Shear viscosity to entropy density ratio below QCD critical temperature

Outline:

1) What is the shear viscosity?2) Background and motivation3) Shear viscosity/Entropy in Pionic gas 4) Summary and outlook

Eiji Nakano

Dept. of Physics, National Taiwan Univ.

- Checking the viscosity/entropy ratio bound conjectured by string theory-

April/21th/2006 at IoP, AS

1) What is the shear viscosity?

Shear viscosity (coefficient) is one of transport coefficients in macroscopic hydrodynamic equations for non-equilibrium systems:

,0 T

0 nV

21

)(,1)(

V

V

xxVWhere Local collective flow velocity:

2) Number conservation:

Energy-momentum conservation:1)

)(xVElementary volume:

Basic equations:

.)(3

1

2

)()(

xV

xVxVT ij

ijjiij

appears in spatial traceless part (dissipative):

y

x

)(yVx

Stress pressure (friction) in shear flow ( :coefficient of frictional force)

,)( )0(

TTxT

)()()()()()0( xPxVxVxPxT

The first term describes Perfect fluid dynamics (dissipationless) :

Let’s remember,

1) Isotropic pressure

xp

xv

Unit cross-section

The number of particle reflected by the cross-section per second:

2/xnv

Thus the isotropic pressure becomes

Tnnv

pPP Bx

xi

ii 2

23

1

31

y

x

)(yVx

2) Anisotropic(stress) pressure

yv

Tm

yVTml

pnv

yVT

B

xBxy

xxy

y

)(

yy

)(

Momentum transfer of x comp. per sec. across unit area normal to y direction: a frictional force facing -x direction

Mean-free path:n

l1

Scattering cross-section

Maxwell formula

Viscos dynamics

e.g., Diffusion equation for transverse momentum :

02 Tt pD

Tp

TsPD

1

diffusion constant:

yp

xp

relaxes transverse fluctuation, in other words, diminishes the velocity gradient (shear flow).

Tp

Hierarchy in theories for space-time scales,

mesoscopic

macro

micro

Kinetic theories

Fluid dynamics

scales theories

Hamiltonian • Liouville eq. • Linear response theory

• Boltzmann eq. • GL eq.• Langevin eq.

• Fluid eqs, e.g. , in Navier-Stokes eq.

0r

l

sw

Our attempt (T<m_pi)

Jeon-Yaffe (1996)

~1fm

~100fm

~10^4fm

Basic properties of shear viscosity

This can be also seen from more microscopic theory, Kubo formula: Auto correlation function of : ijT

)0()(lim20

1 4

0ij

ijipx

pTxTedx

T

4.,. ge

3T

Im

3T

2

Keep in mind that large cross section gives small viscosity.

by S-G. Jeon (1995)

nl

1 is proportional to the mean-free path 　　　　　 : Roughly speaking,

(One has to resum infinite number of diagrams to get LO result even for weak coupling theory).

Maxwell formula:

Scattering cross-section

LO

Tm B

1) A perturbative gravity analysis with a black hole metric corresponding to N=4 supersymmetric gauge field theory in strong coupling (Ads/CFT correspondence) conjectures a lower bound (KSS bound):

08.041

sShear viscosity/entropy ratio :

Kovtun, Son, Starinet, hep-th/0405231

2) Background and motivation

2) Elliptic flow produced just after non-central relativistic heavy ion collisions (RHIC),

Hadronic (chiral broken) phase Quark-Gluon Plasma (QGP),

suggests that the system is near perfect fluid (small viscosity: ).

It implies that expected QGP is in strong coupling regime.

0~

RHIC

)2cos(d

d20

vv

N

)2cos(2)cos(21 φφdφ

dN2v

222 2cos ppppφv yx

Directed flow

1v

y

x

Elliptic flow

QGPHadrons

QCD phase diagram on Density-Temp. plane

Karsch & Laermann, hep-lat/0305025

RHIC

Tc

0~

?

Recent trapped cold atom experiments give an opportunity to investigate strong interacting matter via tunable Feshbach resonance. This dilute and strongly-coupled system of Li6 also behaves hydrodynamically, showing elliptic flow.

O ’Hara et al., Science 298, 2179 (2002)

Time evolution after trap is turned off

Small viscosity is common feature in strongly-coupled systems.

