1

Why is the Quark-Gluon Plasma a “Perfect” Liquid ?

Berndt Mueller – YITP / Duke

RIKEN Workshop8-9 July 2006Special

thanks to M. Asakawa and S.A.

Bass

2

Lecture I

What does Lattice QCD tell us about the QGP ?

What do RHIC experiments tell us about the QGP ?

What is a “perfect” fluid ?

What are the origins of viscosity ?

3

What does Lattice QCD tell us about the

QGP ?A: So far, a lot about thermodynamic properties and response to static probes, a little bit about

spectral functions, (almost) nothing about transport properties.

4

Lattice - EOS

RHIC

2 3

#d.o.f . :

45

2

s

T

2

30

F. Karsch et al.

Indication of weak or strong coupling?

5

Phase coexistence

Study of LQCD at fixed baryon density = B/VDe Forcrand & Kratochvila

hep-lat/0602024

6

Lattice - susceptibilities

R. Gavai & S. Gupta, hep-lat/0510044

2

ln ( , )XY iX Y

i i ii

Z T XY X Y

XS x s n

22

#XS

XS X SC

S S

pQGP

7

Heavy quark potentials

Effective coupling s(r,T)

Kaczmarek et al.

Color singlet potential

Kaczmarek et al.

Important for insight into medium effects on J/,

8

Lattice – spectral functions

Spectral functions via analytic continuation using the maximum entropy method: J/ etc.

(At present only for quenched QCD.)

Karsch et al.

Asakawa & Hatsuda

J/ may survive up to 1.5 – 2 Tc !

9

What do RHIC experiments tell us about

the QGP ?A: So far, a lot about transport properties, a little

bit about thermodynamic properties, and (almost) nothing about spectral functions and

response to static probes.

10

The real road to the QGP

STAR

…is the

Relativistic Heavy Ion Collider

11

Space-time picture

final2

( ) /( )

( ) /

( / )

dN dys

dV dy

dN dy

R

30

0

( 1 fm/c) 30/fm

or ( ) 275 MeV

in Au+Au (200 GeV)

s

T

Bjorken formula

eqPre-equil. phase

Liberation of saturated low-x

glue fields (CGC)

12

RHIC results

Important results from RHIC:

Chemical (flavor) and thermal equilibration Jet quenching = parton energy loss, high opacity Elliptic flow = early thermalization, low viscosity Collective flow pattern related to valence quarks Strong energy loss of c and b quarks Charmonium suppression not significantly increased

compared with lower (CERN) energies Photons unaffected by medium at high pT, medium

emission at low pT in agreement with models

13

0 vs. in Au+Au (vs. p+p)

No suppression for photons

Suppression of hadrons

Cross section in p+p coll’s

Area density of p+p coll’s

in A+A

Yield in A+A

2

2

/( )

/AA T

AA T

AA NN T

d N dp dyR p

T d dp dy

Without nuclear effects:

RAA = 1.

14Hard Probes 2006, June 15, 2006 – G. David, BNL

Photons from the medium

D’Enterria, Peressounko nucl-th/0503054 0 = 0.15 fm/c, T = 570 MeV

Turbide, Rapp, Gale PRC 69 014903 (2004)0 = 0.33 fm/c, T = 370 MeV

Experiment uses internal conversion of

15

qq

Radiative energy loss:

2/ TdE dx L k

Radiative energy loss

Scattering centers = color chargesq q

g

L

2 22

2T

2

ˆf

dq kq dq

dq

Density of scattering centers

Range of color forceScattering power of the QCD medium:

16

q-hat at RHIC

Pion gas

QGP

Cold nuclear matter

sQGP? ??

RHIC data

“Baier plot”

RHIC

Eskola et al.

17

Collision Geometry: Elliptic Flow

Elliptic flow (v2):

• Gradients of almond-shape surface will lead to preferential expansion in the reaction plane• Anisotropy of emission is quantified by 2nd Fourier coefficient of angular distribution: v2

prediction of fluid dynamics

Reaction plane

x

z

y

Bulk evolution described by relativistic fluid dynamics,

F.D assumes that the medium is in local thermal equilibrium,

but no details of how equilibrium was reached.

