Flow Results and Hints of Incomplete Thermalization

Transcription

1 Flow Results and Hints of Incomplete Thermalization for the STAR Collaboration 1

2 The Perfect Liquid Γ s = 4 η(e + p) 3 D. Teaney, PRC (003) Viscosity reduces v Viscosity needs to be small in order to explain data

3 The Hydro Success Y. Lu et al. Journal of Phys. G (006) v /ε approaches the limit of ideal hydrodynamics Hadronic interaction alone does not produce enough v STAR, PRC (00) 3

4 Revisit the v /ε plot Part I : - There are many v methods, what is the relation among them? - There are many ways to calculate the eccentricity, which one to choose? Part II : - Is the hydrodynamic limit really saturated? - What is the trend we should expect if the requirement on local equilibrium is relaxed? STAR, PRC (00) 4

5 Part I: Choose the Right {v,ε} Pairs 5

6 Definition of Planes ψ RP - Reaction plane. Defined by the direction of the impact parameter. ψ PP - Participant plane. Defined by the principle axis of the participant zone. ψ EP - Event plane. Defined by the flow vector Q = {Q x, Q y }. Q n cos(nψ n ) = X n = w i cos(nφ i ) i Q n sin(nψ n ) = Y n = w i sin(nφ i ) i 6

7 Definition of Eccentricities Define ε = {ε x,ε y } = { σ y σ x σ x + σ y We have ε x ε RP part, σ xy σ x + σ y part } ε x ε optical ε part ε PP = ε x + ε y The angle between Ψ RP and Ψ PP is given by ΔΨ = 1 arctan( ε y ε x ) 7

8 ε x and ε y ε x ε y When small, the distribution of ε x and ε y can be well approximated by Gaussian, and it is found that the width of them are very close to each other. 8

9 The distribution of participant eccentricity Given ε x ~ ε y ~ σ ε σ ε 1 π e 1 π e ( ε x ε RP ) σ ε ε y σ ε The probability for is ε part ε PP = ε x + ε y dn = ε part dε part σ I 0 ε ε part ε RP σ ε ε PP exp ε part + ε RP σ BG ε part; ε RP,σ ε ε ( ) 9

10 Some useful mathematical identities dn = ε part dε part σ I 0 ε ε part ε RP σ ε exp ε part + ε RP σ BG ε part; ε RP,σ ε ε ( ) x = x + σ, x x 4 = x 4 x 6 9 x 4 x +1 x 3 = 4 x 6 10

11 The distribution of participant eccentricity dn = ε part dε part σ I 0 ε ε part ε RP σ ε exp ε part + ε RP σ BG ε part; ε RP,σ ε ε ( ) With this Pdf, both of the 4th order and 6th order cumulant ε part equal to ε RP S. Voloshin, nucl-th/

12 The distribution of participant v Define v ' = cos(φ Ψ PP ) s ' = sin(φ Ψ PP ) = 0 Assuming that on average, flow is proportional to ε, then the distribution of v is given by: dn dv ' = v ' I 0 ( v ' v )exp( v ' + v σ v,dyn σ v,dyn ) σ v,dyn 4 With this Pdf, the 4th order and 6th order cumulant are v and, respectively. Although higher order cumulant v can not give us the unbiased v in the participant plane due to nonflow and fluctuations, they can give us v in the reaction plane (not the participant plane)! 4 v 6 1

13 The q distribution method M q n,x = 1 M cos(nϕ i ) i=1 M q n,y = 1 M sin(nϕ i) i=1 dn dq = q σ I 0 v q M σ exp q + Mv σ q distribution has the same formula as that for v, that means that v obtained from fitting the q-distribution will give us v in the reaction plane (not the participant plane) too. v {qdist}=v {4} Here we see that if we calculate v {q Cumulat} (=1/ Μ ( q - q 4 ) 1/4 ), it will give us the same answer as v {4} 13

14 Check with data v {4}=v {qdist} Data points from STAR, PRC (00) G. Wang (STAR) QM005 Au +Au 00 GeV v {4}=v {ZDC-SMD} STAR preliminary 14

