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 S.G., M. Abdel-Aziz, Phys. Rev. Lett. 97, 162302 (2006) S.G., M. Abdel-Aziz, Phys. Rev. Lett. 97, 162302 (2006) IV. Rapidity + Azimuthal Correlation Data -- in progress Measuring Viscosity at RHIC Sean Gavin Wayne State University

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; Dumitru et al. Additional viscosity probes? What does shear viscosity do? It resists shear flow. Small viscosity or strong flow?

3. Transverse 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. 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() 

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

9. 0.08 <  s < 0.3 Current Data? STAR measures rapidity width of p t fluctuations most peripheral  * ~ 0.45 find width  * increases in central collisions central  * ~ 0.75 J.Phys. G32 (2006) L37 naively identify naively identify  * with  (strictly, ) but but maybe  n  2  * STAR, PRC 66, 044904 (2006) uncertainty range  *    2  * 

10. Part of a   Correlation Landscape STAR focus:near side peak -- “soft ridge” focus: near side peak -- “soft ridge”   behavior: transverse and elliptic flow, momentum conservation plus viscous diffusion near side peak S.G. in progress Theory

11. Soft Ridge: Centrality Dependence STAR, J.Phys. G32 (2006) L37 near side peak pseudorapidity width     azimuthal width   centrality dependence: path length trends: rapidity broadening (viscosity) azimuthal narrowing common explanation of trends? find: requires radial and elliptic flow plus viscosity find:   requires radial and elliptic flow plus viscosity  

12. Flow  Azimuthal Correlations mean flow depends on position blast wave opening angle for each fluid element depends on r correlations: momentum distribution: gaussian spatial r(x 1, x 2 ) :  t -- width in  t -- width in

13. Measured Flow Constrains Correlations diffusion + flow for correlations constraints: constraints: flow velocity, radius from measured v 2,  p t  vs. centrality radial plus elliptic flow: radial plus elliptic flow: “eccentric” blast wave n part STAR 200 GeV Au+AuPHENIX 200 GeV Au+Au Heinz et al.

14. Rapidity and Azimuthal Trends rapidity width: viscous broadening,  s ~  assume local equilibrium for all impact parameters fixes  F  b  azimuthal width: flow dominates viscosity effect tiny agreement with data easier if we ignore peripheral region -- nonequilibrium? STAR data, J.Phys. G32 (2006) L37  

15. current correlation data compatible with comparable to other observables with different uncertainties Summary: small viscosity or strong flow? need viscosity info to test the perfect liquid viscosity broadens momentum correlations in rapidity viscosity broadens momentum correlations in rapidity p t covariance measures these correlationsp t covariance measures these correlations azimuthal and rapidity correlations flow + viscosity works! hydro OK flow + viscosity works! hydro OK relation of  structure to the jet-related ridge? relation of  structure to the jet-related ridge?

16. p t Fluctuations Energy Independent Au+Au, 5% most central collisions sources of p t fluctuations: thermalization, flow, jets? central collisions  thermalized energy independent bulk quantity  jet contribution small

17. Azimuthal Correlations from Flow transverse flow: transverse flow: narrows angular correlations no flow         v rel elliptic flow: v 2 contribution STAR subtracted viscous diffusion: increases spatial widths  t and  t     t    t   - 1/2 momentum conservation:  sin  ; subtracted Borghini, et al

18. Elliptic and Transverse Flow flow observables: flow observables: fix  x,  y, and R vs. centrality n part STAR 200 GeV Au+AuPHENIX 200 GeV Au+Au transverse plus elliptic flow: transverse plus elliptic flow: “eccentric” blast wave blast wave: elliptic flow transverse flow

19. Blast Wave Azimuthal Correlations STAR 200 GeV Au+Au data azimuthal trend :  t  system size  t constant roughly: gaussian spatial r(x 1, x 2 ) :  t -- width in  t -- width in

20. 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

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

22. 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(  )

23. Part of a   Correlation Landscape STAR, J.Phys. G32 (2006) L37 soft ridge: soft ridge: near side peak ridge superficially similar to ridge seen with jets J. Putschke, STAR but at soft p t scales -- no jet near side peak

24. 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