Input: (x,i), P(), (,etc.).

18

initial energy density distribution:

time evolution of the energy density:

spatial eccentricity

momentumanisotropy

Elliptic flow is created early

P. Kolb, J. Sollfrank and U.Heinz, PRC 62 (2000) 054909

Model calculations suggest that flow anisotropies are generated at the earliest stages of the expansion, on a timescale of ~ 5 fm/c if a QGP EoS is assumed.

19

v2(pT) vs. hydrodynamics

Mass splitting characteristic property of hydrodynamics

Failure of ideal hydrodynamics - tells us how hadrons form

20

Quark number scaling of v2

In the recombination regime, meson and baryon v2 can be obtained from the quark v2 :

2 2 2 2v22

v3

v3v Btt

q tM q tp ppp

qqq

qqT,,v

Emitting medium is composed of unconfined,

flowing quarks.

21

From QGP to hadrons

Full 3-d Hydrodynamics

QGP evolution

Cooper-Fryeformula

UrQMD

t fm/c

hadronic rescattering

Monte Carlo

Hadronization

TC TSW

Nonaka & Bass, nucl-th/0510038Hirano et al. nucl-th/0511046 Low (no)

viscosityHigh

viscosity

Agreement with data for QGP = 0

22

Bounds on from v2

D. Teaney 0/ /s s

Boost invariant hydro with T00 ~ 1 requires /s ~ 0.1.

N=4 SUSY YM theory (g2Nc 1):

/s = 1/4(Policastro, Son, Starinets).

Absolute lower bound on /s ?

/s = 1/4 implies f ≈ 0.3 d !QGP(T≈Tc) = sQGP ?

( )

0 with

( trace)P u

T

uT uu Pg

Relativistic viscous hydrodynamics:

23

Ultra-cold Fermi-Gas

• Li-atoms released from an optical trap (J. Thomas et al./Duke) exhibit elliptic flow analogous to that observed in relativistic heavy-ion collisions

24

What is an

“ideal” or “perfect”

liquid ?

25

Ideal gas vs. perfect liquid

An ideal gas is one that has strong enough interactions to reach thermal equilibrium (on a reasonable time scale), but weak enough interactions so that their effect on P(n,T) can be neglected. This ideal can be approached arbitraily by diluting the gas and

waiting very patiently (limit t first, then V ).

A perfect fluid is one that obeys the Euler equations, i.e. a fluid that has zero viscosity and infinite thermal conductivity. There is no presumption with regard to the equation of state. Even an imperfect fluid obeys the Euler equations in the limit of

negligible velocity, density, and temperature gradients.

26

What is viscosity ?

23

and are defined as coefficients in the

expansion of the stress tensor in gradients of the velocity fie

viscosity

ld

Shear b k

:

ul

ik i k i k k i ik ikik i kT u u P u uu u u u

13

tr

3

tr 2

Microscopically, is given by the rate of momentum transport:

Unitarity limit on cross sections suggests that has a lower bound:

3

4

12

f

pnp

p

p

27

Viscosity of plasmas

Shear viscosity of supercooled one-component plasma fluids:

*2

Pmn a

2e

aT

Ichimaru

Interaction measure

28

Lower bound on /s ?

Argument [Kovtun, Son & Starinets, PRL 94 (2005) 111601] based on duality between thermal QFT and string theory on curved background with D-dimensional black-brane metric, e.g.:

14 42 232 2 2 20 0

2 4 2 41

1 1ii

r rr Rds dt dx dr

R r r r

Kubo formula for shear viscosity:

3

0

1lim , , 0,0

2i t

xy xydt d x e T t x T

Dominated by absorption of (thermal) gravitons by the black hole:

3abs 0

(hori8

zon , , 0,0 area)i txy xy

Gdt d xe T t x T a

abs (0)Therefore: becau

16 16se

4 4

a s

Gs

GG

a

29

Lower bound on /s – ctd.

