1

Anomalous Viscosity of theQuark-Gluon Plasma

Berndt Mueller – Duke University

Workshop on Early Time Dynamicsin Heavy Ion Collisions

McGill University, 16-19 July 2007

Special credits to M.

Asakawa S.A. Bass

and A. Majumder

2

Part I

Viscosity

3

What is viscosity ?

( ) ( )23

and are defined as coefficients in the expansion of the stress tensor in gradients of the velocity fie

viscosityld

Shear b k:

ul

ik i k i k k i ik ikik i kT u u P u uu u u uε δ ς δη δ= + + ∇ + +∇ − ∇⋅ ∇ ⋅−

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

412

fpnp

pp

λσ

π

η

η

η

πησ

≈ =

≤ ≥⇒

4

Lower bound on η/s ?

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

( )1 13 12v

vf

fn p sn

λ εη τ⎛ ⎞ ⎛ ⎞≈ ≈⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

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

12 4s sη

π≥ ≈

h h

All known materials obey this condition!

For N=4 SU(Nc) SYM theory the bound is saturated at strong coupling:

( )3/ 22

135 (3)14 8 c

s

g N

ςηπ

⎡ ⎤⎢ ⎥= + +⎢ ⎥⎣ ⎦

L

5

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

η λ

λ σ

πσ α α

α αα

−

≈

=

≈

⎛ ⎞= + + −⎜ ⎟

⎝ ⎠

2 1tr

9y of QGP

100 ln

:

Cs s

T sησ πα α −≈ ≈

Baym, Gavin, HeiselbergDanielewicz & GyulassyArnold, Moore & Yaffe

6

What can cause the very low η/s ratio for the matter produced in nuclear collisions at RHIC?

There are two logical possibilities:(i) The quark-gluon plasma is a strongly coupled

state, not without well defined quasiparticle excitations;

(ii) There is a non-collisional (i.e. anomalous) mechanism responsible for lowering the shear viscosity.

Low T (Prakash et al.)using experimental data for 2-body interactions.

1/4π

High T (Yaffe et al.)using perturbative QCD

RHIC data

QCD matter

7

Part II

Anomalous Viscosity

8

A ubiquitous concept

Google search:Results 1 - 10 of about 571,000 for

anomalous viscosity. (0.24 seconds) 

From Biology-Online.org Dictionary:

anomalous viscosity

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.

9

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).

( ) ( ) ( )22 22 1forx y s zs

p p Q pQ

τΔ = Δ = Δ? ?pz

py

px

beam

Such conditions are satisfied in HI collisions: Longitudinal expansion locally “red-shifts” the longitudinal momentum components of small-x gluon fields released from initial state:

In EM case, instabilities saturate due to effect on charged particles. In YM case, field nonlinearities lead to saturation.

10

Spontaneous color fields

Color correlation

lengthTime

Length (z)

Quasi-abelian

Non-abelian

Noise

M. Strickland, hep-ph/0511212

11

Anomalous viscosity - heuristic

p

pΔaB

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 prg Q B rp

η λ

λ

≈

Δ ≈

≈ ≈Δ

33 94

2 2 2 2 2 2

lds:

3Am m

sTnpg Q B r g Q B r

η ≈ ≈

12

The Logic

(Longitudinal) expansion

Momentum anisotropy

QGPlasma instabilities

Anomalous viscosity

13

Expansion Anisotropy

Perturbed equilibrium distribution:

f ( p) = f0(π) 1+ f1(π) 1± f0(π)( )⎡⎣ ⎤⎦f0(π)=xπ[−uμ π

μ / T]

For σηar fλow of uλrarλa. fλuid:

f1(π)=−Δ(π)EπT

2πiπ j−1

3d ij( ) ∇u( )ij

∇u( )ij= 1

2∇iuj+∇jui( )−1

3d ij∇⋅u

η =−1

15Td 3π(2π)3

π4

Eπ2

∂f0∂Eπ

∫ Δ(π)

Anisotropic momentum distributions generate instabilities of soft field modes. Shear viscosity η and growth rate controlled by f1(p).

QGPX-space

QGP

P-space

14

Random fields 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

p F f r p t C ft E

F gQ E B

p f r p t C ft E

D p

E B

⎡ ⎤∂ + ⋅∇ + ⋅∇ =⎢ ⎥∂⎢ ⎥⎣ ⎦

= + ×

⎡ ⎤∂ ⎡ ⎤+ ⋅∇ − =⎢ ⎥ ⎣ ⎦∂⎢ ⎥⎣ ⎦

⇒

∇ ⋅ ⋅∇

( ) ( )( )

fusion coefficient

.' ( '), ' ,i i jj

t

dt F r t t F rp tD−∞

= ∫

If diffusion is dominated bychromo-magnetic fields:

dt ' B(t ')B(t)∫ ≡ B2 μ

( )r t r=

,a aE B

( ')r t

15

Shear viscosity

Take moments of

with pz2( , , )( )r p p

p

D pp f r p t C ft E

⎡ ⎤∂ ⎡ ⎤+ ⋅∇ −∇ ⋅ ⋅∇ =⎢ ⎥ ⎣ ⎦∂⎢ ⎥⎣ ⎦

( ) ( )22

1

3

2

3

4 ln 111 111

0mc

A Cc

FNON

gsT

gOT

τ

η ηη

−−= ≡ +

−+

M. Asakawa, S.A. Bass, B.M.,

PRL 96:252301 (2006) Prog Theo Phys 116:725

(2007)

Dij ( p) = d' Fia r('),'( )U ab(r,r)Fj

b r,( )−∞

∫

rFa =g

rEa + rv×

rBa( ) = coλor forc

dt ' Fi+(')F+i()∫ = F2 μ ≡q

= j quncηing πaraμ r !!!

