Shear and Bulk Viscosities of Hot Dense

Matter

Joe Kapusta

University of Minnesota

New Results from LHC and RHIC, INT, 25 May 2010

Is the matter created at RHIC a perfect fluid ?

Physics Today, May 2010

Atomic and Molecular Systems

vTls free~

nl

1~freeIn classical transport theory and

so that as the density and/or cross section is reduced(dilute gas limit) the ratio gets larger.

In a liquid the particles are strongly correlated. Momentumtransport can be thought of as being carried by voids insteadof by particles (Enskog) and the ratio gets larger.

Helium

NIST dataL. Csernai, L. McLerran and J. K.

Nitrogen

NIST dataL. Csernai, L. McLerran and J. K.

OH2

NIST data L. Csernai, L. McLerran and J. K.

2D Yukawa Systemsin the Liquid State

radius Seitz-Wigner1

17parameter coupling Coulomb

at located Minimum

2

2

na

aT

Q

Applications to dusty-plasmas andmany other 2D condensed mattersystems.

Liu & Goree (2005)

QCD• Chiral perturbation theory at low T

(Prakash et al.): grows with decreasing T.

• Quark-gluon plasma at high T (Arnold, Moore, Yaffe): grows with increasing T.

4

4

16

15

T

f

s

)/42.2ln(

12.54 ggs

TT

TT

Tgln2ln

9

4ln

8

9

)(

1222

MeV 30T

QCDLow T (Prakash et al.)using experimentaldata for 2-bodyinteractions.

High T (Yaffe et al.)using perturbativeQCD.

L. Csernai, L. McLerran and J. K.

Shear vs. Bulk Viscosity

Shear viscosity is relevant for change in shape at constant volume.

Bulk viscosity is relevant for change in volume at constant shape.

Bulk viscosity is zero for point particles and for a radiationequation of state. It is generally small unless internal degreesof freedom (rotation, vibration) can easily be excited incollisions. But this is exactly the case for a resonance gas –expect bulk viscosity to be large near the critical temperature!

Lennard-Jones potential

Meier, Laesecke, KabelacJ. Chem. Phys. (2005)

Pressure fluctuations give peak in bulk viscosity.

QCD• Chiral perturbation theory at low T

(Chen, Wang): grows with increasing T.

• Quark-gluon plasma at high T (Arnold, Dogan, Moore, ): decreases with increasing T.

4

4

2 8

3ln

4

1ln

8

9

f

T

TTspp

)/34.6ln(5000

4

g

g

s

TT

TT

Tgln2ln

9

4ln

8

9

)(

1222

MeV 30T

QCDLow T (Chen & Wang)using chiral perturbation theory.

High T (Arnold et al.)using perturbativeQCD.

ς/s rises dramatically as Tc is approached from above (Karsch, Kharzeev, Tuchin) Lattice w/o quarks (Meyer) → 0.008 at T/Tc=1.65 and 0.065 at T/Tc=1.24

QCDLow T (Prakash et al.) using experimentaldata for 2-bodyinteractions.

High T (Arnold et al.)using perturbativeQCD.

ς/s rises dramatically as Tc is approached from above (Karsch, Kharzeev, Tuchin) Lattice w/o quarks (Meyer) → 0.008 at T/Tc=1.65 and 0.065 at T/Tc=1.24

Quasi-Particle Theory of Shear and Bulk Viscosity of Hadronic Matter

• Relativistic• Allows for an arbitrary number of hadron species• Allows for arbitrary elastic and inelastic collisions• Respects detailed balance• Allows for temperature-dependent mean fields

and quasi-particle masses• The viscosities and equation of state are

consistent in the sense that the same interactions are used to compute them.

P. Chakraborty & J. K.

P. Chakraborty & J. K.

Linear Sigma Model

MeV 600m MeV 900m

Calculated in the self-consistent Phi-derivable approximation= summation of daisy + superdaisy diagrams= mean field plus fluctuations

v

Go beyond the mean field approximation by averagingover the thermal fluctuations of the quasi-particles asindicated by the angular brackets.

1

1

)2(32

1

1

)2(2

0

2

1

2

1)(

/

2

3

3

22

/

2

3

3

22

2

22

2

22

2222

TE

TE

eE

ppd

m

eE

ppd

m

v

UUm

Um

mmUT

mean field fluctuation

P. Chakraborty & J. K.

Linear Sigma Model

MeV 600m MeV 900m

P. Chakraborty & J. K.

Linear Sigma Model

Solution to the integral equation:

P. Chakraborty & J. K.

Linear Sigma Model

a

aeqaaa

a

TEfEE

ppd

T)/()(

)2(15

12

4

3

3

Relaxation time approximation

P. Chakraborty & J. K.

Linear Sigma Model

Increasing the vacuum sigma mass causes the crossovertransition to look more like a second order transition.

P. Chakraborty & J. K.

Linear Sigma Model

a

aassa

eqa

a

aa

Td

mdTmvpvTEf

E

Epd

T

2

2

222222

31

23

3

)/()(

)2(

1

Violation of conformality

P. Chakraborty & J. K.

Linear Sigma Model

a

aassa

eqa

a

aa

Td

mdTmvpvTEf

E

Epd

T

2

2

222222

31

23

3

)/()(

)2(

1

Violation of conformality

Romatschkes 2007

and /sBoth

Both η/s and ζ/s depend on T – they are not constant.Beam energy scans at RHIC and LHC are necessaryto infer their temperature dependence.

Conclusion

• Hadron/quark-gluon matter should have a minimum in shear viscosity and a maximum in bulk viscosity at or near the critical or crossover point in the phase diagram analogous to atomic and molecular systems.

• Sufficiently detailed calculations and experiments ought to allow us to infer the viscosity/entropy ratios. This are interesting dimensionless measures of dissipation relative to disorder.

Conclusion

• RHIC and LHC are thermometers (hadron ratios, photon and lepton pair production)

• RHIC and LHC are barometers (elliptic flow, transverse flow)

• RHIC and LHC are viscometers (deviations from ideal fluid flow)

• There is plenty of work for theorists and experimentalists!