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On viscosity of Quark Gluon Plasma

Defu Hou

CCNU , Wuhan

RHIC-Star full TOF detector and related physics in China Hangzhou April 27-29

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Introduction and motivation Viscosity from Kubo formula Viscosity from kinetic theory (Boltzmann

Eq) Viscosity from AdS/CFT Summary

Outlines

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QCD under extreme conditions

At very High T or density ( deconfined) High T (Early universe, heavy-ion collisions) High density matter ( in the core of neutron stars)

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@ RHIC

Robust collective flows, well described by ideal hydro with Lattice-based EoS. This indicates very strong interaction even at early time => sQGP

sQGP seems to be the almost perfect fluid known /s>= .1-.2<<1

Motivations

Experiments aspect:

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Study of dissipative effects on <vStudy of dissipative effects on <v22>>

How sensitive is elliptic flow to finite /s?

Z. Xu & C. Greiner, PRL 101(08)

Agreement for s=0.3 – 0.6 /s=0.15 – 0.08

Viscous Hydro Cascade (2<->2,2<->3)

P. Romatschke, PRL99 (07)

Dependence on relaxation time

II0 order expansion with green terms (D. Rischke)

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Theoretic aspect:

• To calculate Trsp. Coefs. in FT in highly nontrivial

(nonperturbative ladder resummation) (c around 5)

• String theory method: AdS/CFT (D.Son et al 2003)

/s = 1/4

. Kinetic theory + uncertainty principle (Gyulassy)

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Main obstacle for theory

QCD in nonperturbative regime (T~200Mev) Pertburb. Expansion of QCD is not well behaved for realistic T

For thermodyn.,one can use lattice and resummation techniques Kinetic coefficients are difficult to extract from lattice

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y

v

A

F xx

Shear Viscosity

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10

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Viscosity from Kubo formula

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Nonlinear Response

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S. Jeon, PRD 52; Carrington, Hou, Kobes, PRD61

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Carrington, Hou, Kobes, PRD64 (2001)

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Hou, hep-ph/0501284

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Viscosity from kinetics theory

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Boltzmann Equation

Fluctuation of distribution (s: species)

Recast the Boltzmann equation

P.Arnold, G.D.Moore and G.Yaffe,

JHEP 0011(00)001

Viscosity of hot QCD at finite density

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Shear viscosity

With a definition of inner product and expanded distribution functions,

where

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Collision terms

Performing the integral over dk’ with the help of

Scattering amplitude

Distribution function term \chi term

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Matrix Element

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Variation method gives

Liu, Hou, Li EPJC 45(2006)

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Computing transport coefficients from AdS/CFT

In the regime described by a gravity dual the correlator can be computed using AdS/CFT

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AdS/CFT at finite temperature

Classical Supergravity on AdS-BH×S5

4dim. Large-Nc strongly coupledSU(Nc) N=4 SYM at finite temperature(in the deconfinement phase).

conjecture

=

Witten ‘98

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Field Theory Gravity Theory=

Gauge TheoriesQCD

Quantum GravityString theory

the large N limitSupersymmetric Yang Mills

Gravitational theory in 10 dimensions

N large

Calculations Correlation functionsQuark-antiquark potential

Holography

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AdS/CFT now being applied to RHIC physics

Viscosity, /s. EOS Jet quenching “Sound” waves Photon production Friction … Heavy quarkonium Hardron spectrum (ADS/QCD)

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Universality of shear viscosity in the regime described by gravity duals

Graviton’s component obeys equation for a minimally coupled massless scalar. But then .

Since the entropy (density) is we get

D. Son, P. Kovtun, A.S., hep-th/0405231

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Shear viscosity in SYM

Correction to : A.Buchel, J.Liu, A.S., hep-th/0406264

P.Arnold, G.Moore, L.Yaffe, 2001

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A viscosity bound conjecture

P.Kovtun, D.Son, A.S., hep-th/0309213, hep-th/0405231

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Universality of

Theorem:

For any thermal gauge theory (with zero chemicalpotential), the ratio of shear viscosity to entropy density is equal to in the regime describedby a corresponding dual gravity theory

Remark:

Gravity dual to QCD (if it exists at all) is currentlyunknown.

