shear and bulk viscosity of qgp in pqcd

Qun Wang/USTC/China 1

Shear and Bulk viscosity of QGP in PQCD

Qun Wang

Univ of Sci & Tech of China

Chen, Dong, Ohnishi, QW, Phys. Lett. B685, 277(2010);

Chen, Deng, Dong, QW, Phys. Rev. D83, 034031(2011);

Chen, Deng, Dong, QW, arXiv:1107.0522

HIC in LHC era, 15-21 July 2012, Quy Nhon, Vietnam

1


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(Kubo formula)

2Shear & Bulk viscosity of QGP in PQCD


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

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η/s around phase transitionη/s around phase transition

Lacey et al, PRL98, 092301(2007)

Csernai, et alPRL97,152303(2006)



What isWhat is bulk viscositybulk viscosity

(breaking of scale symmetry) X (mean free path)X (energy momemtum density)

correlation of trace of energy-momemtum tensor in x and y

low-momentum behavior of correlator (Kubo formula)

Generated by dilatation



Scale anomaly in QCDScale anomaly in QCD

Massless QCD Lagrangian and action

The classical action of massless QCD is invariant under scale transformation

6

(1)

(2)

(3)

Shear & Bulk viscosity of QGP in PQCD


Scale anomaly & bulk viscosityScale anomaly & bulk viscosity

Breaking of scale invariance is proportional to beta-function

Beta-function

7

(4)

(5)



ζ/s around phase transitionζ/s around phase transition

Karsch, Kharzeev, Tuchin, PLB 2008Noronha *2, Greiner, PRL 2009, Chen, Wang, PRC 2009, Li, Huang, PRD 2008, ......

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



Bulk viscosity at TcBulk viscosity at Tc

Helium-3 near the critical point

～ a few millions !

Bulk viscosity near T_c diverges in power law,closely related to fluctuation

Kogan, Meyer, J.Low.Temp.Phys.110,899(1998)

3-D Ising model9Shear & Bulk viscosity of QGP in PQCD


Previous Previous worksworks on on ηη,,ζζ for QGP for QGP Shear viscosity: ►PV: Perturbative and Variational approachDanielewicz, Gyulassy, PRD31, 53(1985) ; Arnold, Moore and Yaffe, JHEP 0011, 001 (2000), 0305, 051 (2003).►Transport model : Xu, Greiner, PRL 100, 172301 (2008). ►Lattice : Meyer, PRD 76, 101701 (2007); NPA 830, 641C (2009). ►Anomalous: Asakawa et al, PRL 96, 252301(2006); Mujumder et al, PRL 99, 192301(2007).

Bulk viscosity: ►PV: Perturbative and Variational approachArnold, Dogan, Moore, PRD74, 085021(2006).►Sum rule and spectral desityKharzeev, Tuchin, JHEP 0809,093(2008); Moore, Saremi, PRD JHEP 0809, 015(2008); Romatschke, Son, PRD80, 065021 (2009).►LatticeMeyer, JHEP 1004, 099 (2010); PRL 100, 162001 (2008). ►Models near T_cNoronha*2, Greiner, PRL103,172302(2009) ; Li, Huang, PRD80,034023(2009).



ContradictingContradicting results on results on ηη ff or gluon plasma or gluon plasma

►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

1) A parton cascade model for solving the Boltzmann equation. Gluons are treated as a Boltzmann gas (i.e. a classical gas).

2) Gunion-Bertsch formula (soft emission) for gg↔ggg process.

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XG

AMY1) The Boltzmann equation is solved in a variation method. Gluon as quantum gas.

2) For number changing processes Ng↔ (N+1)g: using collinear splitting g↔gg



Our goal and strategyOur goal and strategy

Goal:

to calculate the shear/bulk viscosity in the leading order using Exact Matrix Element for gg↔ggg process to test previous results

Elements:

■Variational method and Linearized Boltzmann equation for 22+23 processes (AMY)■ 22: Hard Thermal Loop approximation for 22 (AMY)■ 23: Exact Matrix Element (new) + Gunion-Bertsch (XG) ■ 23: LPM effects for Exact Matrix Element (new) + Gunion-Bertsch (XG)



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 Matrix elements: gg↔gg (HTL)(HTL)

where q is momentum transferred

and gluon HTL self-energy

q31

2 4

Heiselberg, Wang, NPB 462,389(1996)

(6)

(7)


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

Ellis, Sexton, NPB 269, 445 (1986)

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(8)



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

NOTE (important!!)Gunion-Bertsch formula is only valid for soft region

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and

(13)

(14)



Linearized Linearized Boltzmann equationBoltzmann equation

Perturbation in distribution function

Right-hand-side or collision part of Boltzmann Eq.

