energy-momentum tensor correlators and viscosity · 2008. 7. 22. · energy-momentum tensor...

Energy-momentum tensor correlatorsand viscosity

Harvey Meyer

The XXVI International Symposium on Lattice Field TheoryCollege of William & Mary, 18 July 2008

Phys.Rev.D76:101701,2007; Phys.Rev.Lett.100:162001,2008arXiv:0805.4567; arXiv:0806.3914

Harvey Meyer Energy-momentum tensor correlators and viscosity

Phase diagram of QCD

liq

T

µ

gas

QGP

CFL

nuclearsuperfluid

heavy ioncollider

neutron star

non−CFLhadronic

Alford, Schmitt, Rajagopal, Schäfer 0709.4635


Heavy Ion collisions at RHIC


Heavy ion collisions and elliptic flow

��

��

��

��

��

��

�� −pp

~13fm

1.2fmtransv. section of thecollision of two gold nuclei

impact vector ~b determinedexperimentally by φdistribution of particles

pressure gradient isgreater in the (~b, z) plane

⇒ excess of particlesproduced in that plane


Hydrodynamics, ideal and viscous

0 1 2 3 4p

T [GeV]

0

5

10

15

20

25

v 2 (p

erce

nt)

STAR non-flow corrected (est.)STAR event-plane

Glauber

η/s=10-4

η/s=0.08

η/s=0.16

0 1 2 3 4p

T [GeV]

0

5

10

15

20

25

v 2 (p

erce

nt)

STAR non-flow corrected (est).STAR event-plane

CGCη/s=10

-4

η/s=0.08

η/s=0.16

η/s=0.24

Teaney ’03; P. & U. Romatschke, ’07; Luzum, Romatschke, ’08;Heinz, Song ’07; Dusling, Teaney ’07

anisotropic flow incompatible with η/s & 0.2


Shear viscosity of ordinary substances

1 10 100 1000T, K

0

50

100

150

200

Helium 0.1MPaNitrogen 10MPaWater 100MPa

Viscosity bound

4π ηsh

from Kovtun, Son, Starinets PRL 94:111601,2005


Shear viscosity of a dilute gas

��

��

��L

F

S

V

F/S = ηV/L

the molecules flying through ahorizontal slice have a longitudinalmomentum characteristic of theirsurface of last scattering

those moving ↓ have greater longit.momentum than those moving ↑⇒ transfer of longit. momentum fromtop to bottom

η = 13

pσ , σ = cross-section

η is independent of the density for a dilute gas

bulk viscosity: ζ ∝ density (requires 3-body collisions).


Pure gauge theory vs. full QCD in LO PT

G. Moore, SEWM 04


Modes of linearized hydrodynamics

Macroscopic form of the energy-momentum tensor:

T µν = −Pgµν + (ε + P)uµuν + ∆T µν

∆T µν = η(∆µuν + ∆νuµ) + (23η − ζ)Hµν∂ρuρ

uµ is the velocity of energy-transportη, ζ = shear and bulk viscosities respectivelyHµν = uµuν − gµν , ∆µ = ∂µ − uµuβ∂β


Kubo formula for the shear viscosity

couple T0i (x) to a small external velocity field vi (x)

switch off the external field⇒ the system relaxes to equilibrium

the Kubo formula is a matching condition between the linearizedhydro. description and the linear response formalism of the QFT:

η = − limω→01ω Im DR

12,12(ω,0), DR the retarded correlator of T12

Dispersion relation and analytic continuation: (Karsch, Wyld, ’86)

DR12,12(k) = −

∫dω ρ(ω)

ω−k0−iε , Im DR12,12(k) = −πρ(k)

D12,12(ωn,k) =∫

dω ρ(ω,k)ω+iωn

(the Euclidean correlator)

D12,12(τ,k) =∫∞

0 dω coshω(L0/2−τ)sinhωL0/2 ρ(ω,k), η = π limω→0 ρ(ω,0)/ω


The QCD energy-momentum tensor

Separating the traceless part θµν from the trace part S forgluons, denoted ‘g’, and quarks, denoted ‘f’,

Tµν ≡ θgµν + θf

µν + 14δµν(θg + θf),

θgµν = 1

4δµνF aρσF a

ρσ − F aµαF a

να,

θfµν = 1

4∑

f ψf↔Dµ γνψf + ψf

↔Dν γµψf − 1

2δµνψf↔Dρ γρψf ,

θg = β(g)/(2g) F aρσF a

ρσ, θf = [1 + γm(g)]∑

f ψf mψf

↔Dµ=

→Dµ −

←Dµ

β(g) is the beta-functionγm(g) is the anomalous dimension of the mass operatorall expressions are written in Euclidean space.


