Experimental signatures of perfect fluidity at RHIC. Derek Teaney Arkansas State University

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1 Experimental signatures of perfect fluidity at RHIC Derek Teaney Arkansas State University

2 Extracting Transport from the Heavy Ion Data Derek Teaney Arkansas State University

3 Observation: y There is a large momentum anisotropy: b φ x v 2 p x 2 p y 2 p x 2 + p y 2 20% Interpretation The medium responds as a fluid to differences in X and Y pressure gradients Hydrodynamic models work well enough. Is the system Large enough? Does it live Long enough for hydro?

4 How Long and Large is Long/Large Enough? What is the mean free path? l mfp η e+p The mean free path should be less than the expansion rate τ : η e + p }{{} l mfp τ Then using the relation: (e + p) = st. η }{{} s Liquid parameter τt }{{} Experimental parameter:. η/s needs to be small to have interacting QGP at RHIC. 2. Even if η/s is small, dissipative effects are significant!

5 Perturbative estimate of η/s: 2 η e + p nσ = }{{} n }{{} σ α 2 s/t 2 T 3 α 2 st So the Figure of Merit: /α 2 s {}} T { η/(e + p) α 2 s }{{} η/s Liquid Parameter /τ {}}{ expansion rate }{{} τt Experimental Parameter Arnold, Moore, Yaffe found η 50 T 3 /g 4, and taking α s /2 ( η 0.5 s 0.3 α s ) 2

6 Estimates of η/s. Perturbative QCD Kinetic Theory Arnold, Moore, Yaffe. η 50 T 3 g 4. Set α s /2 and m D a reasonable value η e + p τ 0.3 }{{} η/s τt }{{} 2. Strongly Coupled conformal N=4 SYM AdS/CFT Son, Starinets, Policastro No kinetic theory exists. η e + p τ = }{{} 4π η/s τt }{{} N = 4 calculation was important. It showed that η/s can be small enough to have hydro at RHIC, at least theory in some field theories

7 Outline Where does hydro break down?. Small Systems Implications of CuCu Data on Viscosity 2. Large Viscosity and large p T Efforts to simulate RHIC collisions with viscous hydro 3. Heavy Particles Heavy Quarks and implications for Hydro

8 Observing the mean free path in the Cu-Cu Run: Based on Bhalero, et al., Phys. Lett. B. ; see also Heiselberg; Voloshin and Poskanzer For peripheral AuAu they argue η e + p τ Inverse Knudsen number the typical number of scatterings ) K R = σ l mfp ( ) dn = σ S dy τf τ o log ( S dn dy {}}{ n(τ ) dτ ( τf In the low density limit K then expect v 2 K v 2 S dn dy τ o )

9 Comparing S dn dy ( v2 ɛ in CuCu and AuAu ) CuCu 0.5 b=5.5 fm ( v2 ɛ ) AuAu b=8.0 fm Low density limit prediction!

10 velops; this time is of the order of the transverse size R. Quite remarkably, the density thus defined varies little with centrality, and has almost the same value in Au-Au using the event plane from directed flow in a sensitive What toiseccentricity ɛ in smallfluctuations. systems? Dynamically ɛ should be understood as ɛ part. (Phobos) y y x x Two particle correlations yield v 2 {2} = v 2 2 To estimate then v 2 /ɛ part we should estimate (Bhalero and Olltirault) FIG. : Schematic view of a collision of two identical nuclei, in the plane transverse to the beam direction (z-axis). The v x- and y-axes are drawn as per the 2 standard v 2{2} convention. The dots indicate the positions of ɛ participant ɛ{2} nucleons. Due to fluctuations, the overlap zone could be shifted and tilted with respect tofortunately the (x, y) frame. x and y are the principal axes of inertia of the dots. with ɛ{2} ɛ 2 ɛ part ɛ 2 part part

11 of N part measured at midrapidity ( η < ), for Au+Au Cu+Cu data are from Ref. [4] and [5], respectively. The e boxes are 90% C.L. systematic errors. PHOBOS v 2 / ε (b) <ε standard > Au-Au <ε part > Au-Au <ε standard > Cu-Cu <ε part > Cu-Cu PHOBOS Preliminary Low density prediction: ( v2 ɛ CuCu 0.5 AuAu ) ( ) v2 0.5 ɛ N p = N p = N part cipant eccentricities, calculated using a Glauber model two eccentricities, for Au+Au and Cu+Cu collisions at articipants N part. Only the σ statistical errors of v 2 are Deviations from low density limit?