Jeon, PRD52, 3591(1995)Jeon, Yaffe, PRD 53,5799(1996)

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(15)

(16)


Linearized Linearized Boltzmann equationBoltzmann equation

Left-hand-side or kinetic part of Boltzmann Eq.

Linearized Boltzmann Eq.

Collision integral of B(p) shear

Collision integral of A(p) bulk

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(17)

(18)



ShearShear/bulk/bulk viscosity viscosity in stress tensorin stress tensor

Solve A(p) and B(p) from linearized Boltzmann Eq. → shear/bulk viscosity

Stress tensor perturbation

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(19)

(20)



ConstraintsConstraints

■ Invariance of energy density in local rest frame

■ Speed of sound

■ Mass

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Functional basis for A(p)Functional basis for A(p)

■ Our choice

■ ADM’s choice

■ Results do not depend on bases, on a certain basis,

22

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(31)

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Evaluating bulk viscosityEvaluating bulk viscosity

■ Having A(p), we can evaluate ζ

■ So ζ is proportional to β(g)

23

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Analytic and numerical Analytic and numerical resultsresults



Leading-Log result:Leading-Log result: ζζ//ss

Weinberg, Astrophys. J. 168, 175 (1971)

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(35)

(36)



N_c scalingN_c scaling

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(37)

(38)



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

HTL: hard-thermal-loopAMY: Arnold-Moore-Yaffegluon mass = m_∞

Chen, Deng, Dong, QW, Phys. Rev. D83, 034031(2011);Erratum: 84, 039902;



Numerical results: Numerical results: ζζ for for 22+2322+23




Comparison of Comparison of ζζ and and ηη


Weinberg, Astrophys. J. 168, 175 (1971)



Comparison with lattice result and dataComparison with lattice result and data




Why different between XG and AMY?Why different between XG and AMY?

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1. For gg ↔ gg:

2. The integration can be done by including the symmetry factor 2 if we constrain the phase space to the region to the t-channel,

•We can also carry out the integration over the full phase space of the final state without the symmetry factor

(39)

(40)

(41)



1

2

3

4

1

3

4

2

qT →0, q →0, t-channel

qT →0, q → ∞, u-channel




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A caveat for gg ↔ ggg:

1. GB holds for constrained phase space in CMF:

2. GB has a symmetry for perm (3,4,5) in CMF, there is a symmetry factor for constrained phase space. On the other hand we can put GB out of summation and obtain the full space expression:

(43)

(42)




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Lesson:

Two ways of integration over final state phase space are equivalent:

1. Use symmetry factor × GB for constrained final state phase space in integration

2. Use GB for full phase space for final state gluons in integration

equivalent




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Introducing the symmetry factor while integrating over full phase space of final state gluons (= without constraining the phase space) multiple counting !



Why GB (soft) ≈ AMY (collinear)?Why GB (soft) ≈ AMY (collinear)?

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GB: soft emission (in CM frame) AMY: collinear splitting (in heat bath frame)



Conclusion and outlookConclusion and outlook

■ We calculate leading order shear and bulk viscosity of gluon plasma in PQCD. EXACT matrix element for 23 process is used for 23 process with m_D as regulator, HTL is used for 22 process.The LPM effects are also included.

■ Our results for shear and bulk viscosities agree with those of Arnold, Moore and Yaffe (AMY) within errors.

■ The difference between AMY’s and XG’s results is clarified. The lesson is to do collisional integration in two equivalent ways: (a) constrained phase space with symmetry factor; (b) full phase space without symmetry factor; otherwise it would lead to multiple counting.

■ The equivalence at LO between AMY’s collinear splitting and GB’s soft gluon bremsstrahlung has been demonstrated.

■ Outlook (undergoing project): (1) Include quark flavor; (2) With chemical potential



THANK YOU !THANK YOU !


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