A Standard Model for the Little Bang How far are we from the goal?

Transcription

1 A Standard Model for the Little Bang How far are we from the goal? Ulrich Heinz (he Ohio State University) Hydrodynamics of Strongly Coupled Fluids EC*, rento, 2-6 May 24 Supported by the U.S. Department of Energy

2 he Big Bang U. Heinz EC*, 5/2/24 (62)

3 he Little Bang (credit: Paul Sorensen/Chun Shen) U. Heinz EC*, 5/2/24 2(62)

4 Big Bang vs. Little Bang Similarities: Hubble-like expansion, expansion-driven dynamical freeze-out chemical freeze-out (nucleo-/hadrosynthesis) before thermal freeze-out (CMB, hadron p -spectra) initial-state quantum fluctuations imprinted on final state Differences: Expansion rates differ by 8 orders of magnitude Expansion in 3d, not 4d; driven by pressure gradients, not gravity ime scales measured in fm/c rather than billions of years Distances measured in fm rather than light years Heavy-Ion Standard Model still under construction = this talk U. Heinz EC*, 5/2/24 3(62)

5 Relativistic Nucleus-Nucleus Collisions Animation: P. Sorensen Collision of two Lorentz contracted gold nuclei U. Heinz EC*, 5/2/24 4(62)

6 Relativistic Nucleus-Nucleus Collisions Animation: P. Sorensen Collision of two Lorentz contracted gold nuclei U. Heinz EC*, 5/2/24 5(62)

7 Relativistic Nucleus-Nucleus Collisions Animation: P. Sorensen Collision of two Lorentz contracted gold nuclei U. Heinz EC*, 5/2/24 6(62)

8 Relativistic Nucleus-Nucleus Collisions 5 Animation: P. Sorensen -5 Collision of two Lorentz contracted gold nuclei U. Heinz EC*, 5/2/24 7(62)

9 Relativistic Nucleus-Nucleus Collisions Animation: P. Sorensen 5-5 Collision of two Lorentz contracted gold nuclei U. Heinz EC*, 5/2/24 8(62)

10 Relativistic Nucleus-Nucleus Collisions Animation: P. Sorensen 5-5 Collision of two Lorentz contracted gold nuclei U. Heinz EC*, 5/2/24 9(62)

11 Relativistic Nucleus-Nucleus Collisions Animation: P. Sorensen 5 Produced fireball is ~ -4 meters across and -5 lives for ~5x -23 seconds 7 Collision of two Lorentz contracted gold nuclei -5 5 U. Heinz EC*, 5/2/24 (62)

12 Relativistic Nucleus-Nucleus Collisions Animation: P. Sorensen 5 Produced fireball is ~ -4 meters across and -5 lives for ~5x -23 seconds 8 Collision of two Lorentz contracted gold nuclei -5 5 U. Heinz EC*, 5/2/24 (62)

13 he Big Bang vs the Little Bangs he Universe HIC kinetic freeze-out distributions and correlations of produced particles lumpy initial energy density hadronization QGP phase quark and gluon degrees of freedom Credit: NASA WMAP HIC P. Sorensen NSAC Subcommittee 22 U. Heinz EC*, 5/2/24 2(62) 8

14 Big vs. Little Bang: he fluctuation power spectrum Mishra, Mohapatra, Saumia, Srivastava, PRC77 (28) 6492 and C8 (2) 3493 Mocsy & Sorensen, NPA855 (2) 24, PLB75 (2) Big Bang temperature power spectrum (Planck 23) Angular scale Flow power spectrum for ultracentral PbPb Little Bangs (Data: CMS, Quark Matter 22; heory: OSU 23) CMS data MCGlb. η/s =.8 MCGlb. η/s =.2 Dl[µK 2 ] v n { 2 } LHC.3 <p <3 GeV (a) Multipole moment, l n Higher flow harmonics get suppressed by shear viscosity A detailed study of fluctuations is a powerful discriminator between models! U. Heinz EC*, 5/2/24 3(62)

15 Each Little Bang evolves differently! Density evolution of a single b = 8 fm Au+Au collision at RHIC, with IP-Glasma initial conditions, Glasma evolution to τ =.2 fm/c followed by (3+)-d viscous hydrodynamic evolution with MUSIC using η/s =.2 =.5/(4π) Schenke, ribedy, Venugopalan, PRL 8 (22) 2523: x [fm] U. Heinz y [fm] x [fm] y [fm] τ=5.2 fm/c ε [GeV/fm ] τ=.2 fm/c ε [GeV/fm ] τ=. fm/c ε [GeV/fm ] 2-3 x [fm] EC*, 5/2/ y [fm] 4(62)

16 Event-by-event shape and flow fluctuations rule! (Alver and Roland, PRC8 (2) 5495) Each event has a different initial shape and density distribution, characterized by different set of harmonic eccentricity coefficients ε n Each event develops its individual hydrodynamic flow, characterized by a set of harmonic flow coefficients v n and flow angles ψ n At small impact parameters fluctuations ( hot spots ) dominate over geometric overlap effects (Alver & Roland, PRC8 (2) 5495; Qin, Petersen, Bass, Müller, PRC82 (2) 6493) U. Heinz EC*, 5/2/24 5(62)