16

Part III

Formalities

17

Vlasov-Boltzmann eq. for partons

vμ ∂

∂xμ f r,π,( )+ gFa ⋅∇π fa r,π,( )+C f⎡⎣ ⎤⎦=0

f a p, x( )=−igC2

Nc2 −1

d 4k

2π( )4∫ d 4 ′x U ab x, ′x( )

ik⋅x− ′x( )

v⋅k+ iFb ′x( )⋅∇π f π( )∫

gFa x( )⋅∇π fa π,x( )=−ig

C2

Nc2 −1

Fa x( )⋅∇π

d 4k

2π( )4∫ d 4 ′x U ab x, ′x( )

ik⋅x− ′x( )

v⋅k+ iFb ′x( )⋅∇π f π( )∫

( ) ( )( ) ( ), , , ,

, , ,

,

, ,a af

f

t dQQ f Q t

t dQ f Q t

=

=

∫∫

r p r

r r

p

p p

%%

a a a= + ×F vE B

parton distribution functions: color Lorentz force:

Perturbative solution for octet distribution:

yielding a diffusive Vlasov term:

18

Random (turbulent) color fields

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), ,ab aba b a b

i j i jfU x x UF x F x F x F x fx x′ ′=′ ′p p

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( )

(el)

(ma

(el)

(magg) )

,

,

, 0

a b a ai j i j

a b a ai j i j

a bi j

ab

ab

ab

U x x

U x x

U

x x

x x

x

t t

t

x

t

x x

σ

σ

τ

τ

′ ′′ ΦΦ −

′Φ −

−

′Φ −

=

′ =

′ =

′

′

′

x x

x x

%%

E E E EB B B BE B

Assumption of color chaos:

Short-range, Gaussian correlations of fields with functions Φel and Φmag :

Explicit form of Vlasov diffusion term:

with the memory time:

( ) ( )( ) ( )

( ) ( )

2 2el mag2

2 1p m ma a a a a

i j i j p pi jc i j

a

p p

x fg CgN p p

f

D f

τ τ⎡ ⎤∂⋅∇ = − + ×∇ ×∇⎢ ⎥

− ∂ ∂⎢ ⎥⎣ ⎦≡ −∇ ⋅ ∇

v vF p

pp

E E B B

( ) ( )( )(el/mae (l/mag el/m g) g)a12m tt t t td τ στ

∞

−∞′ ′Φ −′= Φ −∫ v%

19

Example: Transverse Ba only

( )1 ; 02

a a a ai j ij iz jz i jδ δ δ= − ≈2B B B E E

−∇π ⋅Δ π( )∇π f π( )=

3C2Δ g2 B2 μμ ag

Nc2 −1( )EπT

2f0 1± f0( ) πiπj ∇u( )ij

( ) ( )2 2

2 mag2

1

3c p

m

N E T

gp

C τ

−=Δ 2B

( )( )(gluo6

an)

2

2 m g

16 6 1c

c mA

N TN gπ

ης

τ−

= 2B( ) 2

(quark)6

2 2 mag

62 6 f

mA

cN N Tg

ης

π τ= 2B

Additional assumption: (satisfied at early times)

Diffusive Vlasov term:

Balance between drift and Vlasov term gives:

Anomalous viscosities for gluons and quarks:

20

Complete Shear Viscosity

W f1⎡⎣ ⎤⎦≡

d 3π

2π( )3f1 π( ) vμ

∂f0 π( )∂xμ +

12

−∇π ⋅Δ ⋅∇πd f π( )+ IC f1⎡⎣ ⎤⎦( )⎡

⎣⎢⎢

⎤

⎦⎥⎥∫ =μ in

Δ π( )=A

π

T, A=

Aq

Ag

⎛⎝⎜

⎞⎠⎟ ⇒ aA + aC( )A=r

r =32ς 5( )3π 2

Nc2 −1

158NcN f

⎛

⎝⎜⎜

⎞

⎠⎟⎟

aA =

32ς 4( )5π 2

g2 B2 μμ ag

T 3

Nc 0

078N f

⎛

⎝⎜⎜

⎞

⎠⎟⎟

aC =π Nc

2 −1( )45

g4 λng−1 29

2Nc + N f( )Nc 0

078N f

⎛

⎝⎜⎜

⎞

⎠⎟⎟+

π 2N f Nc2 −1( )

