Viscous Hydrodynamics for
Relativistic Heavy Ion Collisions
Huichao SongThe Ohio State University
Presented atATHIC2008,Tsukuba, Japan, Oct. 13-15, 2008
Supported by DOE

in collaboration with Ulrich Heinz
References:
H. Song and U. Heinz, Phys.Rev.C78:024902 (2008). arXiv:0805.1756 [nucl-th].
H. Song and U. Heinz, Phys.Rev.C77:064901 (2008).
H. Song and U. Heinz, Phys. Lett. B658, 279 (2008).

10/13/2008

What is viscosity
Shear viscosity –measures the resistance to flow
the ability of momentum transfer

Bulk viscosity –measure the resistance to expansion-volume viscosity
Determines the dynamics ofcompressible fluid

The QGP viscosity
Kubo formulas:1lim ∫ d 4 xeiωt < [T ij ( x)T ij (0)] > θ (t )20 ω→0
1bulk viscosity: ζ = lim ∫ d 4 xeiωt < [Tii ( x)Tii (0)] > θ (t )18 ω→0shear viscosity: η =Shear viscosity: uncertainty principle requires a lower limit for η / sArnold, Moore & Yaffe, 00,03
η/s~1-weakly coupled QCD:
Meyer, PRD 07-lattice SU(3) gluon dynamics : η / s < 1-strongly coupled AdS/CFT prediction : η / s ≥ 1 / 4π ~ 0.08 D.T. Son et al. ‘01,’05
Bulk viscosity: zero for classical massless particles, ζ / sreaches a peak near Tc

Arnold, Dogan & Moore, PRD06
ζ / s << 1Meyer, PRL08
ζ / s ~T = 0.73 02..50-lattice SU(3) gluon dynamics :-LET+ assum. of spectral fun. + Lattice data: ζ / s ~T ~ 0.8 Kharzeev, et al. 07-08-strongly coupled AdS/CFT prediction:ζ / s > 2η / s(1 / 3 − c s2 ) Buchel, 07
Gubser, et al . 0806 ..
ζ / s ~T ~ 0.05[]-weakly coupled QCD prediction:

ccc

To extract the QGP viscosity from experimental data, we need viscous hydrodynamics

Viscous hydrodynamics

S.BassConservation laws:∂ μ T μν ( x ) = 0T μν = ( e + p + Π ) u μ u ν − ( p + Π ) g μν + π μν

Evolution equations for shear pressure tensorπτπ ΔαμΔβνπ&αβ + π μν = 2ησ μν& + Π = −ζ (∂ ⋅ u )
τΠΠμν

:

-simplified Israel-Stewart eqn.

∂μSμ ≥ 0

initial conditions and final conditions
Input: “EOS” ε = ε ( p, n)With η , ζ → 0 viscous hydrodynamics reduces to ideal hydrodynamics
Bjorken appro. : v z = z / t

reduces (3+1)-d hydro to (2+1)-d hydro (τ , x , y , η )(2+1)-d viscous hydrodynamics
Song & Heinz
PLB08

P.&U. Romatschke
PRL07
25--30%Dusling & Teaney
PRC08

Molnar & Huovinen
QM08(2+1)-d viscous hydrodynamics-Romatschke & Romatschke: full I-S eqn. EOS I EOS L*
PRL’07Au+Au, Tdec = 150MeV (EOS L* here is the quasi-particle one based on lattice QCD)-Song & Heinz: simplified I-S eqn. & full I-S eqn. EOS I SM-EOS Q EOS L
PLB’08 & PRC08 Cu+Cu, simplified I-S eqn., Tdec = 130MeV
PRC08 (Au+Au, Cu+Cu, system size effects, full I-S eqn. vs. simplified I-S eqn., EOS L etc)

-Dusling & Teaney: Őttinger-Grmela (O-G) eqn. EOS I
PRC’08 Au+Au, decoupling by scattering rate, Nucl. Phys. A08(dilepton production)-Huovinen & Molnar: full I-S eqn. EOS I
QM08 talk: comparing the results from viscous hydro and from transport model-Chaudhuri: simplified I-S eqn.

EOS I EOS Q

arXiv:0708.1252 [nucl-th], arXiv:0801.3180 [nucl-th], arXiv:0803.0643 [nucl-th] Au+Au

Code Checking between Song & Heinz
VISH2+1 and Romatschke Code (Nov.2007)(2+1)-d viscous hydrodynamics-Romatschke & Romatschke: full I-S eqn. EOS I EOS L*(EOS L* here is
the quasi-particle
one based on lattice QCD)
Code Checking
withinTECHQM
PRL’07 Au+Au, Tdec = 150MeV