12 CuCu Conclusions Estimate Knudsen number from this data K R l mfp < 3 Estimate the η/s from the Knudsen number R K = l mfp = η st Substitute R =.0 fm and T = 200 MeV η > s 0.33 * I Initially lost interest in CuCu because v 2 {2} v 2 {4}. * v 2 {2} v 2 {4} is understood as flucts in ɛ part. I need to see that v 2 {2} ɛ{2} v 2{4} ɛ{4}

13 Viscous Simulations of the Heavy Ion Reaction Viscous simulations are starting to appear. A. Muronga, D. Rischke; D. Teaney; Chaudhuri, Heinz; R. Baier, P. Romatschke; Koikidee Normally in viscous hydrodynamics the stress tensor is written T ij = pδ ij η i u j i u j i u j + j u i 2 3 δ ij u All schemes to solve viscous hydro involve a relaxation time approximation T ij = p δ ij + π ij t π ij = π ij + η i u j τ R How different is the viscous solution from the ideal solution?

14 Bjorken Solution with transverse expansion η/s = 0.2 ) 2 e τ (GeV/fm τ= v 0.8 Viscous Euler 0 8 τ= τ =, 3.5, 6, 8, 0 6 τ= τ=8.2 τ= First the viscous case does less longitudinal work R (fm) Then the transverse velocity grows more rapidly because the transverse pressure is larger. The larger transverse velocity then reduces the energy density more quickly than ideal hydro. Viscous corrections do NOT integrate to give an O() change to the flow.

15 Similar results by other groups: R. Baier, P. Romatschke, nucl-th/06008 Temp. vs. Rad. for different τ τ=5. fm/c τ=2.2 fm/c η/s=0.000 η/s=0.08 η/s=0.3 T [GeV] τ=9. fm/c r [fm]. Viscosity doesn t change the solution particularly much. FIG. 4: Temperature profile for calculations with different η/s (dashed, dotted and solid lines, respectively) for three different times (see text for details). As expected, for larger values of η/s, differences to ideal hydrodynamics are biggest and viscous hydrodynamics initially cools slower than ideal hydrodynamics. However, note that in certain regions and at later times, viscous hydrodynamics turns out to give temperatures smaller than the corresponding ideal hydrodynamic calculation. 2. There is a consensus on this result.

16 Freezeout & Viscous Corrections to Spectra () The solution is the same as ideal case Viscosity also changes the thermal distribution function f f o + δf δf pi p j T 2 π ij The viscous correction grows with momentum δf l mfp L ( pt T ) 2 Substitute this viscous correction when performing freezeout integrals for spectra E dn d 3 p p µ dσ µ (f o + δf)

