Shear and Bulk viscosity of QGP in PQCD

Shear and Bulk viscosity of QGP in PQCD Qun Wang Univ of Sci & Tech of China Chen, Dong, Ohnishi, QW, Phys. Lett. B 685, 277(2010); Chen, Deng, Dong, QW, Phys. Rev. D 83, 034031(2011); Chen, Deng, Dong, QW, ar. Xiv: 1107. 0522 HIC in LHC era, 15 -21 July 2012, Quy Nhon, Vietnam Qun Wang/USTC/China 1

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 (Kubo formula) Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 2

Shear viscosity in ideal gas and liquid • ideal gas, high T • liquid, low T Frenkel, 1955 • lower bound by uncertainty principle Qun Wang/USTC/China Danielewicz, Gyulassy, 1985 Policastro, Son, Starinets, 2001 3

η/s around phase transition Lacey et al, PRL 98, 092301(2007) Csernai, et al PRL 97, 152303(2006) Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 4

What is bulk viscosity (breaking of scale symmetry) X (mean free path) X (energy momemtum density) correlation of trace of energy-momemtum tensor in x and y Generated by dilatation low-momentum behavior of correlator (Kubo formula) Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 5

Scale anomaly in QCD Massless QCD Lagrangian and action (1) (2) The classical action of massless QCD is invariant under scale transformation (3) Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 6

Scale anomaly & bulk viscosity Breaking of scale invariance is proportional to beta-function (4) Beta-function (5) Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 7

ζ/s around phase transition Bernard et al, (MILC) PRD 2007, Karsch, Kharzeev, Tuchin, PLB 2008 Cheng et al, (RBC-Bielefeld) PRD 2008, Noronha *2, Greiner, PRL 2009, Bazavov et al, (Hot. QCD), ar. Xiv: 0903. 4379 Chen, Wang, PRC 2009, Qun Wang/USTC/China 8 Li, Huang, PRD 2008, . . . Shear & Bulk viscosity of QGP in PQCD

Bulk viscosity at Tc Helium-3 near the critical point Kogan, Meyer, J. Low. Temp. Phys. 110, 899(1998) ～ a few millions ! Bulk viscosity near T_c diverges in power law, closely related to fluctuation 3 -D Ising model Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 9

Previous works on η, ζ for QGP Shear viscosity: ►PV: Perturbative and Variational approach Danielewicz, Gyulassy, PRD 31, 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, 641 C (2009). ►Anomalous: Asakawa et al, PRL 96, 252301(2006); Mujumder et al, PRL 99, 192301(2007). Bulk viscosity: ►PV: Perturbative and Variational approach Arnold, Dogan, Moore, PRD 74, 085021(2006). ►Sum rule and spectral desity Kharzeev, Tuchin, JHEP 0809, 093(2008); Moore, Saremi, PRD JHEP 0809, 015(2008); Romatschke, Son, PRD 80, 065021 (2009). ►Lattice Meyer, JHEP 1004, 099 (2010); PRL 100, 162001 (2008). ►Models near T_c Noronha*2, Greiner, PRL 103, 172302(2009) ; Li, Huang, PRD 80, 034023(2009). Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 10

Contradicting results on η for gluon plasma ►PV: Perturbative and Variational approach Danielewicz, Gyulassy, Phys. Rev. D 31, 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 Multi. Parton Scatterings Xu and Greiner, Phys. Rev. Lett. 100, 172301(2008) Shear viscosity in a gluon gas Xu, 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) Qun Wang/USTC/China σ(23) (AMY) << σ(23) (XG) Shear & Bulk viscosity of QGP in PQCD 11

Difference: AMY vs XG 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. 1) The Boltzmann equation is solved in a variation method. AMY Gluon as quantum gas. 2) For number changing processes Ng↔ (N+1)g: using collinear splitting g↔gg Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 12

Our 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) Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 13

Boltzmann equation for gluon plasma gluon distribution function phase-space measure gg↔gg collision terms matrix element gg↔ggg collision terms delta function EM conservation [ gain - loss ] Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 14