Tµν on the lattice

Thermodynamics

0.0

0.2

0.4

0.6

0.8

1.0

1.0 1.5 2.0 2.5 3.0 3.5 4.0

T/Tc

p/pSB

3 flavor2 flavor

pure gauge

Karsch, Hard Probes ’06

T00 = θ00 + 14θ,{

ε− 3P = 〈 θ 〉T − 〈 θ 〉0,ε+ P = 4

3〈 θ00 〉T

Renorm. factor, plaq. discretization

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2

Normalization of the traceless energy-momentum tensor

Z(g0)

g02

1-loop [karsch ’82]

data [karsch et al. ’99]

non-pert. fit

(HM, ’07)

0

5

10

15

20

25

30

bare-plaq hyp-plaq bare-clover hyp-clover

100

2.5

14.6

1.2

<Q2>/<Q>2-1

Relative variance

of other discretizations

(HM, Negele ’07)


Cutoff effects in lattice perturbation theory

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

C(N

τ) /

Cco

nt -

1

(1 / Nτ)2

x0 = L0 / 2

Nτ=4Nτ=6Nτ=8

∇ → 14 〈(T11−T22)T11−T22)〉plaq

�→ 14 〈(T11−T22)T11−T22)〉clov

⊗ → 〈T12T12〉clover

clover & plaquette discretizations have comparable cutoff effects

Nτ ≥ 8 is mandatory

use these results to remove treelevel cutoff effects on non-pert. correlators.


Perturbative cutoff effects: anisotropic lattice

-0.1

0

0.1

0.2

0 0.2 0.4 0.6 0.8 1

C(N

τ) /

Cco

nt -

1

(1 / ξ)2

x0 = L0 / 2

1 / (Taσ) = 8

1 / (Taσ) = 16

ξ= 1ξ= 2ξ= 3

�→ 14 〈(T11−T22)T11−T22)〉clov

⊗ → 〈T12T12〉clover

for fixed L0/aσ , ξ = 2 or 3 is optimal

(Nτ = 16, ξ = 2) is as good as (Nτ = 16, ξ = 1) and saves a factor 8in the spatial volume.


The finite T Euclidean correlators (Nτ = 8, ξ = 1)

〈T12T12〉c

10

100

0.2 0.3 0.4 0.5

Cs(

x 0)

T x0

1.02Tc, LT=3

1.24Tc,LT=5/2

1.65Tc, LT=5

2.22Tc, LT=6

3.22Tc, LT=6

〈θ θ〉c

1

10

100

0.2 0.3 0.4 0.5

Cθ(

x 0)

T x0

1.02Tc, LT=31.24Tc,LT=5/2

1.65Tc, LT=52.22Tc, LT=63.22Tc, LT=6

near-conformal behaviour in one channel,large deviations in the othersmall errors thanks to multi-level algorithm (HM ’03)


Taking the continuum limit

〈T12T12〉c 〈θ θ〉c

0

2

4

6

8

10

12

0 0.005 0.01 0.015 0.02 0.025

Nτ=6Nτ=8Nτ=10Nτ=12

n=0 1.65n=0 1.24

0

5

10

15

20

25

0 0.005 0.01 0.015 0.02 0.025

(a/L0)2

n=0 1.65n=0 1.24n=1 1.65n=1 1.24

〈ω2n〉 ≡∫ ∞

0dωω2n ρ(ω)

sinhω/2T= T 5 d2nC

dx2n0

∣∣∣∣x0=L0/2

. . . tree-level improvement works well.


Reconstructing the dinosaur


Expected form of the spectral functions

/GeVω

ρ(ω,Τ=0)

1

ω4

/GeVω1

4ω

ρ(ω,T=2Tc)

/GeVω

ρ(ω,Τ=οο)

1

ω4

Free theory: ρµν,ρσ(ω)↔ 〈TµνTρσ〉

ρ12,12(ω,T = 0) = dA10(4π)2ω

4

ρθ,θ(ω,T = 0) =( 11αsNc

6π

)2 dA4(4π)2ω

4

ρ12,12(ω,T ) = dA10(4π)2

ω4

tanh ω4T

+( 2π

15

)2dAT 4 ωδ(ω)

ρθ,θ(ω,T ) = dA4(4π)2

( 11αsNc6π

)2 ω4

tanh ω4T

.