17 Freezeout & Viscous Corrections to Spectra (2) R. Baier, P. Romatschke nucl-th/06008 Chaudhuri, Phys. Rev. C. (2006) 2 Freezeout T = 0.35 Freezeout T = T C tiny d 3 N/d 3 p [a.u.] dn/p T dp T dy (GeV -2 ) 0 2 η/s=0.000 π s, scaled by 2.64!/s=0 η/s=0.08 0!/s=0.08 η/s= !/s=0.35 η/s=0.3 K s, scaled by p s, 0 scaled -4 by d 3 N/d 3 p [a.u.] 00 0 V π + π K + K -!/s=0!/s=0.08!/s=0.35 pbar p scaled by 0.7 η/s=0.000 η/s=0.08 η/s= p T [GeV] pt (GeV) p T [GeV] pt (GeV) FIG. 7: P T distribution of pions. The solid line is for ideal FIG. 9: Elliptic flow as a function of transverse momentum. The correction solid line is for to ideal the fluid. distribution The long-dashed, function Thefluid. correction The long-dashed comes and medium-dashed entirely from lines are the for viscous viscous fluid with ADS/CFT (η/s=0.08) and perturbative medium-dashed and short-dashed lines are for viscous fluid FIG. 5: Particle (η/s=0.35) spectra estimate in ideal of viscosity. and Non-equilibrium viscous hydrodynamics correction with for η/s=0.08 a freeze-out and 0.35 temperature respectively. Non-equilibrium of T f = 0.35 correction to equilibrium distribution function is included. Yuck! to equilibrium distribution function is included. GeV, for fixed initial temperature T 0 = 0.36 GeV at τ 0 = fm/c (left) and different T 0 s at τ 0 = fm/c (right). The value of TWhen 0 in the right viscosity plot is becomes chosen such that large thehydro resulting becomes slope of theunusable. spectra best fits the experimental data [66]. We find that if one adjusts T 0 = 0.33 GeV ceedsat the η/s equilibrium = 0.08 and contribution T 0 = 0.3 at pat T =4.5 η/s GeV. = 0.6, Thus.2 with perturbative estimate of viscosity, hydrodynamic the resulting particle spectra are nearly!/s=0.08 indistinguishable, anddescription there is break good down agreement above p T with 4.5GeV experimental. The blast.0 data. We remind that the decay feed-down!/s=0.35 wave model analysis [? ] on the otherhand indicated has been neglected here, which is expected to significantly affect that viscous dynamics get invalidated beyond p T.7

18 Viscous corrections to the spectra seen in the Data? Expect the spectrum to change as a function of mean free path to system size f f o + δf δf l mfp L ( pt T ) 2 This leads to the prediction that v 2 (p T ) is not a universal function of p T v 2 (p T ) v v 2(p T ) 2 CuCu v 2 PHENIX, nucl-ex/ plots AuAu

19 4 [5, 23, 26, 29] are that: (i) v 2 scalr a broad range of impact parameters entricity varies, i.e. v 2 (p T )/ɛ should centrality; (ii) v 2 (p T ) should be indeg system size for a given eccentricity; ent particle species, v 2 (KE T ) at midale with the transverse kinetic energy here m T is the transverse mass of the v % 0-20% 20-30% 30-40% 40-50% Au+Au (a) tistics v 2 data to test these scaling prere constraints for the range of validity drodynamics. The measurements were 200 GeV with the PHENIX detec- Approximately Au+Au and Cu minimum-bias collisions were an- 004 and 2005 running periods, respecon vertex z, along the beam direction o be within z < 30 cm. The event Au collisions was determined via cuts am-beam Counter (BBC) versus Zero er analog response [3]. For Cu+Cu e of the BBC analog response was used. were detected in the two central arms k reconstruction was accomplished usbers and two layers of multi-wire prors with pad readout (PC and PC3) 2 m, 2.5 m and 5 m, respectively [30]. ht (TOF) detector positioned at a ra- 06 m, was used to identify pions (π ± ), nti)protons (p)p. The BBCs and TOFded the global start and stop signals. nts were used in conjunction with the um and flight-path length to generate a ribution [32]. A momentum dependent ch peak in this distribution was used and (p)p in the range 0.2 < p T < 2.5 < 2.5 GeV/c and 0.5 < p T < 4.5 v 2 v 2 (p T,centrality) k v 2 (centrality) % 0-20% 20-30% 30-40% Cu+Cu (b) (GeV/c) p T FIG. : v 2 vs. p T for charged hadrons obtained in (a) Au+Au and (b) Cu+Cu collisions for the centralities indicated. (c) v 2 (centrality, p T ) divided by k=3. (see text) times the p T - integrated value v 2 (centrality) for Au+Au and Cu+Cu. I find this universal p T dependence difficult to reconcile with viscous hydro (c)

20 Viscous Hydro Summary Viscous hydro is growing up. There are still tough issues to be resolved. Viscosity will place boundaries on where to believe ideal hydro. Low p T How low? Not too early How early is too early? Not too late How late is too late?