Collaboration (started on May, 2008)-Song & Heinz: simplified I-S eqn. & full I-S eqn. EOS I SM-EOS Q EOS L
PLB’08 & arXiv:0712.3715[nucl-th] Cu+Cu, simplified I-S eqn., Tdec = 130MeV(Au+Au, Cu+Cu, system size effects, full I-S eqn. vs. simplified I-S eqn., EOS L etc, in preparation)-Dusling & Teaney: Őttinger-Grmela (O-G) eqn. EOS I
PRC’08 Au+Au, decoupling by scattering rate, arXiv:0803.1262 [nucl-th], (dilepton production)-Huovinen & Molnar: full I-S eqn. EOS I
QM08 talk: comparing the results from viscous hydro and from transport model-Chaudhuri:

O-G eqn:
simplified I-S eqn. EOS I EOS Q
full I-S eqn:
https://wiki.bnl.gov/TECHQM/index.php/Main_Page

arXiv:0708.1252 [nucl-th], arXiv:0801.3180 [nucl-th], arXiv:0803.0643 [nucl-th] Au+Au
Issues:https://wiki.bnl.gov/TECHQM/-VISH2+1 (Song & Heinz) vs. Romatschke code: (Nov. 2007)-VISH2+1 (Song & Heinz) vs. Dusling & Teaney code: (May, 2008-)

(2+1)-d viscous hydrodynamics-Romatschke & Romatschke: full I-S eqn. EOS I EOS L*
PRL’07Au+Au, Tdec = 150MeV (EOS L* here is the quasi-particle one based on lattice QCD)-Song & Heinz: simplified I-S eqn. & full I-S eqn. EOS I SM-EOS Q EOS L
PLB’08 & PRC08 Cu+Cu, simplified I-S eqn., Tdec = 130MeV
PRC08 (Au+Au, Cu+Cu, system size effects, full I-S eqn. vs. simplified I-S eqn., EOS L etc)

-Dusling & Teaney: Őttinger-Grmela (O-G) eqn. EOS I
PRC’08 Au+Au, decoupling by scattering rate, Nucl. Phys. A08 (dilepton production)-Huovinen & Molnar: full I-S eqn. EOS I
QM08 talk: comparing the results from viscous hydro and from transport model-Chaudhuri: simplified I-S eqn.

EOS I EOS Q

arXiv:0708.1252 [nucl-th], arXiv:0801.3180 [nucl-th], arXiv:0803.0643 [nucl-th] Au+Au
Issues:Shear viscosity effects:
Ideal hydro vs. viscous hydroViscous vs. ideal hydro – spectra & elliptic flow
Simplified I-S eqn.

Song & Heinz PLB 08

minimal shear viscosity

η / s = 1 / 4π
ζ /s=0⎛ 1 p α p β π αβ ( x ) ⎞dNp ⋅ d 3σ ( x )p ⋅ d 3σ ( x )⎟⎟E 3 =∫
[ f eq ( x, p ) + δf ( x, p )] = ∫f eq ( x, p )⎜⎜1 +332ΣΣ
d p2π( 2π )⎝ 2 T ( x ) (e + p )( x ) ⎠-More radial flow, flatter spectra;-elliptic flow is very sensitive to shear viscosityComparison with Romatschke 07 resultsSong & Heinz
PLB 08

P. &U.Romatschke
PRL 0725--30%70--80%simplified I-S eqn.

full I-S eqn.

- different systems & EOS: Cu+Cu, b=7, SM-EOS Q vs. Au+Au, min bias, EOS Lattice- different Isreal-Stewart eqns. used: simplified I-S eqn. vs. full I-S eqn.

Effect of using different I-S eqns.?

Simplified I-S eqn. vs. full I-S eqn.:μα νβsimplified I-S eqn.: Δ Δ Dπ αβ = −[π − 2ησ ]τΔμα Δνβ Dπ αβ = −[π − 2ησ ]τfull I-S eqn.:

1μνμνπ1μνμνπ+1 μνπ [ 5 D ln T − ∇ α u α ] − 2π α ( μ ω αν )2important for preserving the conformal symmetry(Baier et al. ‘07)

simplified I-S eqn. vs. full I-S eqn.
EOS I

Song & Heinz PRC 08

SM-EOS Q-for EOS I, the difference between simplified I-S eqn and full I-S eqn could reach30-50% for larger relaxation times.

- for realistic EOS with a phase transition, the difference between simplified
and full I-S for viscous v2 suppression are small-numerical simulations also show that simplified I-S eqn. and full I-S eqn. approach
the same Navier-Stokes limit as τ π → 0-full I-S eqn. shows much weaker dependence to τ πSystem size effects to viscous v2 suppression
system size, EOS, different I-S equations:

Simplified I-S eqn

Simplified I-S eqn

40%
70%

-system size: CuCu b=7fm vs. AuAu b=7fm:~50-100% effectEOS effects to viscous v2 suppression
system size, EOS, different I-S equations:Song & Heinz PRC 08

Simplified I-S eqn

40%

EOS
Simplified I-S eqn

full I-S eqn

30%-system size: CuCu b=7fm vs. AuAu b=7fm:~50-100% effect-EOS: SM-EOS Q vs. EOS L: ~25% effect-different I-S eqns: simplified I-S eqn. vs. fullI-S eqn.: ~5-10% effects (EOS Q and EOS Lonly )Different contributions to the suppression of v2
System size, EOS, different I-S equations:simplified I-S eqn.

simplified I-S eqn.

