1. Perfect Fluid: flow measurements are described by ideal hydro Problem: all fluids have some viscosity -- can we measure it? I. Transverse 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 Using Correlations 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.; Lappi & Venugopalan; Adil et al.; Dumitru; Torrieri; Fries. 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 flow in each event shear viscosity drives velocity toward the average damping of transverse 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 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 

10. 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  * 

11. Further Experimental Information STAR, J.Phys. G32 (2006) L37 near side correlations  peak pseudorapidity width     azimuthal width   centrality dependence: path length trends: rapidity broadening (viscosity) azimuthal narrowing common explanation of trends? expect: requires radial and elliptic flow plus viscosity expect:   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.; Trainor

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  Lacy’s talk 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 ridge?  Pruneau’s talk relation of  structure to the ridge?  Pruneau’s talk

16. Contributing Effects transverse flow: narrows angular correlations no flow     1/2        1/2  v rel elliptic flow: sinusoidal v 2 contribution (subtr.) “nonflow-flow” narrows correlations viscous diffusion: increases spatial widths  t and  t      v rel 2   t    t   

17. 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

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

19. 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(  )