21 Will heavy quarks thermalize? Collisions scarcely change the direction of the heavy quark δθ T δθ~ m T p~ T m Thus for heavy quarks random walk 2 ( θ) ~ N kick 2 (δθ) ~ N kick m T The equilibration time is then τ heavy R M T τ light R M T η e + p Can we learn something about the mean free path from heavy quarks?

22 Langevin description of heavy quark thermalization: Write down an equation of motion for the heavy quarks. dx dt dp dt = p M The drag and the random force are related = η D p + }{{} ξ(t) }{{} Drag Random Force ξi (t)ξ j (t ) = κ 3 δ ij δ(t t ) η D = κ 2MT η D is rate of momentum loss. /η D is what we intuitively called τ charm R. Einstein relation between the drag and diffusion coefficients. η D = M T D All parameters are related to the heavy quark diffusion coefficient

23 Want to describe heavy quark energy loss and flow STAR: nucl-ex/ Phenix: Phys. Rev. Lett. 96 (2006). R AA STAR charged hadrons p T > 6 GeV/c I: D G LV R (c+b) II: BDMPS (c+b) III: DGLV R+EL (c+b) IV: H e es/rapp EL (c+b) V: B D MPS (c only) R AA.4 non-photonic (e +e )/2.2 (a) (d) Au+Au (0-5%) (2a) (2b) (b) (c) p T (GeV/c) [GeV/c] p T These data are at high momentum. The Langevin description and Diffusion is strictly speaking non-relativistic. Need to extrapolate to learn about transport coefficients

24 Extrapolating Langevin results to the experimentally relevant momentum range Assumes lots and lots of small changes rather than a few hard collisions. The drag and gaussian fluctuations becomes a function of momentum. dp dt p Extreme Model For a given diffusion D, this model will have the smallest R AA. Drag increases with momentum Relaxation time independent of momentum! 2. Fluctuations are minimal at large p T Can safely use this model to give an upper bound on D

25 Connections with strongly coupled N = 4 Super Yang Mills An example of a field theory where this model is realized 0.4 dp dt = with λ = g 2 N c is the coupling constant F λπt u Langevin Figure approach 2: A plot appears of the function relatively F (u) naturally which determines the string profile, in units where u h P M v v C. Figure Herzog 3: et Schematic al.; DT and drawing J. Casalderrey; of the physical S. solution Gubser (left) in which energy flows toward the h and the unphysical energy solution (right) in which energy flows away from the horizon. This function is plotted in Figure 2. It vanishes as u and diverges to as u

26 A Langevin model for the heavy quarks in the medium Hydro + Heavy Quarks All parameters are related to the diffusion coefficient of the heavy quark, τ charm R = M T D Initial spectrum of heavy quarks from Caciarri et. al. Fragmentation and Decay of hadrons to get electrons Similar Langevin approach followed by Rapp & van Hees Different Momentum dependence

27 Summary. Suppression and Elliptic Flow are intimately related. 2. From the suppression pattern, we estimate that D < 2 2πT 3. With this diffusion coefficient, I can t produce enough elliptic flow. 4. Rapp and van Hees come closest to data with coalescence Increases R AA Increases v 2

28 STAR Data R AA STAR charged hadrons p T > 6 GeV/c I: D G LV R (c+b) II: BDMPS (c+b) III: DGLV R+EL (c+b) IV: H e es/rapp EL (c+b) V: B D MPS (c only) 0. (d) Au+Au (0-5%) p T (GeV/c) Much more to life than Langevin; apologies to many theorists Wicks, Horowitz, Djordjevic, Armesto, Mustafa, Zhang, Molnar...

29 . Small Systems Summary Where does hydro breakdown? Implications of CuCu Data on Viscosity Does not support a viscosity which is too small η s > 0.33 More experimental and theoretical work is needed 2. Large Viscosity and large p T Efforts to simulate RHIC collisions with viscous hydro Need η/s small (say η/s = 0.3) in order for hydro to be self-consistent. Work is in progress to settle remaining issues

30 3. Heavy Particles Heavy Quarks and implications for Hydro The data are interesting but confusing Taken at face value D < 2 2πT Difficult to accommodate the large elliptic flow of heavy flavors Same problem in the light quark case Looking forward to LHC