40%
Considering all of theseeffects,
thefinal
suppres70%
sion of v2 for Au+Au with EOS L and the full I-S
eqn., for minimal shear viscosity η / s = 0.08, is ~25%,
approaching the results of P. & U. Romatschke(PRL 99, 172301 (2007)).
simplified I-S eqn.

full I-S eqn.

- system size: CuCu b=7fm vs. AuAu b=7fm:~50-100% effect30%- EoS: SM-EOS Q vs. EOS L: ~25% effect- different I-S eqn.: simplified vs. full I-S eqn.:~5-10% effect (EOS Q and EOS L only )Comment: To extract QGP viscosity fromexp. data by using viscous hydro, one
needs a better description of EoS (LatticeEoS + chemical non-equil. HRG EoS)

System size effects

Multiplicity scaling of v2/ε EOS ISong & Heinz PRC 08

ideal hydrov2 scaling line
full I-S eqn

Ideal hydrodynamics: multiplicity scaling of v2/ε is weakly broken:- freeze-out condition introduces time scale, breaking scale invariance of id. hydro eqns.
- Initial profiles for Cu+Cu and Au+Au systems are not identical after a rescaling
Viscous hydrodynamics: additional scale breaking by shear viscosity, resulting in finestructure of v2/ε:- for similar initial energy density, Cu+Cu curves are slightly below the Au+Au curves
1 dN ch- at fixed, the e0 = 15GeV/fm3 curves are slightly above the e0 = 30GeV/fm3 ones
Sdy

Viscous effects are larger for smaller systems and lower collision energies

Multiplicity scaling of v2/ε EOS LSong & Heinz PRC 08

hydroregion

full I-S eqn

dilutegases- experimental data show qualitatively similar fine ordering as viscous hydro prediction- to reproduce slope of v2/ε vs. (1/S)dN/dy, a better description of the highly viscous
hadronic stage is needed: T-dependent η / s , viscous hydro + hadron cascade- the experimental v2/ε vs. (1/S)dN/dy scaling (slope and fine structure) is anothergood candidate to constrain η / s (insensitive to Glauber-type vs. CGC initialization)- this requires, however, experimental and theoretical improvements: reduced error bars,
accounting for T-dependence of η/ s, ζ / s near Tc, modeling hadronic phase with realistic cascadeA first attempt to extract η / sfrom experimental data
-Luzum & Romatschke,
arXiv:0804.4015 [nucl-th]Extracting η / s from elliptical flow data
Glauber

Luzum & Romatschke, arXiv:0804..

CGC-Glauber vs.CGC ~100% effects on the extracted value of η / s , (highly viscous latehadronic stage, bulk viscosity , non-eq. chemistry in HG, etc are not included )- η/ s ≤5×(1 4π) (estimated from these data and results by other groups)

Effects from bulk viscosity-Preliminary resultsmin. Ads/CFT results

Ideal vs.Viscous hydro – spectra & elliptic flow
Viscous hydro: bulk viscosity only, using C=1 here

-negative bulk pressure could effectively soften the EoS near phase transition, which
prohibits the development of radial flow and momentum anisotropy.
-spectra becomes steeper, v2 is suppressed at lower pT regionViscous v2 suppression: shear and bulk viscosity
ideal hydrovisc. hydro: η / s20%30%

ζ /s
1/ 4π, c=0
1/ 4π, c=1
1/ 4π, c=2-2 x min bulk viscosity could result in ~50% additional v2 suppression-when extracting theη / s from exp. data, bulk viscous effects cannot be neglectedSummary and discussion- v2 is sensitive to η / s- use full I-S to minimize sensitivity to τ π- multiplicity scaling of v2 / ε is a good candidate to extract the QGP viscosity:- larger viscous effects in smaller systems and at lower collision energies- multiplicity scaling of v2 / ε is insensitive to Glauber model vs. CGC initialization.
A first attempt to constrain η/ s from RHIC data indicatesη / s ≤ 5×(1 4π ) ,BUT:
to extract QGP viscosity, one must consider (at least) all the following aspects:- a realistic EOS:

EOS L vs. SM-EOS Q ~25% (for v 2 and v2 / ε )- initial conditions: CGC initialization vs. Glauber initialization ~15-30% (forv 2 )- bulk viscosity:with vs. without bulk viscosity ~?%- hadronic stage : viscous hydro+ hadron cascade in the furthure ?

……

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Thank You

EOSEOSEOS