 21,1

2

2

)()1(

2 dxdxrrN

NNNR eqnnN

)()()()( 2121 xnxnxnxnrn −=

where:

130 GeV

s1/2=200 GeV

blue-shift: • average increases

• enhances equilibrium contribution

€

∝Teff2 ∝

1+ vr

1− vr

thermalization

flow added

M. Abdel-Aziz & S.G.

participants

Thermalization Thermalization + Flow + Flow

20 GeV€

δpt1δpt 2 ∝ δT1δT2 r12∫

• blue-shift cancels in ratio

€

δpt1δpt 2

pt

2 ≈ constant

€

pt ∝Teff ∝1+ vr

1− vr

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

Rapidity Dependence of Momentum Rapidity Dependence of Momentum Fluctuations Fluctuations

momentum correlation function near midrapiditymomentum correlation function near midrapidity

€

rg − rg,eq ∝ e−yr2 / 2σ 2

e−ya2 / 2Σ2

• relative rapidity yr = y1-y2

Abdel-Aziz, S.Gfluctuationsfluctuations in rapidity window

• weak dependence on ya = (y1+y2)/2

€

C ∝ (rg − rg,eq ) dy1dy2Δ

∫∫

take σo ~ 0.5, fo 2

we want: 2

2

1t

jitjti ppp

NC −= ∑

≠

Uncertainty RangeUncertainty Range

STAR measures:

density correlation functiondensity correlation function rn rn r n,eq may differ from rg

maybe σn 2σ* STAR, PRC 66, 044904 (2006)

uncertainty range σ* σ 2σ* 0.08 s 0.3

€

N Δσ p t :n = pti − pt( ) ptj − pt( )i≠ j

∑

momentum density momentum density correlationscorrelations

€

= dx1dx2 Δrg (x1, x2) − pt

2Δrn (x1, x2)[ ]∫

density density correlationscorrelations

observable:observable:

€

C = pt1pt 2 − pt

2

Transverse Momentum CovarianceTransverse Momentum Covariance

€

pt1pt 2 ≡1

N2 pti ptj

pairs i≠ j

∑

€

pt ≡1

Npti∑

vanishes in equilibriumvanishes in equilibrium

€

N 2 − N2

= N ⇒

€

pti ptj

pairs i≠ j

∑eq

= N(N −1) pt

2= N

2pt

2

€

rg = gt (x1)gt (x2) − gt (x1) gt (x2)measures correlation functionmeasures correlation function

€

C = 1

N2 dx1dx2∫ rg − rg, eq( )

viscous diffusionviscous diffusion

V = 2ν/0

0 2 4 6

/0

€

∂gt

∂τ=

ν

τ 2

∂2gt

∂y 2

€

V = 2ν1

τ 0

−1

τ

⎛

⎝ ⎜

⎞

⎠ ⎟ V

random walk in rapidityrandom walk in rapidity y vs. proper time

€

V = y 2∫ gt dy

€

d

dτΔV =

2ν

τ 2,

kinematic viscosity ν Ts

formation at 0

Viscosity Broadens Rapidity Viscosity Broadens Rapidity DistributionDistribution