A heuristic argument for (/s)min is obtained using s ~ 4n :

1 13 12v

vf

fn p sn

But the uncertainty relation dictates that f (/n) , and thus:

12 4s s

(It is unclear whether this relation holds in the nonrelativistic domain, where s/n can be much larger than 4. But is obeyed by all known substances.)

For N=4 susy SU(Nc) Yang-Mills the bound is saturated at strong coupling:

3/ 22

135 (3)1

4 8 c

s

g N

30

Exploring strong coupling

Ability to perform analytical strong coupling calculations in N=4 susy SU(Nc) YM and success with have motivated other applications:

Equation of state [Gubser, Klebanov, Tseytlin, hep-th/9805156] Spectral densities [Teaney, hep-ph/0602044] Jet quenching parameter [Liu, Rajagopal, Wiedemann, hep-ph/0605178] Heavy quark energy loss [Herzog, Karch, Kovtun, Kozcaz, Yaffe,hep-

th/0605158] Heavy quark diffusion [Casalderrey-Solana, Teaney, hep-ph/0605199] Drag force on heavy quark [Gubser, hep-th/0605182 ] …and continuing!

31

Some results from duality

0 3/ 22

3/ 22

3/ 4 31/ 24 2 3

3/ 25 24

Equation of state:

Shear viscosity:

Jet

3 45 (3)

4 32

135 (3)1

4 8

1.7652...ˆ quenching parameter:

Heavy quark drag force:

12

c

c

c

c

g N

s

g N

q g N Tg N

dp

dt

1/ 22 2

2 c

pg N T

m

horizon

(3+1)-D world

r0

0

1r

T

32

Lecture II

Does “perfect” fluidity imply “strong coupling” ?

What is “anomalous” viscosity ?

Derivation of the anomalous viscosity

Manifestations of anomalous QGP transport processes

33

Today…

…we ask the question:

Is strong coupling really necessary for

small /s ?

34

What is viscosity ?

23

and are defined as coefficients in the

expansion of the stress tensor in gradients of the velocity fie

viscosity

ld

Shear b k

:

ul

ik i k i k k i ik ikik i kT u u P u uu u u u

31

tr3 2tr

2

tr 2

Micro

4

3

scopically, is given by the rate of momentum transport:

a small value of implies strong coupling

12

!?!

f

s

p pnp

p

p

35

Stellar accretion disks

“A complete theory of accretion disks requires a knowledge of the viscosity. Unfortunately, viscous transport processes are not well understood.

Molecular viscosity is so small that disk evolution due to this mechanism of angular momentum transport would be far too slow to be of interest. If the only source of viscosity was molecular, then ~ / ~ vT, where is the particle mean free path and vT is the thermal velocity. Values appropriate

for a disk around a newly formed star might be r ~ 1014 cm, n ~ 1015 cm-3, ~ 10-16 cm2, so that ~ 10 cm, and vT ~ 105 cm/s . The viscous accretion

time scale would then be r2/(12) > 1013  yr! Longer by a factor of 105 - 106 than the age conventionally ascribed to such disks. Clearly if viscous

accretion explains such objects, there must be an anomalous source of viscosity. The same conclusion holds for all the other astronomical objects

for which the action of accretion disks have been invoked.”

(From James Graham – Astronomy 202, UC Berkeley)http://grus.berkeley.edu/~jrg/ay202/lectures.html

The solution is: String theory?Unfortunately, NO.

36

Anomalous viscosity

B

Differentially rotating disc with weak

magnetic field B shows an instability

(Chandrasekhar)

2Magnetic tension /(4 )T B

Spontaneous angular momentum transfer from inner mass to outer mass is amplified by interaction with the rotating disk and leads to instability (Balbus & Hawley – 1991).

