Viscosity of quark gluon plasma DONG Hui ( 董 辉 ) Shandong University ( 山东大学 ) in collaboration with J.W. Chen(NTU), Q. Wang(USTC), K. Ohnishi(NTU) / J. Deng(SDU) / Y.-f. Liu (NTU) based on Phys. Lett. B685, 277 (2010) Phys.Rev.D83, (2011)

What is quark gluon plasma

What is viscosity viscosity = resistance of liquid to viscous forces (and hence to flow) Shear viscosity Navier 1822 Generally, the shear viscosity coefficient (η) indicates how fast the fluid equilibrates due to momentum transport between the different velocity domains.

Shear viscosity in ideal gas and liquid ideal gas, high T liquid, low T lower bound by uncertainty principle or Ads/CFT Danielewicz, Gyulassy, 1985 Policastro,Son,Starinets, 2001 Kovtun,Son,Starinets,2005 Frenkel, 1955

η/s around phase transition Lacey et al, PRL98, (2007) Csernai, et al PRL97, (2006) Relativistic viscous hydrodynamics calculations have confirmed very low values of η/s in order to reproduce the RHIC elliptic flow (v 2 ) data.

An access to the viscosity is provided by the Kubo formula (in which the transport coefficients can be derived from equilibrium correlation functions of the EM tensor). → turns out to be somewhat complicated.  One can use effective kinetic theory to get the leading term of obsevables (which are dominantly sensitive to the dynamics of typical excitations). These two approaches are equivalent at leading order in the cases of scalar theory and QED. General approaches

Previous results on shear viscosity for GP ► Variational solution of Boltzmann equation G. Baym et al., Phys. Rev. Lett. 64, 1867 (1990); Nucl. Phys. A525, 415C (1991) H. Heiselberg, Phys. Rev. D49, 4739 (1994) Arnold, Moore and Yaffe, JHEP 0011, 001 (2000) Arnold, Moore and Yaffe, JHEP 0305, 051 (2003) Transport coefficients in high temperature gauge theories: (II) Beyond leading log ► BAMPS: Boltzmann Approach of MultiParton Scatterings (microscopic transport model) 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: ~ ( 10-20)% (AMY) ~ (70-90)% (XG) XG showed that the inelastic process can contribute about eighty percent to the final shear viscosity over entropy density. consistent with that indicated by relativistic viscous hydrodynamics calculations for RHIC It raises again the question whether PQCD interactions can in fact explain that quark gluon plasma behaves like a “strongly coupled” system with a small shear viscosity to entropy ratio.

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 gluon approximation ---> factorized form Gunion-Bertsch, PRD 25, 746(1982)

Importance of phase space for gg↔ggg ■ almost 3-body (3-jet) phase space (used by Xu-Greiner) treated as equal footing phase space dim: ~ 3X3-4= 5

■ 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) soft The point: If the soft bremsstrahlung limit can be used as an indeed good approximation for the inelastic process, the results should not change significantly if we impose the same limit to the phase space.

■ almost 2-body phase space (used by Arnold-Moore-Yaffe) colinear splitting function is used phase space dim: ~ 2X3-4=2 polar and azimuthal angles, (θ,φ)

LL result for gg↔gg : Bose –Einstein vs. Boltzmann gas We reproduced AMY's leading-log(LL) result, For Boltzmann gas, LL result: Our numerical results show good agreement to LL result in weak coupling

Shear viscosity from 22 and 23 process Using the same set of and from XG, we approximately reproduced their results, except that our calculation is based on variational method. It’s interesting to note the good agreement using two different cut-offs for. Changing to two-body-like phase space, the correction from the number changing process is small. This is close to AMY’s conclusion.

Comparison : AMY, XG, Our work η/smain ingredients gg↔ggg effect, 1- η / 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 ~10% Xu, Greiner BAMPS, numerical, Boltzmann gas, gg↔ggg (GB), LPM ( rate), 3-body phase space ~[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) 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 remarks ■ We have bridged to some extent the gap between AMY and XG. ■ To our understanding, their main difference lies in the phase space for number changing processes. There are much more 3-body-like 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-like configuration? or equivalently: Does GB formula over-estimate the rate of the general 3-body-like configuration? If so, how to understand it? How about the bulk viscosity of gluon gas? How after adding quarks?

THANK YOU !