Shear viscosity of a gluon plasma in heavy ion collisions

Qun Wang

Univ of Sci & Tech of China

J.W. Chen, H. Dong, K. Ohnishi, QW

Phys.Lett.B685, 277(2010)

J.W. Chen, J.Deng, H. Dong, QW

arXiv: 1011.4123

AdS/CFT program, KIPTC, Oct 11-Dec05,2010 (Nov 22)

What is viscosity related to HICWhat is viscosity related to HIC

viscosity = resistance of liquid to viscous forces (and hence to flow)

Shear viscosity

Bulk viscosity

Navier 1822

What isWhat is shear shear viscosity viscosity

(mean free path)x (energy momemtum density)

correlation of energy-momemtum tensor in x and y

low-momentum behavior of correlator of energy-momemtum tensor in x and y(Kubo formula)

D.F. Hou talk

Shear viscosity in ideal gas and liquidShear viscosity in ideal gas and liquid

• ideal gas, high T

• liquid, low T

• lower bound by uncertainty principle

Danielewicz, Gyulassy, 1985Policastro,Son,Starinets, 2001

Frenkel, 1955

η/s around phase transitionη/s around phase transition

Lacey et al, PRL98, 092301(2007)

Csernai, et alPRL97,152303(2006)

ζ/s around phase transitionζ/s around phase transition

Karsch, Kharzeev, Tuchin, PLB 2008Noronha *2, Greiner, 2008, Chen, Wang, PRC 2009, B.C.Li, M. Huang, PRD2008, ......

Bernard et al, (MILC) PRD 2007, Cheng et al, (RBC-Bielefeld) PRD 2008, Bazavov et al, (HotQCD), arXiv:0903.4379

Previous results on shear viscosity for QGP Previous results on shear viscosity for QGP

►PV: Perturbative and Variational approachDanielewicz, Gyulassy, Phys.Rev.D31, 53(1985) Dissipative Phenomena In Quark Gluon PlasmasArnold, Moore and Yaffe, JHEP 0011, 001 (2000),0305, 051 (2003)Transport coefficients in high temperature gauge theories: (I) Leading-log results (II): Beyond leading log ...........

►BAMPS: Boltzmann Approach of MultiParton ScatteringsXu and Greiner, Phys. Rev. Lett. 100, 172301(2008)Shear viscosity in a gluon gasXu, Greiner and Stoecker, Phys. Rev. Lett. 101, 082302(2008)PQCD calculations of elliptic flow and shear viscosity at RHIC

►Different results of AMY and XG for 2↔3 gluon process:

~(5-10)% (AMY) ~ (70-90)% (XG)

η (23) (AMY) >> η (23) (XG) σ(23) (AMY) << σ(23) (XG)

Difference: AMY vs XG Difference: AMY vs XG

Both approaches of XG and AMY are based on kinetic theory. However, the main points of differences are:

1) A parton cascade model is used by XG to solve the Boltzmann equation. Since the bosonic nature of gluons is hard to implement in real time simulations in this model, gluons are treated as a Boltzmann gas (i.e. a classical gas). For AMY, the Boltzmann equation is solved in a variation method without taking the Boltzmann gas approximation.

2) The Ng↔ (N+1)g processes, N=2,3,4,..., are approximated by the effective g↔gg splitting in AMY with 2-body-like phase space, while the Gunion-Bertsch formula for gg↔ggg process is used in XG with 3-body-like phase space.

Our goal and strategyOur goal and strategy

Goal:

to calculate the shear viscosity in a different way, to understand the nature of the difference between two results

Strategy:

1) We use variational method as AMY

2) We use the Gunion-Bertsch formula for gg↔ggg process as XG

3) For evaluating collisional integrals we treat phase space for 3 gluons in two ways: (a) 3 body state as XG; (b) 2+1(soft) state, almost 2 body state, close to AMY. We call it the soft gluon approximation;

Boltzmann equation for gluon plasma Boltzmann equation for gluon plasma

gluon distribution function

gg↔gg collision terms

gg↔ggg collision terms

matrixelement

delta functionEM conservation

phase-spacemeasure

[ gain - loss ]

Matrix elements: gg↔gg and gg↔ggg Matrix elements: gg↔gg and gg↔ggg

q

qk

Soft-collinear approximation gg↔ggg,factorized form,Gunion-Bertsch, PRD 25, 746(1982)

Shear viscosity: variational method Shear viscosity: variational method

perturbation in distribution function

linear in χ(x,p)

Shear viscosity: variational method Shear viscosity: variational method

S. Jeon, Phys. Rev. D 52, 3591 (1995); Jeon, Yaffe, Phys. Rev. D 53, 5799 (1996).

solve χ(x,p) by Boltzmann eq. → the constraint for B(p)

shear viscosity in terms of B(p)

Shear viscosity: variational method Shear viscosity: variational method

Inserting eq for B(p) into shear viscosity, quadratic form in B(p)

B(p) can be expanded in orthogonal polynomialsorthogonal condition

Shear viscosity: variational method Shear viscosity: variational method

Inserting eq for B(p) into shear viscosity, quadratic form in B(p)

Collisional rateCollisional rate

Boltzmann equation written in

Collisional rate is defined by

Regulate infrared and collinear divergence for kRegulate infrared and collinear divergence for kT in gg↔gggT in gg↔ggg

■ Landau-Pomeronchuk-Migdal (LPM) effect by cutoff (used by Xu-Greiner and Biro et al)

■ Debye mass m_D as the gluon mass or regulator (used by Arnold-Moore-Yaffe)

Importance of phase space Importance of phase space for gg↔gggfor gg↔ggg

■ almost 3-body (3-jet) phase space (used by Xu-Greiner)

