Strongly Interacting Low Viscosity Matter Created in Heavy Ion Collisions

Joe Kapusta*

University of Minnesota

Quark Matter 2006, Shanghai, China

* Original work done in collaboration with Laszlo Csernai and Larry McLerran.

The phase transition is 2nd

order for 2 massless flavorsand 1st order for 3, otherwisea rapid crossover.

Karsch, Laermann, Peikert

For realistic quark massesthere may be a line of 1st

order transition terminatingat a critical point.

de Forcrand, Philipsen

What has RHIC told us about the equation of state?

Big Experimental Motivation!PHENIX data + Huovinen et al. PHENIX: First Three Years of

Operation of RHIC

.correlated are and But fT T

Assume thermalization between 0.15 and 1 fm/c.

Agreement provides strong indication for early thermalization and collective flow.

Numerical Hydrodynamics(Huovinen, Kolb, Heinz, Hirano, Teaney, Shuryak, Hama, Morita, Nonaka, Bass)

Big Theoretical Motivation!

Viscosity in Strongly Interacting Quantum Field Theories from Black Hole Physics

Kovtun, Son, Starinets PRL 94, 111601 (2005)

Using the Kubo formula )0(),(1

lim20

1tracelesstraceless

4

0

ijijti TxTexd

the low energy absorption cross section for gravitons on blackholes, and the black hole entropy formula they found that

4/1/ s and conjectured that this is a universal lower bound.

Is the RHIC data, in the formof elliptic and radial flow,telling us that the matter hasvery small viscosity, a perfect fluid ?

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 data

Nitrogen

NIST data

OH2

NIST data

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

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.

η/s≈1/2 just above Tc

from lattice (Nakamura, Sakai)and classical quasiparticle model (Gelman, Shuryak, Zahed)

Large Nc Limit at Low T

• Baryon masses are proportional to Nc and can be neglected, meson masses are essentially independent of Nc. Hagedorn temperature and critical temperature should not change by much. Meson-meson cross sections scale as 1/ Nc

2, therefore η/s should scale as Nc

2 in the hadronic phase.

• From Yaffe et al. η/s = A/[(g2 Nc)2 ln(Bg2 Nc)] with A and B known constants, therefore η/s has a finite limit as Nc becomes large in the plasma phase.

• Implication: There is a jump in η/s of order Nc2 in going from the

low to the high temperature phases.

Huot,Jeon,Moore

Policastro,Son,Starinets & Buchel,Liu,Starinets

SYM •SYM has no running coupling and no phase transition•SYM has many more d.o.f. as scattering targets than QCD

BBB JunJ

TuuwPgT

QuHuHuHuuT 3

2

uTuTQuuguuH ,,

TuT

sus 1

22

22

32

2 kkk

kk

kiji

jj

i uTTT

uT

uuuT

s

Relativistic Dissipative Fluid Dynamics

In the Eckart approach u is the velocity of baryon number flow.

BBB JunJ

TuuwPgT

uHuuT 32

uTuTQuuguuH ,,

BBBB

B JT

susTw

TnJ

,

2

22

22

32

2 kkk

kk

kiji

jj

i uTTT

uT

uuuT

s

Relativistic Dissipative Fluid Dynamics

In the Landau-Lifshitz approach u is the velocity of energy transport.

How is this relevant for RHIC?

For baryon-free matter: transverse waves

sound waves

02 kiDt

0222 kiDv ls

TsD

TsD lt

3/4Momentum diffusion constants:

Bulk viscosity is generally small unless internal degreesof freedom (rotation, vibration) can easily be excited incollisions.

How is this relevant for RHIC?

• Solve relativistic viscous fluid equations, with appropriate initial conditions and with a hadron cascade afterburner, over a range of beam energies and nuclei and extract η(T)/s(T) from comparison with data.

• An analogous program was successful in obtaining information on the compressibility of nuclear matter and on the momentum-dependence of the nuclear mean-field at low beam energies.

m

TmT

1

03

2

Viscous Heating of Expanding Fireballs JK, PRC 24, 2545 (1981)

Viscosity smoothesout gradients intemperature, velocity, pressure,etc.

Extracting η/s from RHIC data

• Elliptic flow (Teaney,…)

• HBT (Teaney,…)

• Momentum spectra (Teaney, Baier & Romatschke,…)

• Momentum fluctuations (Gavin & Abdel-Aziz,…)

• Fluctuations in v2 (Csernai,…)

• Photon & dilepton spectra• Jet quenching

RHIC.at 5.01.0/at suggest th studiesy Preliminar s

Work in progress…and complications

• Numerical relativistic viscous fluid dynamics (Baier & Romatschke; Heinz & Song, Chaudhuri; Muronga &

Rischke) Large gradients (Baier & Romatschke) may require second-order Israel-Stewart equations (Muronga).

• Initial conditions (Lappi & Venugopalan CGC,…)

• Hadron afterburner (Hirano, Heinz, Kharzeev, Lacey, Nora; Bass & Nonaka)

• Turbulent plasmas (Asakawa, Bass, Muller) Charged particles scatter coherently from

dynamically generated color fields leading to “anomalous” viscosity.

Conclusion

• Hadron/quark-gluon matter should have a minimum 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 ratio. This is an interesting dimensionless measure of dissipation relative to disorder.

Conclusion

• RHIC is a thermometer (hadron ratios, photon and lepton pair production)

• RHIC is a barometer (elliptic flow, transverse flow)

• RHIC may be a viscometer (deviations from ideal fluid flow)

• There is plenty of work for theorists (and experimentalists)!