Hints of incomplete thermalization in RHIC data

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1 Hints of incomplete thermalization in RHIC data Nicolas BORGHINI CERN in collaboration with R.S. BHALERAO Mumbai J.-P. BLAIZOT ECT J.-Y. OLLITRAULT Saclay N. BORGHINI p.1/30

2 RHIC Au Au results: the fashionable view Ideal fluid dynamics reproduce both p t spectra and elliptic flow v 2 (p t ) of soft (p t 2 GeV/c) identified particles for minimum bias collisions, near central rapidity. This agreement necessitates a soft equation of state, and very short thermalization times: τ thermalization < 0.6 fm/c. strongly interacting Quark-Gluon Plasma N. BORGHINI p.2/30

3 Ideal fluid dynamics in heavy-ion collisions A few reminders on fluid dynamics Fluid dynamics in heavy ion collisions: theory Overall scenario General predictions of ideal fluid dynamics Anisotropic flow Fluid dynamics and heavy ion collisions: theory vs. data Reconciling data and theory (?) (including predictions for Cu Cu@RHIC and Pb Pb@LHC) R.S. Bhalerao, J.-P. Blaizot, N.B., J.-Y. Ollitrault, nucl-th/ N. BORGHINI p.3/30

4 Fluid dynamics: various types of flow Thermodynamic equilibrium? mean free path 3 Knudsen number Kn = λ system size L Kn 1: Free-streaming limit Kn 1: Thermalization: Fluid (hydro) limit Reynolds number Re = Lv fluid viscosity η Re 1: Ideal (non-viscous) flow η λc s Re 1: Viscous flow Viscous or Ideal? Mach number Ma = v fluid speed of sound c s Ma 1: Incompressible flow Ma > 1: Compressible (supersonic) flow Compressible or Incompressible? N. BORGHINI p.4/30

5 Fluid dynamics: various types of flow Three numbers: Kn = λ L, Re = Lv fluid η, Ma = v fluid c s an important relation: Kn Re = λv fluid η v fluid c s = Ma Compressible fluid: Thermalized means Ideal Viscosity departure from equilibrium N. BORGHINI p.5/30

6 Ideal fluid picture of a heavy-ion collision 0. Creation of a dense gas of particles 1. At some time τ 0, the mean free path λ is much smaller than all dimensions in the system thermalization (T 0 ), ideal fluid dynamics applies 2. The fluid expands: density decreases, λ increases (system size also) 3. At some time, the mean free path is of the same order as the system size: ideal fluid dynamics is no longer valid (kinetic) freeze-out Freeze-out usually parameterized in terms of a temperature T f.o. If λ varies smoothly with temperature, consistency requires T f.o. T 0 analytical predictions, see N.B. & J.-Y. Ollitrault, nucl-th/ N. BORGHINI p.6/30

7 Heavy-ion observable: Anisotropic flow Non-central collision: b Initial anisotropy of the source (in the transverse plane) anisotropic pressure gradients, larger along the impact parameter b E dn d 3 p y z x φ - Φ R anisotropic emission of particles: anisotropic (collective) flow dn [ ] 1 + 2v 1 cos(φ Φ R ) + 2v 2 cos 2(φ Φ R ) +... p t dp t dy directed elliptic Flow : misleading terminology; does NOT imply fluid dynamics! N. BORGHINI p.7/30

8 Non-central collisions: parameters Initial conditions in non-central collisions, will be characterized by a parameter measuring the shape of the overlap region: y 2 x 2 spatial eccentricity ɛ = y 2 + x 2 two numbers measuring the size of the overlap region: reduced radius 1 R 1 = x y 2 (anisotropic flow caused by pressure gradients) transverse area of the collision zone S = 2π x 2 y 2 N. BORGHINI p.8/30

9 Dependence of v 2 on centrality The natural time scale for v 2 is R/c s : v 2 scaled by initial excentricity y 2 x 2 y 2 +x 2 Ε v 2 b 2 Impact parameter dependence massless particles c s = c s t R N. BORGHINI p.9/30

10 Dependence of v 2 on centrality The natural time scale for v 2 is R/c s : v 2 scaled by initial excentricity y 2 x 2 y 2 +x 2 Ε v 2 b 4 b 2 Impact parameter dependence massless particles c s = c s t R N. BORGHINI p.9/30

11 Dependence of v 2 on centrality The natural time scale for v 2 is R/c s : v 2 scaled by initial excentricity y 2 x 2 y 2 +x 2 Ε v 2 b 6 b 4 b 2 Impact parameter dependence massless particles c s = c s t R N. BORGHINI p.9/30

12 Dependence of v 2 on centrality The natural time scale for v 2 is R/c s : v 2 scaled by initial excentricity y 2 x 2 y 2 +x 2 Ε v 2 b 8 b 6 b 4 b 2 Impact parameter dependence massless particles c s = c s t R N. BORGHINI p.9/30