15 Choose the right {v, ε} pairs v that are sensitive to anisotropy w.r.t. the Reaction Plane v : v {4}, v {qdist}, v {qcumulant4}, v {ZDCSMD} ε that are sensitive to anisotropy w.r.t. the Reaction Plane: ε{std}, ε{4} v that are sensitive to anisotropy w.r.t. the Participant Plane : v {},v {EP},v {uq} etc. ε That are sensitive to anisotropy w.r.t. the Participant Plane: ε{part} ε{} In this slide (and throughout this talk as well), I assume that nonflow has been suppressed by external techniques (such as pseudorapidity gap etc.) in v measurements that are based on two particle correlations (v {},v {EP},v {uq} ). S.Voloshin, A.Poskanzer,A.Tang and G.Wang, Phys. Lett. B 659 (008) 537 R.Bhalerao and J-Y. Ollitrault, Phys. Lett. B 614 (006) 60 15

16 Flow Increases Y. Bai, Ph.D. Thesis, STAR. v {4}/ε std increases with centrality over large p t range (v {} did not allow for this study due to strong nonflow at high p t ). Peak position of v {4} moves to higher transverse momentum with increasing centrality 16

17 Part II: What if we relax the requirement for local equilibrium? 17

18 Is hydro limit saturated? Let s check a classical example A jet of sand deforms into an extraordinarily thin symmetric granular sheet clearly resembling a spreading liquid. A sharply focused azimuthal pattern is seen if the target has a rectangular shape. v /ε = 0.6 ~ comparable to central AuAu collisions at RHIC! (shall we believe that it behaves like ideal hydro as well? ) X. Cheng, G. Varas, D. Citron, H. Jaeger and S. Nagel, Phys. Rev. Lett (007) 18

19 Is hydro limit saturated? Let s check different EoSs P. Houvine Nucl. Phys. A (005) An EoS with a rapid crossover over predicted the flow 19

20 How to view the hydro behavior better? - Move away from it - Ideal fluid and low viscosity local equilibrium (small λ or large σ) - To study the local equilibrium, we have to move away from it, say, check what if we relax the constraint of local equilibrium - How to get a complete view? Study Boltzman equation for diluted system. It recovers Hydro when λ becomes small. To have a complete view of Lu Mountain, one has to move away from it. - Shi Su (1037~1101) 0

21 Transport Theory and Hydrodynamics Transport Theory Hydrodynamics Microscopic Applicable out of equilibrium Cannot describe phase transition D<<1 Macroscopic Local equilibrium Can treat phase transition K<<1 D (Dilution parameter) = Typical distance between two particles Mean free path K (Knudsen number) = Mean free path System size Boltzmann Equation will be reduced to Hydrodynamics when both D<<1 and K<<1 1

22 Connecting Pieces D n 1/3 λ = σn/3 n: particle density σ: parton cross section R: system size λ: mean free path 1 K R λ λ = 1 σn n = 1 1 ct S t R / c s 1 K = σ 1 S dn dy dn dy Number of collisions. Local thermal equilibrium is achieved if k -1 >>1 c s c

23 v from Solving the Boltzmann Equation Hydro limit is recovered when D<<1 and K <<1 C. Gombeaud and J-Y Ollitrault, nucl-th/ v 1/K, and v saturates eventually when the system reaches local equilibrium v hydro ε = v ε K / K 0 See next slide 3

24 How much deviation from ideal hydro? v hydro ε = v ε K / K 0 K = λ / R 1 K = σ S dn dy C s For the case with Standard ε : σ=4.9mb, v /ε=0.38. For the case with CGC ε : σ=6.mb, v /ε=0.. Fitting function from Drescher, Dumitru, Gombeaud, J.Ollitrault, Phys. Rev. C76, 04905(007) CGC ε obtained from A.Adil, H-J Drescher,A.Dumitru, A.Hayashigaki and Y.Nara, Phys. Rev. C (006) Dashed lines are hydro limit from fitting the data (as opposed to a pure theoretical calculation as as adopted before) ~40% away from ideal hydro even in central collisions 4

25 How much deviation from ideal hydro? Boltzmann Curves : J-Y Ollitrault. Y. Bai, Ph.D. Thesis, STAR. J. Mlynarz 07 An improved analysis since QM06 Considerable deviation from ideal Hydro 5

26 v 4 Systematics From v Y. Bai, Ph.D. Thesis, STAR. The first order systematics in v 4 is from flow*nonflow term The nonflow term is from v nonflow (not v 4 ) The difference between v {FTPC} and v {4} is used in the estimation of nonflow of v 4. 6

27 An Inconvenient Truth (not really related to global warming) - While it is generally accepted that Hydrodynamics did a good job, for the first time, in describing RHIC s data, there are features that are not consistent with a complete thermalization, and they cannot be easily dismissed. THE END 7

28 Backup Slides 8

29 Eccentricity Definitions 9