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Possible Mechanisms for Low viscosity Large cross-section, strong coupling

Anomalous viscosity: turbulence

M. Asakawa, S.A. Bass, B.M., hep-ph/0603092, PRLSee Abe & Niu (1980) for effect in EM plasmas

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Take moments of

with pz2( ) ( , , )r p p

p

pD p f r p t C f

t E

2

23

4

2

12

3

11

110

1 ln 1mc

c A C

g gO

T

g BNO

N sT

M. Asakawa, S.A. Bass, B.M., hep-ph/0603092

See Abe & Niu (1980) for effect in EM plasmas

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Low viscosity due to Anderson Local. AL effect renders infinite reduces viscosity

significantly even at weak coupling

Mechanism:coherent backscattering (CBS) effect

Ginaaki, Hou , Ren PRD 77(2008)

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Summary

Kubo formula: via correlation functions of currents

Transport theory: Boltzmann Eqs. (for weak scattering)

ADS/CFT(strongly coupled) Lattice calculation (noisy)

Approches to calculate viscosity

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Thanks

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Renormalized diffusion

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Weak Localization (WL)

Anderson proposed (‘58) that electronic diffusion can vanish in a random potential (AL)

Experiments detected ( Ishimaru 1984,Wolf Maret 1985) Mechanism:coherent backscattering (CBS) effect

after a wave is multiply scattered many times ， its phase coherence is preserved in the backscattering direction ， the probability of back scattering is enhenced via constructive interference

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Viscosity with random mediumSystem: quasi-particles in random potential

Candidate disorder in sQGP ?1. The islands of heavy state; bound states

(Shuryak); 2. The reminiscent of confinement vaccum,

say the domain structure of 't Hooft's monopole condensation;

3. The disoriented chiral condensate (DCC);4. CGC

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Response function

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BS Eq. In Diagrams

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Localization length

Itinerant states ---- Localized States

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II Some applications to N=4 SUSY YM Plasma:

Equation of state in strong coupling: Plasma temperature = Hawking temperature

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222

22

2

14

1 )1(

xdz

dtz

dds

zz

hh

h

Near Schwarzschild horizon

Continuating to Euclidean time, it

hhh zd

zd

zdds

2 , scoordinatepolar 2d

14 22

222

22 x

To avoid a conic singularity at 0 , the period of hz

Recalling the Matsubara formulation

hzT

1

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Free energy = temperature X (the gravity action without metric fluctuations) E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998), hep-th/9803131.

Consider a 4D Euclidean space of spatial volume V_3 atThe EH action of AdS-Schwarzschild:

z

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5

3

053

5

11

8)1220(

16

10

h

z

EH zG

V

z

dzdtV

GI

h

The EH action of plain AdS

4

5

3

053

5

)0( 1

8)1220(

16

10

G

V

z

dzdtV

GI EH

----- To eliminate the conic singularity,----- To match the proper length in Euclidean time

nz

00 2

1 )0(4

4

GHGHh

IIz

f

Plasma free energy:

342

2

45

3)0(

0 81600lim

1VTN

zG

VIIF c

hEHEH

Plasma entropy:

332

2

23

VTNT

FS c

V

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Bekenstein-Hawking entropy:

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areahorizon

4

1unitsPlanck in measured area)(horizon

4

1

PBH l

S

8

1

length Planck d10 where GlP

------ The metric on the horizon ：

3365

33

25

22

2

) of angle solid the( areahorizon The

1

VTSz

V

ddz

ds

h

h

x

------ The gravitational constant of the dual: 2

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10 2 cP N

lG

plasmacBH SVTNS 3322

2

1

agree with the entropy extraced from the gravity action.

Gubser, Klebanov & Pest, PRD54, 3915 (1996)

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The ratio 3/4:

The plasma entropy density at and cN322

3 2

1/ TNVSs c

The free field limit:

322322

30

7

240

78 TNTN cc

the contents of N=4 SUSY YM number entropy density

gauge potential 1

real scalars 6

Weyl spinors 4

322322

5

1

30

16 TNTN cc

322

30

1TN c

222)0(

3

2TNs c

The lattice QCD yields

75.04

3)0(

s

s

.8.00

s

s

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Shear viscosity in strong coupling:

Kubo formula

Policastro, Son and Starinets, JHEP09, 043 (2002)

where

)0(),()(),(

)0,(Im1

lim

,

,0

xyxyxitiR

xyxy

Rxyxy

TxTtedtdG

G

qxq

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Gravity dual: the coefficient of term of the gravity action

