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
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