“Anomalous”, i.e. non-

collisional viscosity

37

Anomalous viscosity:

A ubiquitous phenomenon

38

Anomalous viscosity on the WWW

Chaotic Dynamics, abstract chao-dyn/9509002

Anomalous Viscosity, Resistivity, and Thermal Diffusivity of the Solar Wind Plasma

Authors: Mahendra K. Verma (IIT Kanpur, India)

In this paper we have estimated typical anomalous viscosity, resistivity, and thermal difffusivity of the solar wind plasma. Since the solar wind is collsionless plasma, we have

assumed that the dissipation in the solar wind occurs at proton gyro radius through wave-particle interactions. Using this dissipation length-scale and the dissipation rates

calculated using MHD turbulence phenomenology [Verma et al., 1995a], we estimate the viscosity and proton thermal diffusivity. The resistivity and electron's thermal diffusivity have also been estimated. We find that all our transport quantities are several orders of

magnitude higher than those calculated earlier using classical transport theories of Braginskii. In this paper we have also estimated the eddy turbulent viscosity.

Google search:

Results 1 - 10 of about 322,000 for anomalous viscosity. (0.22 seconds) 

39

Anomalous viscosity - origins

40

Anomalous viscosity - usage

Plasma physics: A.V. = large viscosity induced in nearly collisionless plasmas by

long-range fields generated by plasma instabilities.

Astrophysics - dynamics of accretion disks: A.V. = large viscosity induced in weakly magnetized, ionized

stellar accretion disks by orbital instabilities.

Biophysics: A.V. = The viscous behaviour of nonhomogenous fluids or

suspensions, e.g., blood, in which the apparent viscosity increases as flow or shear rate decreases toward zero. (From: http://www.biology-online.org/dictionary)

41

Can the QGP viscosity be anomalous?

Can the extreme opaqueness of the QGP (seen in experiments) be explained without invoking super-strong coupling ?

Answer may lie in the peculiar properties of turbulent plasmas.

Plasma “turbulence” = random, nonthermal excitation of coherent field modes with power spectrum similar to the vorticity spectrum in a turbulent fluid [P(k) ~ 1/k2].

Plasma turbulence arises naturally in plasmas with an anisotropic momentum distribution (Weibel-type instabilities).

Expanding plasmas (such as the QGP at RHIC) always have anisotropic momentum distributions.

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

42

QGP viscosity – collisions

13

1( )tr

2tr 2

Classical expression for shear viscosity:

Collisional mean free path in medium:

Transport cross section in QCD medium:

Collisional shear

5

11

viscos

2 l

i

2

t

n 1

f

Cf

s s

s ss

np

n

Ip

I

2 1tr

9

y of QGP

100 ln

:

Cs s

T s

Baym, Heiselberg, ….

Danielewicz & Gyulassy, Phys. Rev. D31, 53 (85)

43

QGP viscosity – anomalous

p

paB

mr

13

2 2( )

2 2 2 2

Classical expression for shear viscosity:

Momentum change in one coherent domain:

Anomalous mean free path in medium:

Anomalous viscosity due to random color fie

f

a am

Af m

m

np

p gQ B r

p pr

g Q B rp

33 94

2 2 2 2 2 2

lds:

3A

m m

sTnp

g Q B r g Q B r

44

Color instabilitiesSpontaneous generation of color fields requires infrared instabilities. Unstable modes in plasmas occur generally when the momentum distribution of a plasma is anisotropic (Weibel instabilities – 1959).

22 22 1forx y s z

s

p p Q pQ

pz

py

px

beam

Such conditions are satisfied in HI collisions:

Longitudinal expansion locally “red-shifts” the longitudinal momentum components of released small-x gluon fields (CGC) from initial state:

In EM case, instabilities saturate due to effect on charged particles. In YM case, field nonlinearities lead to saturation (competition with Nielsen-Olesen instability?)

45

Weibel (two-stream) instability

B

v

v

v v

46

HTL formalism

Nonabelian Vlasov equations describe interaction of “hard” and “soft” color field modes and generate the “hard-thermal loop” effective theory:

( ) ( ) ( ) ( ( ))

ˆ(1, v) light-like

i i i

i

i

J x d Q u x x

u

D F gJ

k ~ gT (gQs) k ~ T (Qs)

21

HTL 4 2 2( )

( )cg Na a a b

ab

p pdpL F F f p F F

p p D

Effective HTL theory permits systematic study of instabilities of “soft” color fields.

