1. Perfect Fluid: flow measurements are described by ideal hydro Problem: all fluids have some viscosity -- can we measure it? I. Radial flow fluctuations: Dissipated by shear viscosity II. Contribution to transverse momentum correlations III. Implications of current fluctuation data Phys. Rev. Lett. 97 162302 (2006) Measuring Viscosity at RHIC Sean Gavin Wayne State University Mohamed Abdel-AzizInstitut für Theoretische Physik, Frankfurt

2. Measuring Shear Viscosity flow v x (z) shear viscosity  Elliptic and radial flow suggest small shear viscosity Teaney; Kolb & Heinz; Huovinen & Ruuskanen Problem: initial conditions & EOS unknown CGC  more flow? larger viscosity? Hirano et al. Kraznitz et al.; Lappi & Venugopalan; Adil et al. Additional viscosity probes? What does shear viscosity do? It resists shear flow. Small viscosity or strong flow?

3. Radial Flow Fluctuations neighboring fluid elements flow past one another  viscous friction small variations in radial flow in each event shear viscosity drives velocity toward the average damping of radial flow fluctuations  viscosity

4. Evolution of Fluctuations diffusion equation diffusion equation for momentum current kinematic viscosity shear viscosity  entropy density s, temperature T r z momentum current momentum current for small fluctuations momentum conservation u u(z,t) ≈ v r  v r  shear stress

5. viscous diffusion viscous diffusion drives fluctuation g t    V  = 2 /  0 /0/0 VV random walk in rapidity random walk in rapidity y vs. proper time  kinematic viscosity  Ts formation at  0 Viscosity Broadens Rapidity Distribution  rapidity broadening increase

6. Hydrodynamic Momentum Correlations momentum flux density correlation function   satisfies diffusion equation  r g  r g  r g,eq satisfies diffusion equation Gardiner, Handbook of Stochastic Methods, (Springer, 2002) fluctuations diffuse through volume, driving r g  r g,eq width in relative rapidity grows from initial value  

7. How Much Viscosity? viscosity in collisions -- Hirano & Gyulassy supersymmetric Yang-Mills:  s   pQCD and hadron gas:  s ~ 1 viscous liquid pQGP ~ HRG ~ 1 fm nearly perfect sQGP ~ (4  T c ) -1 ~ 0.1 fm Abdel-Aziz & S.G kinematic viscosity effect on momentum diffusion: limiting cases: wanted: rapidity dependence of momentum correlation function

8. observable: Transverse Momentum Covariance propose: propose: measure C(  ) to extract width  2 of  r g measures momentum-density correlation function C depends on rapidity interval  C()C() 

9.  s ~ 0.08 ~ 1/4  Current Data? STAR measures rapidity width of pt fluctuations most peripheral  * ~ 0.45 find width  * increases in central collisions central  * ~ 0.75 naively identify naively identify  * with  freezeout  f,p ~ 1 fm,  f,c ~ 20 fm  ~ 0.09 fm at T c ~ 170 MeV  J.Phys. G32 (2006) L37

10. we want: Uncertainty Range STAR measures: maybe  n  2  * STAR, PRC 66, 044904 (2006) uncertainty range  *    2  *  0.08   s  0.3 density correlation function density correlation function may differ from  r g

11. Current Data  Viscosity Hints!

12. Measure p t Covariance for Clearer Picture

13. current fluctuation data compatible with compare to other observables with different uncertainties Summary: small viscosity or strong flow? test the perfect liquid  viscosity info viscosity broadens momentum correlations in rapidity viscosity broadens momentum correlations in rapidity p t covariance measures these correlationsp t covariance measures these correlations

14. sQGP + Hadronic Corona viscosity in collisions -- Hirano & Gyulassy supersymmetric Yang-Mills:  s   pQCD and hadron gas:  s ~ 1 Broadening from viscosity H&G  Abdel-Aziz & S.G, in progress QGP + mixed phase + hadrons  T(  )

15. Centrality Dependence freezeout time model Teaney, Lauret & Shuryak rapidity width STAR, J.Phys. G32 (2006) L37 centrality: mean parton path length Abdel-Aziz & S.G, in progress

16. covariance Covariance  Momentum Flux unrestricted sum: correlation function: C =0 in equilibrium 

17. p t Fluctuations Energy Independent Au+Au, 5% most central collisions STAR, Phys.Rev. C72 (2005) 044902

18. Flow Contribution  2 =  p t     p t   PHENIX explained by blueshift? thermalization flow added dependence on maximum dependence on maximum p t, max needed: more realistic hydro + flow fluctuations

19. want: Covariance  Momentum Flux STAR measures: two correlation functions: where:

20. 130 GeV s 1/2 =200 GeV blue-shift: average increases enhances equilibrium contribution thermalization flow added M. Abdel-Aziz & S.G. participants Thermalization + Flow 20 GeV blue-shift cancels in ratio

21. Rapidity Dependence of Momentum Fluctuations momentum correlation function near midrapidity relative rapidity y r = y 1 -y 2 Abdel-Aziz, S.G fluctuations fluctuations in rapidity window  weak dependence on y a = (y 1 +y 2 )/2 take  o ~ 0.5,  f  o  2 

22. we want: Uncertainty Range STAR measures: density correlation function    density correlation function  r n  r n  r n,eq may differ from  r g maybe  n  2  * STAR, PRC 66, 044904 (2006) uncertainty range  *    2  *  0.08   s  0.3 momentum density correlations density correlations

23. observable: Transverse Momentum Covariance vanishes in equilibrium measures correlation function

24. viscous diffusion  V  = 2 /  0 /0/0 VV random walk in rapidity random walk in rapidity y vs. proper time  kinematic viscosity  Ts formation at  0 Viscosity Broadens Rapidity Distribution