The reconstructed spectral functions (2007)

ρ(ω) = m(ω)[1 +∑N`=1 c`u`(ω)], m(ω � T ) as predicted by PT

〈T12T12〉c (Nτ = 8, ξ = 1)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25

ρ(ω) K(x0=1/2T,ω)/T4

ω/TT=1.24Tc

T=1.65Tc

〈θ θ〉c (Nτ = 12, ξ = 1)

-0.5

0

0.5

1

1.5

2

2.5

0 5 10 15 20 25

ρθ(ω)/(T4sinh(ω/2T))

ω/T

T=1.24Tc

T=1.65Tc

η/T 3 = π2 × intercept

black curve = normalizedN = 4 SYM spectral function.

ζ/T 3 = ( π18 × intercept)increasing for T → Tc

black curve = δ(0, ω)


2007 Results

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1 1.5 2 2.5 3 3.5 4

T / Tc

LHC

RHIC

1/(4π)

η/s [0704.1801]ζ/s [0710.3717]

Perturbative and AdS/CFT calculations:

η/s, ζ/s =

{0.484

π2α2s log(0.608/αs)

,1.25α2

sπ2 log(4.06/αs)

Nf = 0 PT1/(4π) , 0 N = 4 SYM.

Arnold, Moore, Yaffe ’03; Arnold, Dogan, Moore ’06;

Policastro, Son, Starinets ’01; Kovtun, Son, Starinets ’04


The 〈θθ〉 correlator at T = Tc(1± ε) in SU(2)

Hübner, Karsch, Pica; see C. Pica’s talk

0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1

T/Tc

0

10

20

30

40

50

GΘ

Θ/T

5

τΤ=1/2τΤ=1/4

0.95 1 1.05 1.1 1.15

T/Tc

0

10

20

30

Gxx

/T5

GΘΘGεεGεp

Gpp

beautiful confirmation of the 3d Ising class critical behavior

what happens to ζ at the phase transition is still an openquestion.


Coming back to the 〈T12T12〉 correlator

10

100

0.2 0.3 0.4 0.5

Cs(

x 0)

T x0

1.02Tc, LT=3

1.24Tc,LT=5/2

1.65Tc, LT=5

2.22Tc, LT=6

3.22Tc, LT=6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1 1.5 2 2.5 3 3.5

<ω2n> / <ω2n>free

T/Tc

n=0

n=1

AdS/CFT

AdS/CFT

〈ω2n〉 ≡∫ ∞

0dωω2n ρ(ω)

sinhω/2T= T 5 d2nC

dx2n0

∣∣∣∣x0=L0/2

due to the large UV contribution, the Euclidean correlator is notvery sensitive to interactions. . .


Universal properties of spectral functions

Ward Identities: if q = (0,0,q), ∂µTµν = 0⇒

ρ00,00(ω,q) =q4

ω4 ρ33,33(ω,q) ⇒ ρ00,00(ω,q)ω→∞∼ q4

ρ01,01(ω,q) =q2

ω2 ρ13,13(ω,q) ⇒ ρ01,01(ω,q)ω→∞∼ q2ω2.

Hydrodynamics prediction at small (ω,q):

shear channel:ρ01,01(ω,q)

ω

ω,q→0∼ η

π

q2

ω2 + (ηq2/(ε+ P))2 ,

sound channel:ρ00,00(ω,q)

ω

ω,q→0∼ Γs

π

(ε+ P) q4

(ω2 − v2s q2)2 + (Γsωq2)2

,

Γs =43η + ζ

ε+ P=

43η + ζ

Tsfor a derivation see D. Teaney, 2006


Sound channel spectral function ρ00,00(ω,q)

0

0.5

1

1.5

2

2.5

3

3.5

0 1 2 3 4 5 6

ρ(ω

) / (

dA T

4 tanh

(ω/ 2

T)

)

ω / T

Hydrodynamics spectral function ρ00,00(ω)

hydro q=0.3 (2πT)

hydro q=0.5 (2πT)

