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 datadata

S.G., M. Abdel-Aziz, Phys. Rev. Lett. 97, S.G., M. Abdel-Aziz, Phys. Rev. Lett. 97, 162302 (2006)162302 (2006)

IV. Rapidity + Azimuthal Correlation Data IV. Rapidity + Azimuthal Correlation Data -- in progress-- in progress

Measuring Viscosity at RHIC Measuring Viscosity at RHIC

Sean Gavin Sean Gavin Wayne State University Wayne State University

Measuring Viscosity at RHIC Measuring Viscosity at RHIC

Sean Gavin Sean Gavin Wayne State University Wayne State University

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; Dumitru et al.

Additional viscosity probes?

What does shear viscosity do? It resists shear flow.

Small viscosity or strong flow?

Transverse Flow FluctuationsTransverse 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

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 STAR measures rapidity width of pptt fluctuations 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

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

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

at Tc ~ 170 MeV

J.Phys. G32 (2006) L37

naively identifynaively identify σ* with σ

0.08 < s < 0.3

Current Data?Current Data?

STAR measures rapidity width of STAR measures rapidity width of pptt fluctuations 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

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

J.Phys. G32 (2006) L37

naively identifynaively identify σ* with σ (strictly, )

€

σ p t :n = N C + corrections

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

uncertainty range σ* σ 2σ*

Part of a Part of a Correlation Landscape Correlation Landscape

STAR

focus:focus: near side peak -- “soft ridge”near side peak -- “soft ridge”

behavior: transverse and elliptic flow, momentum conservation plus viscous diffusion

near side near side peak peak

-4 -2 0 2

S.G. in progress

Theory

€

φ=φ2 − φ1

Soft Ridge: Centrality DependenceSoft Ridge: Centrality Dependence

€

centrality, 2nbin /npart

STAR, J.Phys. G32 (2006) L37near side peaknear side peak

• pseudorapidity width σ σ

• azimuthal width σ

centrality dependence:path length

€

~ 2nbin /npart

trends: trends: • rapidity broadening (viscosity)• azimuthal narrowing

common explanation of trends? common explanation of trends?

find:find: σ requires radial and elliptic flow plus requires radial and elliptic flow plus viscosityviscosity

σ

σ

Flow Flow Azimuthal Azimuthal CorrelationsCorrelations

€

rv r ~ λ

r r

€

φ

€

⇒ σφ = φ2 ∝ λ−2(Σt2 −σ t

2 /4)−1

mean flow depends on position

blast wave

opening angle for each fluid element depends on r

€

φ ~ v th vr ~ (λr)−1

correlations:correlations:

momentum distribution:

€

f = e−γ (E−r p ⋅

r v ) /T

gaussian spatial r(x1, x2):

σt -- width in

t -- width in

€

rr =

r r t 2 −

r r t1

€

rR = (

r r t1 +

r r t 2) /2

€

= f p1,x1( ) f p2, x2( )r(x1, x2)x1 ,x2

∫∫

€

vr(r)

( ) 221 )singles(pairs, −=ppr

Measured Flow Constrains CorrelationsMeasured Flow Constrains Correlations

diffusion + flow for correlationsdiffusion + flow for correlations

€

∂gt

∂τ+

r v r ⋅

r ∇ tgt + gt

∂v t

∂r= ν∇ 2gt

€

t (τ ), σ t (τ )

constraints:constraints: flow velocity, radius from measured v2, pt vs. centrality

€

rv r = εx (τ ) xˆ x + εy (τ ) yˆ y radial plus elliptic radial plus elliptic

flow:flow: “eccentric” blast wave

0

0.1

0.2

0.3

0.4

0.5

0.6

0 100 200 300 4000

0.02

0.04

0.06

0.08

0.1

0 100 200 300 400

€

v2

€

pt

npart npart

STAR 200 GeV Au+Au PHENIX 200 GeV Au+Au

Heinz et al.

Rapidity and Azimuthal TrendsRapidity and Azimuthal Trends

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1 2 3 4 5 6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1 2 3 4 5 6

€

centrality, 2nbin /npart

rapidity width:rapidity width:• viscous broadening, s ~

• assume local equilibrium for all impact parameters

• fixes Fb

azimuthal width:azimuthal width:• flow dominates• viscosity effect tiny

agreement with data easier if

we ignore peripheral region

-- nonequilibrium?

STAR data, J.Phys. G32 (2006) L37

σ

σ

current correlation data compatible with current correlation data compatible with

comparable to other observables with different comparable to other observables with different uncertaintiesuncertainties

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

€

0.08 < η /s < 0.3

need viscosity info to test the perfect need viscosity info to test the perfect liquidliquid

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

• pptt covariance covariance measures these correlations measures these correlations

azimuthal and rapidity correlationsazimuthal and rapidity correlations

• flow + viscosity works! hydro OKflow + viscosity works! hydro OK

• relation of relation of structure to the jet- structure to the jet-related ridge? related ridge?

pptt Fluctuations Energy IndependentFluctuations Energy Independent

Au+Au, 5% most central collisions

sources of pt fluctuations: thermalization, flow, jets?

• central collisions thermalized

• energy independent bulk quantity jet contribution small

-4 -3 -2 -1 0 1 2 3 4

Azimuthal Correlations from FlowAzimuthal Correlations from Flow

€

−

€

€

φ=φ2 − φ1

€

−

€

€

φ

transverse flow:transverse flow: narrows angular correlations

• no flow σ σ vrel

elliptic flow:elliptic flow:

• v2 contribution

• STAR subtracted

viscous diffusion:viscous diffusion:

• increases spatial widths t and σt

σ t

2 σt2-1/2

momentum conservation:momentum conservation: sin ; subtracted Borghini, et al

Elliptic and Transverse FlowElliptic and Transverse Flow

flow observables:flow observables: fix x, y, and R vs. centrality

0

0.1

0.2

0.3

0.4

0.5

0.6

0 100 200 300 4000

0.02

0.04

0.06

0.08

0.1

0 100 200 300 400

€

v2

€

pt

npart npart

STAR 200 GeV Au+Au PHENIX 200 GeV Au+Au

€

rv r = εx xˆ x + εy yˆ y

transverse plus elliptic transverse plus elliptic flow:flow: “eccentric” blast wave

€

n( p)∝ f p,x( )ρ(x)∝ e−γ (E−r p ⋅

r v ) /T∫∫ e−r 2 / 2R 2

blast wave:blast wave:

elliptic flowelliptic flow transverse flowtransverse flow

Blast Wave Azimuthal Blast Wave Azimuthal CorrelationsCorrelations

€

r( p1, p2) = f p1,x1( ) f p2,x2( )r(x1,x2)x1 ,x2

∫∫

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1 2 3 4 5 6

€

centrality, 2nbin /npart

STAR 200 GeV Au+Au data

azimuthal trend : t system size σt constant

• roughly:

€

φ2 ∝ λ−2(Σt2 −σ t

2 /4)−1

gaussian spatial r(x1, x2):

σt -- width in

t -- width in

€

rr =

r r t 2 −

r r t1

€

rR = (

r r t1 +

r r t 2) /2

€

σφ

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

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

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()

Part of a Part of a Correlation Landscape Correlation Landscape

STAR, J.Phys. G32 (2006) L37

soft ridge:soft ridge: near side peak

superficially similar to ridge ridge seen with jets J. Putschke, STAR

but at soft pt

scales -- no jet

near side near side peak peak

€

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