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

What is viscosity related to HIC viscosity = resistance of liquid to viscous forces (and hence to flow) Shear viscosity Bulk viscosity Navier 1822

What is shear 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 liquid ideal gas, high T liquid, low T lower bound by uncertainty principle Danielewicz, Gyulassy, 1985 Policastro,Son,Starinets, 2001 Frenkel, 1955

η/s around phase transition Lacey et al, PRL98, (2007) Csernai, et al PRL97, (2006)

ζ/s around phase transition Karsch, Kharzeev, Tuchin, PLB 2008 Noronha *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:

Previous results on shear viscosity for QGP ► PV: Perturbative and Variational approach Danielewicz, Gyulassy, Phys.Rev.D31, 53(1985) Dissipative Phenomena In Quark Gluon Plasmas Arnold, 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 Scatterings Xu and Greiner, Phys. Rev. Lett. 100, (2008) Shear viscosity in a gluon gas Xu, Greiner and Stoecker, Phys. Rev. Lett. 101, (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 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 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 gluon distribution function gg↔gg collision terms gg↔ggg collision terms matrix element delta function EM conservation phase-space measure [ gain - loss ]

Matrix elements: gg↔gg and gg↔ggg q q k Soft-collinear approximation gg↔ggg, factorized form, Gunion-Bertsch, PRD 25, 746(1982)

Shear viscosity: variational method perturbation in distribution function linear in χ(x,p)

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 Inserting eq for B(p) into shear viscosity, quadratic form in B(p) B(p) can be expanded in orthogonal polynomials orthogonal condition

Shear viscosity: variational method Inserting eq for B(p) into shear viscosity, quadratic form in B(p)

Collisional rate Boltzmann equation written in Collisional rate is defined by

Regulate infrared and collinear divergence for kT 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 for 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 used phase space dim: ~ 2X3-4=2 polar and azimuthal angles, (θ,φ)

■ Soft gluon approximation in our work (as one option of our calculation) Importance of phase space for 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↔gg This 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, (2005).

Soft gluon approximation in cross section of gg↔ggg two roots: y' (forward), -y' (backward) keep only positive root for y': a factor 1/2 Eq.(D5), Xu & Greiner, PRC71, (2005) Biro, et al, PRC48, 1275 (1993)

Our results-with GB formula

Leading-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 result in weak coupling

η22: Bose and Boltzmann gas

Collisional rates

Shear viscosity from 22 and 23 process

Effects of 23 process

Comparison: AMY, XG, Our work (GB) η/smain ingredientsLL gg↔ggg effect, 1- η (22+23)/ η 22 α_s < 0.01α_s > 0.01 Arnold, Moore, Yaffe pQCD, analytic, variational, boson, g↔gg, LPM (m_D), dominated by 2-body phase space Yes~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) YesLPM (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 remarks: 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 23 Exact matrix element in vacumm for massless gluons all momenta are incoming or outgoing exact matrix element for massless gluon is invariant for

Regulating 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-Bertsch Using light-cone variable Gluon momenta are

Exact 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 22 LL : the leading log result HTL: hard-thermal-loop MD: m_D as regulator AMY: Arnold-Moore-Yaffe normalized by η_22 (m_D) gluon mass = m_D

Numerical results: η/s for LL : the leading log result HTL: hard-thermal-loop MD: m_D as regulator AMY: Arnold-Moore-Yaffe gluon mass = m_D

Numerical results: error estimate XG

Numerical results: η_{22}/η_{22+23}

Conclusion 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

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