“soft”

“hard”

a a

ab c

abc

dpgQ F u

d

dQgf A u Q

d

47

HTL instabilities

Wavelength and growth rate of unstable modes can be calculated perturbatively:

2 2eq

For any there always exist modes with for some .

For (oblate), there exist two such unstable modes;

for (prolate),

0 ( ) (

only

)

0

1 o- 0

Find HTL modes for axisymme (tric ) ( )

k i k k

f p f p p n

ne unstable mode exists.

2 32 2

D 2 3( ) ( / ; )

(2 ) D

g d pk m f p k m

p

=10

=/8

Mrowczynski, Strickland et al., Arnold et al.

48

From instability to “turbulence”

Turbulent power spectrum

Kolmogorov-type power spectrum of coherent field excitations

[Arnold, Moore, Yaffe, hep-ph/0509226]

Exponential growth saturates when

B2 > g2 T4.

Quasi-abelian growth

Non-abelian growth

49

Space-time picture

Color correlation

lengthTime

Length (z)

Quasi-abelian

Non-abelian

Noise

M. Strickland, hep-ph/0511212

50

Anomalous viscosity

Formal derivation

(Sorry – using Chapman-Enskog)

51

Expansion Anisotropy

0 01

0

132

1 12 3

1

Perturbed equilibrium distribution:

For shear flow of ultrarelat. flu

( ) ( ) 1 1 ( )

(

( )

( )

) exp[

i

/

5

:

/

d

]

i jij ij

p

i j j i ijij

f p f p f p

f p u p T

sp p u

E T

u u u

f p

f p

u

Anisotropic momentum distributions generate instabilities of soft field modes. Growth rate ~ f1(p).

Shear flow always results in the formation of soft color fields;

Size controlled by f1(p), i.e. (u) and /s.

QGP

X-space

QGP

P-space

52

Turbulence Diffusion

Vlasov-Boltzmann transport of thermal partons:

with Lorentz force

Assuming , random Fokker-Planck eq:

w

( , , )

v

( ,

ith f

)( ) ,

di

p p

r pp

a a a

rp

pF f r p t C f

t E

F gQ E B

pf r p t C f

t ED p

E B

( )

fusion coefficient

.' ( '), ' ,i i jj

t

dt F r t t F rp tD

2

Diffusion is dominated by

chromo-ma

'

gnetic fields:

( ') ( ) mdt B t B t B

( )r t r

,a aE B

( ')r t

53

Shear viscosity

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

54

ηA - the feedback loop

pz

p• Longitudinal flow induces momentum anisotropy:

Anisotropy grows with shear viscosity η• Soft color fields are proportional to Δ:

• The anomalous viscosity is inversely proportional to B2:

• Shear viscosity η stabilizes due to:

21 10zz

xx yy

uT

T T s T

2 2 3B g Ts

3

2 2

1A

m

sT

g B

1 1 1

A C

_

1/ 2

3A T

s g u

Self-consistency

55

Time evolution of viscosity

Initial stateCGC ?

QGP andhydrodynamic

expansionHadronization

Hadronic phaseand freeze-out

A C HG viscosity: ? ?

Smallest viscosity dominates in system with several sources of viscosity

1/ 2

A3

Anomalous visco y

s

t

1

si

g u

4 1

Collisional viscosit

5 n

y

36

2 lC

ss g

56

Manifestations?

Possible effects on QGP probes:

Longitudinal broadening of jet cones (observed – “ridge”)

Anomalous diffusion of charm and bottom quarks (observed)

Synchrotron-style radiation of soft, nonthermal photons ?

Field induced quarkonium dissociation ?

No unstable modes for quarks: quasi-particle picture of QGP is compatible with low viscosity

z

yx

B

JetJet

Au+Au 20-30%

a

b

c c

b

57

Summary

Conclusion:

Because the matter created in heavy ion collisions expands rapidly, it forms a turbulent color plasma, which has an anomalously small shear viscosity. Transport phenomena involving quarks and gluons are strongly influenced by the turbulent color fields, especially at early times, when the expansion is rapid.