13 Dependence of v 2 on centrality The natural time scale for v 2 is R/c s : v 2 scaled by initial excentricity y 2 x 2 y 2 +x 2 Ε v 2 b 10 b 8 b 6 b 4 b 2 Impact parameter dependence massless particles c s = c s t R N. BORGHINI p.9/30

14 Dependence of v 2 on centrality The natural time scale for v 2 is R/c s : v 2 scaled by initial excentricity y 2 x 2 y 2 +x 2 Ε v 2 b 12 b 10 b 8 b 6 b 4 b 2 Impact parameter dependence massless particles c s = 1 3 v 2 ɛ v 2 knows nothing about early times! c s t R N. BORGHINI p.9/30

15 Dependence of v 2 on the speed of sound v 2 Ε 0.8 Variation with c s c s c s t R N. BORGHINI p.10/30

16 Dependence of v 2 on the speed of sound v 2 Ε 0.8 Variation with c s c s 0.2 c s c s t R N. BORGHINI p.10/30

17 Dependence of v 2 on the speed of sound v 2 Ε 0.8 Variation with c s c s 0.3 c s 0.2 c s c s t R N. BORGHINI p.10/30

18 Dependence of v 2 on the speed of sound v 2 Ε 0.8 Variation with c s c s 0.4 c s 0.3 c s 0.2 c s c s t R N. BORGHINI p.10/30

19 Dependence of v 2 on the speed of sound v 2 Ε 0.8 Variation with c s c s 0.5 c s 0.4 c s 0.3 c s 0.2 c s c s t R N. BORGHINI p.10/30

20 Dependence of v 2 on the speed of sound v 2 Ε c s 1 3 c s 0.5 c s 0.4 c s 0.3 c s 0.2 c s 0.1 Variation with c s c s t R one can increase v 2 by increasing c s N. BORGHINI p.10/30

21 Anisotropic flow: predictions of hydro Characteristic build-up time of v 2 is R/c s v 2 /ɛ constant across different centralities v 2 roughly independent of the system size (Au Au vs. Cu Cu) v 2 increases with increasing speed of sound c s Mass-ordering of the v 2 (p T ) of different particles (the heavier the particle, the smaller its v 2 at a given momentum) Relationship between different harmonics: v 4 (v 2 ) 2 = 1 2 N. BORGHINI p.11/30

22 RHIC Au Au data: a personal choice [1/4] v 2 (p t ) at midrapidity, minimum bias collisions: STAR Collaboration, Phys. Rev. C 72 (2005) STAR Data π ± K S 0 p Λ+Λ v Hydrodynamic results π K p Λ π ± systematic error Transverse momentum p t (GeV/c) N. BORGHINI p.12/30

23 RHIC Au Au data: a personal choice [2/4] v 2 (p t ) for various centralities (impact parameters): STAR Collaboration, Phys. Rev. C 72 (2005) (a) pion ± b (fm) (b) anti-proton Hydro Model α = 0.0 v (c) pion ± (d) anti-proton Hydro Model α = 0.02 fm Transverse momentum p t (GeV/c) N. BORGHINI p.13/30

24 RHIC Au Au data: a personal choice [2/4] v 2 (p t ) for various centralities (impact parameters): STAR Collaboration, Phys. Rev. C 72 (2005) (a) pion ± b (fm) (b) anti-proton Hydro Model α = 0.0 (c) pion data above ± 0.15 ideal fluid 0.1 hydro yields 0.05 maximum v 2 v (d) anti-proton Hydro Model α = 0.02 fm Transverse momentum p t (GeV/c) N. BORGHINI p.13/30

25 RHIC Au Au data: a personal choice [3/4] (Pseudo)rapidity dependence of v 2 (%) v 2 STAR Collaboration, Phys. Rev. C 72 (2005) Hirano & Tsuda, Phys. Rev. C 66 (2002) STAR PHOBOS v 2 (hydro) flatter than data η N. BORGHINI p.14/30

26 RHIC Au Au data: a personal choice [4/4] v 4 Transverse momentum dependence of (v 2 ) 2 STAR Collaboration, Phys. Rev. C 72 (2005) v 4 /v p t (b) (GeV/c) ideal fluid prediction N. BORGHINI p.15/30

27 Ideal fluid dynamics vs. RHIC data v 2 (p t ) hydro < data v 2 (y) hydro data v 4 hydro < data (v 2 ) 2 what is wrong with ideal fluid scenario? N. BORGHINI p.16/30