0.5 1 1.5 w

2

4

6

8

10

strongly coupled N = 4 SYM2ρ

πdAT 4 for q/(2πT ) = 0.3, 0.6, 1.0, 1.5

(Kovtun, Starinets 2006)

← Hydro. spectral function:v2

s = 13 , s = 3

4 sSB , T Γs = 13π

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 1 2 3 4 5 6

ρ(ω

) / (

dA T

4 tanh

(ω/ 2

T)

)

ω / T

Free spectral function ρ00,00(ω)

free q=0.5 2πT

free q=0.3 2πT

SU(Nc) gauge theory (HM, 2008)


Asymptotic behavior of ρ00,00(ω)

small ω :

large ω :

asymptotic behavior2π

tanh( ω2T ) Γs (ε+P)q4

(ω2−v2s q2)2+(Γsωq2)2

2dAq4

15(4π)2 tanh ω2T


Ansatz

ρ(ω,q,T ) =

2π


(ω2−v2s q2)2+(Γsωq2)2 · (1− tanh2 ω/2T )

+ 2dAq4

15(4π)2 tanh ω2T · tanh2 ω/2T


Ansatz: remarks

ρ(ω,q,T ) =

2π


(ω2−v2s q2)2+(Γsωq2)2 · (1− tanh2 ω/2T )

+ 2dAq4

15(4π)2 tanh ω2T · tanh2 ω/2T

thermodynamics known⇒ only one free parameter, Γs (!)

the tanh2 are arbitrary, but for small enough q, it will not matter

caveat: the behavior for ω ≈ ΛMS ≈ T is not known

we know that the sound peak dominates the Euclideancorrelator C(x0 ≈ 1

2T ,q) for sufficiently small q

it is assumed that ddq2 C( 1

2T ,q = 0) is also dominated by thesound peak rather than the ω ≈ T region.

it is at least true at weak coupling and in strongly coupled SYM.


Energy density two-point function

0.6 0.8

1 1.2 1.4 1.6 1.8

2 2.2 2.4 2.6 2.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

( q/ 2πT )2

C( x0=1/(2T), q ) / dA

s / (

d AT

3 v s2 )

4π2/15 T=1.27 Tc

T=1.49 Tc

T=2.52 Tc

free theory


Sensitivity to Γs

1.6

1.8

2

2.2

2.4

2.6

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

( q/ 2πT )2

C( x0= 1/ (2T), q ) / dA

free theoryTΓs=5.0TΓs=2.0TΓs=0.8TΓs=0.4TΓs=0.2TΓs=0.1

TΓs=0.05


Fitting the small momenta (T = 1.27Tc ,12× 483, ξ = 2)

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

( q/ 2πT )2

C( x0= 1/ (2T), q ) / dA

T=1.27 Tc

fit: TΓs=0.15(10)

TΓs=0.35

TΓs=0.05

used v2s = 0.263(18) [Boyd. et al ’96, CPPACS ’01], s

T 3 = 4.54(14)


Conclusions

1 the observed strong dynamics at RHIC calls for atheoretical understanding of transport coefficientsbeyond PQCD⇒ lattice QCD

2 subpercent-level precision gluonic correlators have beenachieved thanks to multi-level algorithm, variance-reduceddiscretizations of Tµν , treelevel improvement andanisotropic lattices

3 new method based on the energy density correlator at0 < q� 2πT (sound channel)⇒the functional form of ρ is largely known a priori

4 preliminary result: ΓsT =43η+ζ

s = 0.15(10) at 1.3Tc

5 hopefully soon: prediction for the LHC.


Backup slides


The “inverse problem”: solving for ρ(ω)

Given Ci =∫∞

0 dωρ(ω) coshωL0τi/2sinhωL0/2 (i = 1, . . .N), reconstruct ρ(ω).

our estimator for ρ: bρ(ω) = m(ω)[1 + ba(ω)], ba(ω) =PN`=1 c`u`(ω)

m(ω) > 0; m(ω � T ) has the correct perturbative behavior

basis u` determined by singular-value decomposition ofM(x0, ω)

def= K (x0, ω)m(ω): M t = UwV t, u`(ω) =columns of U

we are only able to reconstruct a ’fudged’ version of the genuine a(ω):

ba(ω) =R bδ(ω, ω′)a(ω′)dω′ bδ(ω, ω′) =

PN`=1 u`(ω)u`(ω′).

bδ(ω, ω′) is called the resolution function (how complete is the basis?)

we would like the resolution function to resemble a delta-function.