2xyh

22

2222

2

4

1du

fudfdt

u

Tds x

10 1 22

2

uufz

zu

h

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The metric fluctuation

Substituting into Einstein equation

04 gR and linearize

The Laplace equation of a scalar field

dxdxuzthdu

fudfdt

u

Tds ),,(

4

1 22

2222

2 x

xyxy h

T

uh

xg

xg 22 where 0

1

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Calculation details:

zyxjiuuf

f

f

f

ufTuT

ijiuj

tut

uuuij

uij

utt

,,, 2

1

1

2

1

2

2

1 2 12 22422

------ Nonzero components of the Christofel (up to symmetris):

fuR

u

TRf

u

TR uuijijtt 2

2222 1

4

4

------ Nonzero components of the Ricci tensor:

uyxu

xyuz

yxz

xyz

yxt

xyt

uuxyz

zxy

txy ufT

f

,2

1 ,

2

1

2

1

,2 ,2

1

2

1

:)symmetries to(up components nonzero with

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Linear expansion:

4,2,2

1

component nonzeroonly the with

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uzzyx

u

f

uuu

f

u

Tr

rRR

x

ggxgu

f

uuu

f

u

Thr uzzy

xy

x

2

1,2,

2

14 3

2

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The solution:

Heun equation (Fucks equation of 4 canonical singularities)------trivial when energy and momentum equatl to zero;------low energy-momentum solution can be obtained perturbatively.

The boundary condition at horizon: 1u

correlator advanced waveoutgoing )-(1

correlator retarded waveincoming )1(~),,(

)(ˆ

2

)(ˆ

2

tqzii

tqzii

eu

euuzt

The incoming solution at low energy and zero momentum:

tii

eOu

iuuzt

)ˆ(

2

1ln

2

ˆ1)1(),,( 2

ˆ2

T

qq

Teuuuuzt tqzi

i

2ˆ

2ˆ where)()1()1(),,( )(

ˆ2

1ˆ

2

04

ˆˆ

2

1ˆˆˆ11ˆ)1(1)1(

2222

2

22

uii

qdu

duiui

du

duu

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32, 8

)0,( TNi

G cR

xyxy

32

8

1TN c

Viscosity ratio: 08.04

1

s

Elliptic flow of RHIC:

Lattice QCD: noisy

1.0s

V_4 = 4d spacetime volume

)0,(2

1

16

lim8

1

8

1

)()( of termquadratic The

,432

4

0422

4

1

0

24422

Rxyxyc

ucc

GHEH

GVTNVi

uu

fTNV

uu

fxdduTN

II

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III. Remarks:N=4 SYM is not QCD, since1). It is supersymmetric2). It is conformal ( no confinement )3). No fundamental quarks---- 1) and 2) may not be serious issues since sQGP is in the deconfined phase at a nonzero temperature. The supersymmetry of N=4 SYM is broken at a nonzero T.---- 3) may be improved, since heavy fundamental quarks may be introduced by adding D7 branes. ( Krach & Katz)

Introducing an infrared cutoff ---- AdS/QCD:

2222

2

2

4

5

1

fielddilaton thewhere

1216

1

dzddtz

ds

cz

RegxddzG

I EH

x

----- Regge behavior of meson spectrum ---- confinement;----- Rho messon mass gives ----- Lack of string theory support.

MeV; 338c

Karch, Katz, Son & Stephenov

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Deconfinement phase transition: Herzog, PRL98, 091601 (2007)

Hadronic phase:

Plasma phase:

2222

2

4

5hadronic

1with

1216

1 2

dzddtz

ds

RegxddzG

I czEH

x

dzzTddtzTz

ds

RegxddzG

I czEH

1444224442

2

4

5plasma

111

with

1216

1 2

x

Hawking-Page transition:

---- First order transition with entropy jump

MeV1914917.0

plasmahadronic

cT

II

c

EHEH

2cN

---- Consistent with large N_c QCD because of the liberation of quark-gluon degrees of freedom.

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Epilogue

AdS/CFT gives insights into physics of thermal gauge theories in the nonperturbative regime

Generic hydrodynamic predictions can be used to check validity of AdS/CFT

General algorithm exists to compute transport coefficients and the speed of sound in any gravity dual

Model-independent statements can presumably be checked experimentally

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Mechanisms for Low viscosity Large cross-section, strong coupling

Anomalous viscosity: turbulence

Soft color fields generate anomalous transport coefficients, which may give the medium the character of a nearly perfect fluid even at moderately weak coupling

M. Asakawa, S.A. Bass, B.M., hep-ph/0603092, PRLSee Abe & Niu (1980) for effect in EM plasmas