■ almost 2-body phase space (used by Arnold-Moore-Yaffe)

soft

colinear

treated as equal footing

phase space dim: ~ 3X3-4=5

splitting function is usedphase space dim: ~ 2X3-4=2polar and azimuthal angles, (θ,φ)

■ Soft gluon approximation in our work (as one option of our calculation)

Importance of phase space Importance of phase space for gg↔gggfor gg↔ggg

Emission of the 5th gluon does not influence the configuration of 22 very much, therefore gg↔ggg can be factorized into gg↔gg and g↔ggThis is just the way Gunion-Bertsch got their formula. → Phase space dim: ~ 2X3-4=2, polar and azimuthal angles, (θ,φ)

This is equivalent to exand Jacobian of δ(E1+E2-E3-E4-E5) in large √s limit and keeping the leading order.

For the form of Jacobian, see Appendix D of Xu, Greiner, PRC71, 064901(2005).

Soft gluon approximation in cross section Soft gluon approximation in cross section of gg↔gggof gg↔ggg

two roots: y' (forward), -y' (backward)keep only positive root for y': a factor 1/2

Eq.(D5),Xu & Greiner, PRC71, 064901(2005)

Biro, et al, PRC48, 1275(1993)

Our results-with GB formulaOur results-with GB formula

Leading-Log result for gg↔ggLeading-Log result for gg↔gg

We reproduced AMY's leading-log(LL),

For Boltzmann gas, LL result:

Our numerical results show good agreement to LL resultin weak coupling

η22: Bose η22: Bose andand Boltzmann Boltzmann gas gas

Collisional ratesCollisional rates

Shear viscosity from 22 and 23 processShear viscosity from 22 and 23 process

Effects of 23 processEffects of 23 process

Comparison: AMY, XG, Our workComparison: AMY, XG, Our work (GB) (GB)η/s main ingredients LL gg↔ggg effect, 1-η(22+23)/η22

α_s < 0.01 α_s > 0.01

Arnold,Moore, Yaffe

pQCD, analytic, variational, boson, g↔gg, LPM (m_D), dominated by 2-body phase space

Yes ~10% ~10%

Xu, Greiner

BAMPS, numerical,

Boltzmann gas, gg↔ggg (GB), LPM ( rate), 3-body phase space

No ~[60--80]% ~[80--90]%

Our

work

pQCD, numerical, variational, gg↔ggg (GB), LPM (rate, m_D, 3-body phase space as XG), soft-g approx (2-body phase space, LPM by m_D)

Yes LPM (rate, m_D):

~[30--60]%

soft-g approx:

~[10--30]%, close to AMY

LPM (rate, m_D):

~[60--80]%, close to XG up to 1/2

soft-g approx:

~[10--30]%, close to AMY

Concluding remarksConcluding remarks: results with GB: results with GB

■ We have bridged to some extent the gap between AMY and XG.

■ To our understanding, their main difference is in the phase space for number changing processes, there are much more 3-body configurations in XG approach than in AMY, or equivalently phase space in XG for gluon emission is much larger than in AMY (about dim 5 : dim 2), causing effect of 23 for viscosity in XG is much larger than in AMY.

■ Core question: Is GB formula still valid for general 3-body (3-jet) configuration? or equivalently: Does GB formula over-estimate the rate of the general 3-body (3-jet) configuration? Further study of viscosity using exact matrix element should give an answer to this question.

Exact matrix element for 23Exact matrix element for 23

Exact matrix element in vacumm for massless gluons

1

2

3

4

5

all momenta are incoming or outgoing

exact matrix element for massless gluon is invariant for

Regulating IR/collinear singualrityRegulating IR/collinear singualrity

Matrix element for

can be obtained by flipping signs of (p1, p2)

Internal momenta are all:

so we make substitution in and set gluon mass

Most singular part is regular since

Exact matrix element to Gunion-BertschExact matrix element to Gunion-Bertsch

Using light-cone variable

Gluon momenta are

Exact matrix element to Gunion-BertschExact matrix element to Gunion-Bertsch

Taking large s limit (s→ ) and then small y limit (y→0)

Gunion-Bertsch formula (set m_D=0)

Numerical results: η/s for 22Numerical results: η/s for 22

LL : the leading log resultHTL: hard-thermal-loopMD: m_D as regulatorAMY: Arnold-Moore-Yaffe

normalized by η_22 (m_D)

gluon mass = m_D

Numerical results: η/s for 22+23Numerical results: η/s for 22+23

LL : the leading log resultHTL: hard-thermal-loopMD: m_D as regulatorAMY: Arnold-Moore-Yaffe

gluon mass = m_D

Numerical results: error estimateNumerical results: error estimateXG0.5

0.13

0.076

Numerical results: η_{22}/Numerical results: η_{22}/η_{22+23}η_{22+23}

Conclusion and outlookConclusion and outlook

■ We have calculated η/s to leading order for 22 and 23 process, exact matrix element is used for 23 process with m_D as regulator, HTL is used for 22 process.

■ The errors from not implementing HTL and the Landau-Pomeranchuk-Migdal effect in the 23 process, and from the uncalculated higher order corrections, have been estimated.

■ Our result smoothly connects the two different approximations used by Arnold, Moore and Yaffe (AMY) and Xu and Greiner (XG). However, we find no indication that the proposed perfect fluid limit η/s =1/(4π) can be achieved by perturbative QCD alone.

■ Outlook: (1) Include quark flavor; (2) Bulk viscosity; (3) Beyond the linear Boltzmann equation; (4) Semi-QGP

THANK YOU !THANK YOU !