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

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?

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

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

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

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  

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() 

 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 

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, (2006) uncertainty range  *    2  * 

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

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  

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

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.

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  

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?

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

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

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

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

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, (2006) uncertainty range  *    2  *  0.08   s  0.3 momentum density correlations density correlations

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

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(  )

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

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