Backus & Gilbert, (geophysics, 1968)


Spectral functions at finite q in free theory HM, 2008

1d A

T4

[ρ(ω,q,T

)/ta

nh(ω/2T

)−ρ

(ω,q,0

)]

0

0.05

0.1

0.15

0.2

0.25

0.3

0 5 10 15 20 25

ω / T

Perturbative thermal spectral functions

q=(0,0,πT)

θ,θ12,12

13,13

ρθ,θ(ω,q,T = 0) =(

11αsNc6π

)2 dAθ(ω−q)4(4π)2 (ω2 − q2)2


Correlators in the confined phase

10

100

1000

10000

100000

0 0.1 0.2 0.3 0.4 0.5

Tx0

Treelevel Improved Correlators at T=Tc/2 (β=6.2, 204)

< T12 T12 >< θ θ >

2++ contrib.0+++0++* contrib.

in the scalar channel, the two stable glueballs almost saturatethe correlator beyond 0.5fm

calculation made possible by the multi-level algorithm (HM ’03)


Correlators in the confined phase

10

100

1000

10000

100000

0 0.1 0.2 0.3 0.4 0.5

Tx0

Treelevel Improved Correlators at T=Tc/2 (204, 284)

< T12 T12 >< θ θ >

< T12 T12 >< θ θ >


in the scalar channel, the two stable glueballs almost saturatethe correlator beyond 0.5fm

calculation made possible by the multi-level algorithm (HM ’03)


Spectral functions in free theory: analytic expressions

For a polynomial P, define

I([P], ω, q,T ) = θ(ω−q)

Z 1

0dz

P(z) sinh ω2T

cosh ω2T − cosh qz

2T

+θ(q−ω)

Z ∞1

dzP(z) sinh ω

2T

cosh qz2T − cosh ω

2T

.

Then, for instance,

ρθ,θ(ω,q,T ) =

(11αsNc

6π

)2 dA

4 (4π)2 (ω2 − q2)2 I([1], ω, q,T ) ,

I([1], ω, q,T ) = −ωqθ(q − ω) +

2Tq

logsinh(ω + q)/4Tsinh |ω − q|/4T

in all other channels, spectral function is a linearcombination of polylogarithms

HM, 2008


Lattice perturbation theoryThe “clover” discretization of Tµν :

θ00(x) =1g2

0Re Tr

{Zdτ (g0, ξ0)

∑k

F0k (x)2 − Zdσ(g0, ξ0)∑k<l

Fkl (x)2}.

Perturbation theory on the anisotropic lattice:

a3σ

Xy

eiq·y 〈 bF aµν(0)bF a

ρσ(0) bF bαβ(y)bF b

γδ(y) 〉0

=dA

a5σ

Z π

−π

d3p(2π)3

hφµναβ(p, τ)φρσγδ(p + q, τ) + φµνγδ(p, τ)φρσαβ(p + q, τ)

i,

where

φµναβ(p, τ) ≡ ξ0

Z π

−π

dp0

2πeip0τ

ξ20 p2

0 + p2×h

δνβξδµ+δα0 cos2(pν/2) sin pµ sin pα + δµαξ

δν+δβ0 cos2(pµ/2) sin pν sin pβ

−δµβξδν+δα0 cos2(pµ/2) sin pν sin pα − δναξ

δµ+δβ0 cos2(pν/2) sin pµ sin pβ

i.


Spectrum of the pure gauge theory

0

1

2

3

4

m /

[2π

σ]1/

2

PC

1 GeV

2 GeV

3 GeV

4 GeV

++ -+ +- --

4D SU(3)Glueballs

J=0J=1J=2J=3J=4J=5J=6

HM, Ph.D. thesis 04


Expected form of the spectral function for η

/GeVω

ρ(ω,Τ=0)

1

ω4

/GeVω1

4ω

ρ(ω,T=2Tc)

/GeVω

ρ(ω,Τ=οο)

1

ω4

0

0.05

0.1

0.15

0.2

0 1 2 3 4 5

ρ(ω)/(π2Nc2T4)

ω/T

AdS/CFT

free

at T � Tc , perturbation theorypredicts a peak at ω = 0 of widthO(αsNT )

“strong coupling ↔ smoothspectral function”“weak coupling ↔ spikyspectral function”

in strongly coupled N = 4 SYM,ρ(ω) is very smooth

ρ(ω) for ζ: no peak at ω = 0⇒more favorable.