128Nc

1 −1−1 1

⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Minimization of full Vlasov-Boltzmann functional W[f1]:

Following AMY, make the

variational ansatz:

η =−

115T

d 3π(2π)3

π4

Eπ2

∂f0∂Eπ

∫ Δ(π)

21

Parametric dependence

( )2

21 2 3z

p

pf pE Tξ ⎛ ⎞

= − −⎜ ⎟⎝ ⎠

p

ηA

σ≈

T 3

g2 B2 μ

≈Tσ

g2 ∇uηA

⎛⎝⎜

⎞⎠⎟3/ 2

⇒ηA

σ≈

Tg2 ∇u

⎛⎝⎜

⎞⎠⎟3/5

Romatschke & Strickland convention:

Perturbation of equilibrium distribution: x =2Δ

∇uT

=10ησ∇uT

⇒ g2 B2 μ ≈kinσ3 ≈x3/ 2(gT)3Unstable modes: kinst

2 ≈ xmD2

Saturation condition: g|A| ≈ kinst

22

Who wins?

Smallest viscosity dominates in system with several sources of viscosity

Anomalous viscosity

ηA

σ:

Tg2 ∇u

⎛⎝⎜

⎞⎠⎟3/5

Collisional viscosityηC

σ≈

36π50g4 λng−1

∇u : τ −1 ⇒ Anomalous viscosity wins out at small g and τ

Interestingly, a (magnetic) gauge field expectation value also arises in the linearly expanding N=4 SYM solution (hep-th/0703243):

Time dependence of turbulent color field strength:

F2 : x2(gT)4 : ∇u / T( )4 /5T 4 : −28/15

tr F2 : −10/3

23

Part IV

Is the QGP weakly or strongly

coupled ?What exactly do we mean by this

statement?

24

Connecting η with q^

Hard partons probe the medium via the density of colored scattering centers:

q =r q2dq2 dσ / dq2( )∫

If kinetic theory applies, the same is true for thermal quasi-particles.

Assumptions:- thermal QP have the same interactions as hard partons;- interactions are dominated by small angle scattering.

η ≈13

ρ p λ f (p) =13

pσ tr (p)

σ tr (p) ≈4s

dq⊥2 q⊥

2 dσdq⊥

2 ≈∫4qsρ

With p ~ 3T, s^ ~ 18T2 and s 4r one finds:

ηs

≈54

T 3

q

Then the transport cross section is:

Majumder, BM, Wang,

hep-ph/0703085

25

Examples

From RHIC data:

T0 ≈335 Mς , q0 ≈1−2 ς 2 /fμ ⇒54T0

3

q0

≈0.15 −0.30

ηs

≈54

T 3

q

q=

8πaσNc

3 Nc −1( )2E2 +B2 μ ηA =

16ς(6) Nc −1( )2

πNc

T 6

g2 E2 + B2 μ

Turbulent gluon plasma:

Perturbative gluon plasma:

q =8ς(3)Nc

2

πT 3aσ

2 λn 1 /aσ( ) ηLL =3.81 Nc

2 −1( )π 2Nc

2

T 3

aσ2 λn 1 /aσ( )

s =4π 2 Nc

2 −1( )45

T 3

?

26

Counter-examples ηs

≠54

T 3

q

Strongly coupled N=4 SYM:

q =

π 3/2(3 / 4)(5 / 4)

g2Nc T3 ⇒

54T 3

q≈

0.166

g2Nc

=ησ=

14π

Chiral limit of QCD for T << Tc (pion gas):

q ≈

4π 2C2aσ

Nc2 −1

rπ xπ (x)[ ]x→ 0⇒

54T 3

q≈

0.235aσ xπ (x)[ ]x→ 0

=ησ≈

15 fπ4

16πT 4 From RHIC data:

T0 ≈335 Mς , q0 ≈5−15 ς 2 /fμ ⇒54T0

3

q0

≈0.02 −0.06 ?

27

Strong vs. weak coupling

At strong coupling, is a more faithful measure of medium opacity.

T 3

q

η/s

5T 3

4q

ln T / Tc( )1

(ln T)-4

(fπ/T)4

ln 1 / λ( )~1

λ−2

λ−1/2

4π( )−1

xGπ x( )x→ 0

QCD N=4 SYM

strong strong

weak weak

RHIC data:

q0 ≈1−2 ς 2 /fμ or q0 ≈5−15 ς 2 /fμ ?

28

Conclusion

Summary:The matter created in heavy ion collisions forms a highly (color) opaque plasma, which has an extremely small shear viscosity. The question remains whether the matter is a strongly coupled plasma without any quasiparticle degrees of freedom, or whether it is a marginal quasiparticulate liquid with an anomalously low shear viscosity due to the presence of turbulent color fields, especially at early times, when the expansion is most rapid.Jets constitute the best probe to ascertain the structure of this medium. The extended dynamic range of RHIC II and LHC will be essential to the success of this exploration, but so will be sophisticated 3-D models and simulations of the collision dynamics and their application to jet quenching.