….We investigate how the shear viscosity of QCD (pionic gas) behaves below Tc (chiral / deconfinement transition),

with special attentions: a) How the viscosity behaves 　 in Hadronic phase approaching Tc from below,

b) How about ? Small or Large?s

taking the pionic gas….

Motivation:

3) Shear visc./entropy in pionic gas in Kinetic theory

TT

pxfpxf

pxfxT

p

p

E

ppcdp

E

ppcdp

)0(

)0(

2

2

)],(),([

),()(

3

3

3

3Local equilibrium distribution,

Small deviation

(Dissipationless process)

(Dissipative process)

is given by as a functional of , which we will obtain from Boltzmann eq. .

f

)(3

1

2

)()(

,)()()()()()0(

xVxVxV

T

xPxVxVxPxT

ijijji

ij

1

1),(

/)0(

TEpepxf

at local rest frame: 0)( xV

Bose distribution function

][d

dp

p fCt

f

The distribution function is obtained from Boltzmann eq. for ,

ppp fffffffffC 3213211231111d

2

1][

with collision integral

2p1k

2k 3k

),( pxff p

~Scattering cross-section

Strategy to obtain f(x,p) from Boltzmann eq.

1. Expand to the 1st order

2. parametrize

3. Substitute it into Boltzmann eq.

4. Linearize the eq. in terms of

5. Expand using a set of specific polynomials

6. Linearized Bolzmann = Matrix eq. for

),(),(),( )0( pxfpxfpxf

)()( )( pBbpB r

rr

A polynomial up to rp

)( pB

rb

Finally, the viscosity is given by,

1

][|

..|)(

)0()0(

2)0(

BCB

shlBbpB r

)()(),( )0( xfpBpxf

)( pB

Step

Step

Step

Step

Step

Step

Known (by symmetry)

unknown

Linearized Boltzmann equation for B(p);

ChPT: effective theory on the basis of chiral symmetry

(low energy limit: Weinberg theorem)

Increase with collision energy!

4

42

ChPT 3

23||

F

MΤ

Pion-Pion scattering

444

Isospin

22ChPT

9

1|||| pM

F�ΤΤ

II

vanishes in massless limit!

LO

TT

TT

Highat

Lowat~

3

2/1

coincide with the behavior in 4

!4

by Jeon,Yaffe, Heinz,Wang, etc…

couplingconst.withviscosityShear

(Low energy limit)

42

ChPT 2

23||

F

MΤ )MeV(93

)MeV(139

F

m

From very naïve dimensional analysis, we find a power law in T:

TT

TT

highfor

Lowfor~

1

2/1

χpt

Non- monotonic!

MeV170CT

)MeV(93

)MeV(139

F

m

442ChPT

42ChPT

~||

||

TpT

MT

Universal behavior!

Intensive behavior at low T, divergent at T=0 ! But it seems to be typical for pure NG bosons with derivative couplings.

This aspect is also seen for CFL phonon by Manuel etal (2004).

0m

S: statistical entropy

0.08KSS_bound

35.0~

32

45

2~ Tg

4) Summary and Outlook

We have shown small ratio of the visc./entropy in Chpt approaching Tc of QCD:

.MeV)(140120for32.064.0 Ts

So we conclude that the small viscosity/entropy ratio <1 is not unique only above Tc, but below Tc. But it suggests discontinuity at Tc (~2times larger than KSS bound).

MeV170Tc T

s

Hadron

QGP

KSS

We are interested in shear visc. behavior in BCS-BEC crossover regime, above and below Tc.

Superfluid phonon + ….Quasiparticle with fluctuations

This work is close collaboration with Prof. J-W Chen at NTU.

As future works

Thank you for your attention…

Back up files

Muroya and Sasaki, PRL(2005)

Hadronic gas at finite density 1-2 rho_0

Applicability of ChPT

s/

)MeV(T170~CT

0qq

?

140~ MT

Hadrons QGP

Data

Melting of Chiral cond.

0qq 0qq

T

pxgpxfpxfpxff p

),(),(11),(),( )0()0(

3

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

xVpxVppBxVpApxg

In 1st Chapman-Enskog expansion,

with parametrization

Related to bulk viscosity to shear viscosity

[CMF]

qq

]MeV[q

Scatt. Amp. of ChPT

2,

4,0

444

2ChPT

1|| pM

FΤ

T

TS

LogZ)( 1E(p)/T-e-1LogpdLogZ

with

S: statistical entropy

S

T

32

45

2~ T