Strategy I, pictorially

∆ρ = ρ− ρT =0 + ρ1p

/GeVω1

T=2Tc)ρ(ω,∆

=

/GeVω1

4ω

ρ(ω,T=2Tc)

-

/GeVω

ρ(ω,Τ=0)

1

ω4 +

/GeVω1

ρ (ω)1p

1 compute ρT =0(ω)

2 compute the one-particlecontributions to ρ(ω), ρ1p

3 compute ∆C(x0,T ) ≡ C(x0,T )−RdωK (ωx0, ω/T )[ρ0(ω)− ρ1p(ω)]

4 ∆C =R

dωK (ωx0, ωL0)∆ρ(ω)

5 η = π∆ρ(ω)/ω|ω→0 10

100

1000

10000

100000

0 0.1 0.2 0.3 0.4 0.5

Tx0

Treelevel Improved Correlators at T=Tc/2 (β=6.2, 204)

< T12 T12 >< θ θ >



Subtracting ρ(ω,p 6= 0)

q = (0,0,q): ρ12,12(ω,q,T = 0) = dAθ(ω−q)10(4π)2 (ω2 − q2)2

exploit the linearity of the problemto cancel the ω4 contribution, e.g. via

C(x0,q = 0)− 43 C(x0,q) + 1

3 C(x0,2q)

this inverse problem is betterconditioned

if q = (0, 0, q) large enough,estimate ρ(ω,p)/ω|ω=0 in PT (orneglect it).

if q = (q, 0, 0), hydro predictsρ(ω,q)/ω|ω=0 = 0

N = 4 SYM P. Kovtun, A. Starinets ’06

1 2 3 w

-0.2

0.2

0.4

0.6

0.8

Finite-temperature part of the tensor spectral function(ρ(ω)− ρ0(ω))/ω, plotted in units of π2N2

c T 4 as afunction of ω/2πT . Different curves correspond to valuesof the momentum p/2πT equal to 0, 0.6, 1.0, and 1.5.


Differences of p 6= 0 correlators

-1

-0.5

0

0.5

1

1.5

2

2.5

3

0.2 0.3 0.4 0.5

T x0

Tensor correlators (10x203, ξ=2, clover discretization)

T=1.49Tc LT=4p=(0,0,p3), p3 = πT/2

Cs(x0, 0) - (4/3) Cs(x0, p) +(1/3)Cs(x0, 2p)

Cs(x0, 0) - Cs(x0, p)

Cs(x0, 0) / 100

s


Non-zero momentum correlators (anisotropic lattices)

〈T12 T12〉, 203 × 10, ξ = 2 〈θ θ〉, 163 × 12, ξ = 2


Summary

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1 1.5 2 2.5 3 3.5 4

T / Tc

LHC

RHIC

1/(4π)

η/s [0704.1801]ζ/s [0710.3717]

0.001

0.01

0.1

1

0.1 1

(ε-3P)/(ε+P)

1.02

1.24

1.652.22

T/Tc=3.22

LO PT

LO PT

1/(4π)

ζ/sη/s

Perturbative and AdS/CFT calculations:

η/s, ζ/s =

{0.484

π2α2s log(0.608/αs)

,1.25α2

sπ2 log(4.06/αs)

Nf = 0 PT1/(4π) , 0 N = 4 SYM.

Arnold, Moore, Yaffe ’03; Arnold, Dogan, Moore ’06;

Policastro, Son, Starinets ’01; Kovtun, Son, Starinets ’04


Universal properties of spectral functions

Ward Identities: q = (0,0,q)

ρ00,00(ω,q) =q4

ω4 ρ33,33(ω,q)

Hydrodynamics prediction at small (ω,q):

ρ01,01(ω,q)

ω

ω,q→0∼ η

π

q2

ω2 + (ηq2/(ε+ P))2 ,

ρ00,00(ω,q)

ω

ω,q→0∼ Γs(ε+ P)

π

q4

(ω2 − v2s q2)2 + (Γsωq2)2

,

Γs =43η + ζ

ε+ P=

43η + ζ

TsTeaney, 2006


energy-momentum tensor correlators and viscosity · 2008. 7. 22. · energy-momentum tensor...

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