28 Ideal fluid dynamics vs. RHIC data v 2 (p t ) hydro < data v 2 (y) hydro data v 4 hydro < data (v 2 ) 2 what is wrong with ideal fluid scenario? 0. Creation of a dense gas of particles 1. At some time τ 0 ( 0.6 fm/c in hydro models), the mean free path λ is much smaller than all dimensions in the system (Kn 1) thermalization, ideal fluid dynamics applies 2. The fluid expands: density decreases, λ increases (system size also) 3. At some time, the mean free path is of the same order as the system size: ideal fluid dynamics is no longer valid N. BORGHINI p.16/30

29 Ideal fluid dynamics vs. RHIC data v 2 (p t ) hydro < data v 2 (y) hydro data v 4 hydro < data (v 2 ) 2 what is wrong with ideal fluid scenario? 0. Creation of a dense gas of particles 1. At some time τ 0 ( 0.6 fm/c in hydro models), the mean free path λ is much smaller than all dimensions in the system (Kn 1) Is this really true? N. BORGHINI p.16/30

30 Ideal fluid dynamics vs. RHIC data v 2 (p t ) hydro < data v 2 (y) hydro data v 4 hydro < data (v 2 ) 2 what is wrong with ideal fluid scenario? 0. Creation of a dense gas of particles 1. At some time τ 0 ( 0.6 fm/c in hydro models), the mean free path λ is much smaller than all dimensions in the system (Kn 1) Is this really true? Can we estimate the number of collisions per particle Kn 1? How do v 2, v 4 depend on Kn 1? N. BORGHINI p.16/30

31 Anisotropic flow vs. number of collisions Kn 1 An exact computation of the dependence of v 2, v 4 on the number of collisions per particle Kn 1 requires some cascade model but we can guess the general tendency! in the absence of reinteractions (Kn 1 = 0), no flow develops the more collisions, the larger the anisotropic flow for a given number of collisions, the system thermalizes: further collisions no longer increase v 2 v 2 fully thermalized (hydro) incomplete thermalization 1 1 Kn N. BORGHINI p.17/30

32 Anisotropic flow vs. number of collisions Kn 1 An exact computation of the dependence of v 2, v 4 on the number of collisions per particle Kn 1 requires some cascade model but we can guess the general tendency! in the absence of reinteractions (Kn 1 = 0), no flow develops the more collisions, the larger the anisotropic flow for a given number of collisions, the system thermalizes: further collisions no longer increase v 2 should be quantified! v 2 fully thermalized (hydro) incomplete thermalization 1 1 Kn N. BORGHINI p.17/30

33 Incomplete equilibration & RHIC data [1] v 4 Ideal fluid dynamics predicts (v 2 ) 2 = 1, RHIC data are above ( 1.2) 2 increase can be explained by incomplete equilibration naturally: v n proportional to the number of collisions Kn 1 v 4 (v 2 ) 2 1 v 2 fully thermalized (hydro) Kn 1 1 incomplete thermalization Kn 1 v 4 2 v 2 1/2 v 4 1 Kn 1 1 Kn 1 N. BORGHINI p.18/30

34 Number of collisions Kn 1 : a control parameter The natural time (resp. length) scale for v 2 is R/c s (resp. R) number of collisions per particle to build up v 2 : Kn 1 R ( ) R λ = R σ n c s σ c S σ interaction cross section, n(τ) particle density, S transverse surface 1 S c s dn dy dn dy control parameter for v 2: to vary Kn 1, one can study centrality dependence (using the universality of v 2 /ɛ) beam-energy dependence system-size dependence importance of lighter systems! rapidity dependence transverse momentum dependence (p T σ Kn 1 ) N. BORGHINI p.19/30

35 Control parameter: centrality dependence The number of collisions to build up v 2 is Kn 1 = R λ σ S In Au Au collisions at RHIC: ( ) b (fm) R (fm) dn dy n R c s (fm 3 ) dn dy n( R c s ), hence λ, varies little for b = 0 8 fm, while R varies by 30% centrality-dependence of v 2 ɛ 1 S dn dy -dependence N. BORGHINI p.20/30

36 Incomplete equilibration & RHIC data [2] Centrality and beam-energy dependence: v 2 /ε 0.25 HYDRO limits E lab /A=11.8A GeV, E877 E lab /A=40A GeV, NA49 E lab /A=158A GeV, NA49 s NN =130 GeV, STAR s NN =200 GeV, STAR Prelim (1/S) dn ch /dy NA49 Collaboration, Phys. Rev. C 68 (2003) N. BORGHINI p.21/30

37 Incomplete equilibration & RHIC data [2] Centrality and beam-energy dependence: v 2 /ε E lab /A=11.8A GeV, E877 E lab /A=40A GeV, NA49 E lab /A=158A GeV, NA49 s NN =130 GeV, STAR s NN =200 GeV, STAR Prelim (1/S) dn ch /dy NA49 Collaboration, Phys. Rev. C 68 (2003) Scaling law seems to work for RHIC data (+ matching with SPS) v 2 (Kn 1 ) increases steadily (no hint at hydro saturation in the data) N. BORGHINI p.21/30