58

Additional slides

59

Anisotropic HTL modes

2 3

HTL 3

2 2eq

( )

1

4 2 (2 )

( ) ( )

ˆco

HTL effective action for general parton momentum distribution :

For axisymmetric

the gluon propagator (with ) cans

a a a b

p ab

f p

g d p p pL F F f p F F

E p D

f p f p p n

k n

2 2 2A G

1 2 2A

1 2 2 2 2 2G

be written as:

Collective modes are found as pol

( , , ) ( ) ( )

( , , ) 0

( ,

es of :

For sma

, ) ( )( ) s

ll the functions

in 0

ij ij ij ij ij ij

ij

k A C k B C D

k k

k k

Romatschke & Strickland, hep-ph/0304092

, , , , ,

(

of are explicitly known

up to [ ]) .

k

O

60

Shear perturbation

11 32

21r 0 0 0 3

2 2 2

0 02 3 2

Shear flow: with

Bjorken flow:

Drift term

(

:

) ( ) 5 /

1 1diag 1, 1,2

3

v ( ) (1 )

3( ) ( ) (1 )

1Vlasov term:

i jij ij

p

ij

iji j

p

m

p p

c p

f p p p u sE T

u u

uf p f f p p p

t E T

Q g BD p f p f f u p

N E T

213

210 0 3Collision (1 ) ( )term:

i j

i j

p p

C f f f u p p p I p

61

Estimate of A

2 2

11/ 2

42 2 2 4

2

2 2

Unstable modes are soft:

measures momentum anisotropy

Characteristic domain size:

Saturation condition for unstable modes:

Energy density of turbulent fields:

D

m

k m

r gT

g A k

kB g T

g

g Q B

32 3/ 2

A 3

4

3 / 2

1

Anomalous viscosity:

Compare with collisional viscosi36

25

9

tl

4

ny: C

s

m

s

g

r gT

s

g

62

Turbulence Diffusion

V-B transport of thermal partons:

with Lorentz force

Assuming E, random, V-B eq.

v ( , , , )

F-P eq:

with past trajec

v

( , , )

tory

( )

a ap

a a a

r pp

p

igQ A F f r p Q t C f

F gQ E B

pf r p t C f

t Ep

B

D

, so tha( ') ( )

' ( '), '

t

, .( )t

ij i jD p dt F r t t F r t

r t r t r

2

Most important for diffusion -

chromo-magnetic fiel

'

:

)

d

)

s

( ' ( mdt B t B t B

2 2 2

2

2 2

( )

2 1

p p

m

p

c p

D p

g Q Bp

N E

( )r t r

,a aE B

( ')r t

63

Shear viscosity

32

3

232 2

3

16( , )

(2 ) 45

4( ) ( , )

(2 ) 15

z r

Dz p p m

d pp v f r p u

E

md pp D p f r p B u

E T

2( )Take moments of with( , , )r p p zp

pD p f r p t C f p

t E

2 2

4

15A

D m

T

s m B

hep-ph/0603092

PRL (June30)

5 / s

64

Diffusion Viscosity

Diffusion corresponds to “anomalous” viscosity:

2 2

4

15A

D m

T

s m B

But recall that B2 itself depends on anisotropy f1(p) ~ viscosity !

2 20Take

/ 1u and :A m

D

sB b g

T m

1/ 2

3 3/0

2

4

15 AA T

s b g u

T uu

g

65

ηB - the feedback loop

pz

p

The Münchhausen

effect

• Longitudinal flow induces momentum anisotropy:

Anisotropy grows with shear viscosity η• Soft color fields are proportional to Δ:

• The anomalous viscosity is inversely proportional to B2:

• Shear viscosity η stabilizes due to:

2 8 81

5zz

xx yy

T

T T T T s

2 2 3B g T

3

2 2A

m

sT

g B

1 1 1

A C

_

66

B vs. C

Compare anomalous viscosity

with HTL (weak coupling) result for collisional viscosity

1/ 22

3 3/ 20

( 1) 1(1)

10cB

c

N TO

s b N g g

4 1

5

lnC

s g g

B indeed dominates at weak coupling !

T3

g2g2

g2T3g2

1( )m gT

4C g

2 3B g