Matrix elements: gg↔gg (HTL) (6) where q is momentum transferred Heiselberg, Wang, NPB 462, 389(1996) 1 3 q and gluon HTL self-energy 2 4 (7) Qun Wang/USTC/China 15

Exact matrix element for 23 Exact matrix element in vacumm for massless gluons 5 1 4 (8) 2 3 exact matrix element for massless gluon is invariant for all momenta are incoming or outgoing Ellis, Sexton, NPB 269, 445 (1986) Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 16

Exact matrix element to Gunion-Bertsch Taking large s limit (s→ ) and then small y limit (y→ 0) (13) Gunion-Bertsch formula NOTE (important!!) Gunion-Bertsch formula is only valid for soft region and Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD (14) 17

Linearized Boltzmann equation Perturbation in distribution function Jeon, PRD 52, 3591(1995) (15) Jeon, Yaffe, PRD 53, 5799(1996) Right-hand-side or collision part of Boltzmann Eq. (16) Qun Wang/USTC/China 18

Linearized Boltzmann equation Left-hand-side or kinetic part of Boltzmann Eq. (17) Linearized Boltzmann Eq. Collision integral of B(p) shear (18) Collision integral of A(p) bulk Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 19

Shear/bulk viscosity in stress tensor Stress tensor perturbation (19) Solve A(p) and B(p) from linearized Boltzmann Eq. → shear/bulk viscosity (20) Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 20

Constraints ■ Invariance of energy density in local rest frame (27) ■ Speed of sound (28) ■ Mass (29) Qun Wang/USTC/China 21

Functional basis for A(p) ■ Our choice (30) ■ ADM’s choice (31) ■ Results do not depend on bases, on a certain basis, (32) Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 22

Evaluating bulk viscosity ■ Having A(p), we can evaluate ζ (33) ■ So ζ is proportional to β(g) Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 23

Analytic and numerical results Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 24

Leading-Log result: ζ/s (34) (35) (36) Weinberg, Astrophys. J. 168, 175 (1971) Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 25

N_c scaling (37) (38) Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 26

Numerical results: η/s for 22+23 HTL: hard-thermal-loop AMY: Arnold-Moore-Yaffe gluon mass = m_∞ Qun Wang/USTC/China Chen, Deng, Dong, QW, Phys. Rev. D 83, 034031(2011); Erratum: 84, 039902; Shear & Bulk viscosity of QGP in PQCD 27

Numerical results: ζ for 22+23 Chen, Deng, Dong, QW, ar. Xiv: 1107. 0522 Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 28

Comparison of ζ and η Weinberg, Astrophys. J. 168, 175 (1971) Chen, Deng, Dong, QW, ar. Xiv: 1107. 0522 Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 29

Comparison with lattice result and data Chen, Deng, Dong, QW, ar. Xiv: 1107. 0522 Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 30

Why different between XG and AMY? 1. For gg ↔ gg: (39) 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, (40) 3. We can also carry out the integration over the full phase space of the final state without the symmetry factor (41) Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 31

3 q. T → 0, q → 0, t-channel 1 2 4 4 q. T → 0, q → ∞, u-channel 1 2 3 Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 32

Why different between XG and AMY? A caveat for gg ↔ ggg: 1. GB holds for constrained phase space in CMF: (42) 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) Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 33

Why different between XG and AMY? Lesson: Two ways of integration over final state phase space are equivalent: 1. Use symmetry factor × GB for constrained final state phase space 2. in integration equivalent 2. Use GB for full phase space for final state gluons in integration Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 34

Why different between XG and AMY? Introducing the symmetry factor while integrating over full phase space of final state gluons (= without constraining the phase space) multiple counting ! Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 35

Why GB (soft) ≈ AMY (collinear)? GB: soft emission (in CM frame) AMY: collinear splitting (in heat bath frame) Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 36

Conclusion 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 Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 37

THANK YOU ! Qun Wang/USTC/China Shear & Bulk viscosity of QGP in PQCD 38