38 /2 Incomplete equilibration & RHIC data [3] '/ /d (Pseudo)rapidity dependence of v 2 Steve Manly (PHOBOS Coll.) QM 05 PHOBOS Collaboration Phys. Rev. Lett. 91 (2003) % central part N ch dn GeV 130 GeV 19.6 GeV ' = -ybeam v 2 (η) and dn dy approximately proportional v 2 Kn 1 Hirano, Phys. Rev. C 65 (2002) N. BORGHINI p.22/30

39 Reconciling data and theory In hydrodynamical fits, the speed of sound is constrained by p t spectra, which require a soft equation of state with a hard equation of state, the energy per particle is too high All relies on the assumption that the energy per particle is related to the density, i.e., that chemical equilibrium is maintained chemical equilibrium is more fragile than kinetic equilibrium the only experimental indication of chemical equilibrium is in the particle-abundance ratios (cf. however e + e... ) If there is no chemical equilibrium, energy per particle and density are independent variables, as in ordinary thermodynamics there is no constraint on the equation of state from p t spectra: one can consider a larger c s to increase v 2 in central collisions N. BORGHINI p.23/30

40 Incomplete equilibration: predictions for Cu Cu flow The matching between central SPS and peripheral RHIC suggests that we can even compare systems with different densities, i.e., different σ (and c s ) compare Au Au at b = 8 fm with Cu Cu at b = 5.5 fm (similar centrality) If hydro holds, v 2 should scale like ɛ: v 2 (Cu) = 0.69 v 2 (Au) If thermalization is incomplete, v 2 ɛ 1 S v 2 (Cu) = 0.34 v 2 (Au) dn dy Kn 1, i.e. Cu Cu further from equilibrium than Au Au v 4 (v 2 ) 2 > 1.2 N. BORGHINI p.24/30

41 Cu Cu collisions at RHIC: anisotropic flow [1/2] Steve Manly (PHOBOS QM 05: Cu Cu results seem to be compatible with Kn 1 -scaling N. BORGHINI p.25/30

42 Cu Cu collisions at RHIC: anisotropic flow [2/2] Gang Wang (STAR QM 05: Measurements with different methods give very different v 2 values (not a surprise... ) Wait and see! N. BORGHINI p.26/30

43 Hints of incomplete equilibration in RHIC data R.S. Bhalerao, J.-P. Blaizot, N.B., J.-Y. Ollitrault, nucl-th/ A reminder: the natural time scale for anisotropic flow is R c s no knowledge about early times anisotropic flow cannot conclude on early thermalization Size of v 2 controlled by Kn 1 1 S dn dy incomplete equilibration, but data do not saturate: v 2 overshoots the hydrodynamical prediction... because the latter is over-constrained by a non-existent chemical equilibrium Predictions for Cu Cu collisions at RHIC... data shown at QM 05 too preliminary... and for Pb Pb at LHC! N. BORGHINI p.27/30

44 Predictions for LHC Measuring anisotropic flow at LHC, you will find v 2 larger than at RHIC (getting closer to thermalization) ɛ larger signal, larger statistics easier measurement v 4 (v 2 ) 2 smaller than at RHIC (closer to the ideal fluid value 1 2 ) Well... that definitely means a smaller signal... Smaller systems yield complementary values of 1 dn S dy Kn 1, allowing checks (thermalization or not? onset of equilibration?) N. BORGHINI p.28/30

45 Hints of incomplete equilibration in RHIC data Extra slide N. BORGHINI p.29/30

46 Methods of flow analysis Anisotropic flow is usually measured using two-particle correlations: cos 2(φ 1 φ 2 ) cos 2(φ 1 Φ R ) cos 2(Φ R φ 2 ) = (v 2 ) 2 Assumption: all two-particle correlations are due to flow which is obviously wrong! Non-flow sources of correlations: jets, decays of short-lived particles, global momentum conservation, quantum effects between identical particles, etc. can bias the standard flow analysis The bias is comparatively larger for smaller systems New methods for measuring flow have been developed cumulants of multiparticle correlations, Lee Yang zeroes (N.B., P.M. Dinh, J.-Y. Ollitrault, R.S. Bhalerao, ) N. BORGHINI p.30/30

Lecture 12: Hydrodynamics in heavy ion collisions. Elliptic flow Last lecture we learned:

Lecture 12: Hydrodynamics in heavy ion collisions. Elliptic flow Last lecture we learned: Particle spectral shapes in thermal model ( static medium) are exponential in m T with common slope for all particles.