17 Definition of flow coefficients: How anisotropic flow is measured: dn (i) dyp dp dφ p (b) = dn(i) dyp dp (b) ( +2 n= Define event average {...}, ensemble average... ( v n (i) (y,p ;b)cos n(φ p Ψ (i) n )) ). Flow coefficients v n typically extracted from azimuthal correlations (k-particle cumulants). E.g. k = 2,4: c n {2} = {e ni(φ φ 2 ) } = {e ni(φ ψ n ) }{e ni(φ 2 ψ n ) }+δ 2 = v 2 n+δ 2 c n {4} = {e ni(φ +φ 2 φ 3 φ 4 ) } 2 {e ni(φ φ 2 ) } = v 4 n+δ 4 v n iscorrelatedwiththeeventplanewhileδ n isnot( non-flow ).δ 2 /M,δ 4 /M 3. 4 th -order cumulant is free of 2-particle non-flow correlations. hese measures are affected by event-by-event flow fluctuations: v 2 2 = v 2 2 +σ 2, v 4 2 = v σ 2 v 2 2 v n {k} denotes the value of v n extracted from the k th -order cumulant: v 2 {2} = v 2 2, v 2{4} = 4 2 v v 4 2 U. Heinz EC*, 5/2/24 6(62)

18 Panta rhei: soft ridge = Mach cone =flow! ALAS (J. Jia), Quark Matter 2 ALICE (J. Grosse-Oetringhaus), QM anisotropic flow coefficients v n and flow angles ψ n correlated over large rapidity range! M. Luzum, PLB 696 (2) 499: All long-range rapidity correlations seen at RHIC are consistent with being entirely generated by hydrodynamic flow. in the % most central collisions v 3 >v 2 = prominent Mach cone -like structure! = event-by-event eccentricity fluctuations dominate! U. Heinz EC*, 5/2/24 7(62)

19 Event-by-event shape and flow fluctuations rule! ALICE (A. Bilandzic) Quark Matter 2 in the % most central collisions v 3 >v 2 = prominent Mach cone -like structure! triangular flow angle uncorrelated with reaction plane and elliptic flow angles = due to event-by-event eccentricity fluctuations which dominate the anisotropic flows in the most central collisions U. Heinz EC*, 5/2/24 8(62)

20 Viscous relativistic hydrodynamics (Israel & Stewart 979) Include shear viscosity η, neglect bulk viscosity (massless partons) and heat conduction (µ B ); solve with modified energy momentum tensor µ µν = µν (x) = ( e(x)+p(x) ) u µ (x)u ν (x) g µν p(x)+π µν. π µν = traceless viscous pressure tensor which relaxes locally to 2η times the shear tensor µ u ν on a microscopic kinetic time scale τ π : Dπ µν = τ π ( π µν 2η µ u ν ) +... where D u µ µ is the time derivative in the local rest frame. Kinetic theory relates η and τ π, but for a strongly coupled QGP neither η nor this relation are known = treat η and τ π as independent phenomenological parameters. For consistency: τ π θ (θ = µ u µ =local expansion rate). U. Heinz EC*, 5/2/24 9(62)

21 Numerical precision: Gubser-est Gubser (PRD82 (2) 8527) found analytical solution for relativistic Navier-Stokes equation with conformal EOS, boost-invariant longitudinal and non-zero transverse flow, corresponding to a specific transverse temperature profile. Marrochio, Noronha et al. (arxiv:37.63) found semianalytical generalization of this solution for Israel-Stewart theory. his solution provides a stringent test for numerical Irael-Stewart codes (very rapid and non-trivial transverse expansion!) VISH2+ (C. Shen, 23) U x x (fm) π yy (GeV/fm 3 ) x (fm) π xy (GeV/fm 3 ) x (fm) U. Heinz EC*, 5/2/24 2(62)

22 Initial Condition + = Ω Generate Initial Conditions supermc (C++) Hydrodynamics Flow of time Hydrodynamic simulations VISHNew (FORRAN)? Cascade Particle emission Hadron re-scattering Collect particle information into observables iss (C++) UrQMD (FORRAN) binutilities (Python) U. Heinz EC*, 5/2/24 2(62)

23 package structure generatejobs.py & submitjobs_xxx.py Generate jobs and run them in parallel Job Job 2 Job N supermc Initial condition generator M initial conditions VISHNew Hydrodynamics simulator freeze-out surface info iss Particle emission sampler no Particle space-time and momentum info Finished all events? binutilities Spectra and flow calculator Particle spacetime and momentum info osc2u prepare UrQMD ICs multiple UrQMD Hadron rescattering simulator... yes EbeCollector Collect data into SQLite database zip results and store to results folder Analyze the SQLite databases file: collected.db iebe available at U. Heinz EC*, 5/2/24 22(62)

24 Converting initial shape fluctuations into final flow anisotropies the QGP shear viscosity (η/s) QGP U. Heinz EC*, 5/2/24 23(62)

25 How to use elliptic flow for measuring (η/s) QGP Hydrodynamics converts spatial deformation of initial state = momentum anisotropy of final state, through anisotropic pressure gradients Au+Au RHIC 2~3% Shear viscosity degrades conversion efficiency ε x = y2 x 2 y 2 +x 2 = ε p= xx yy xx + yy of the fluid; the suppression of ε p is monotonically related to η/s p (fm/c) ideal /s =.8 /s =.6 /s =.24 he observable that is most directly related to the total hydrodynamic momentum anisotropy ε p is the total (p -integrated) charged hadron elliptic flow v ch 2 : ε p = xx yy xx + yy i p dp dφp p 2 cos(2φ p) i p dp dφp p 2 dn i dyp dp dφ p dn i dyp dp dφ p v ch 2 U. Heinz EC*, 5/2/24 24(62)

26 How to use elliptic flow for measuring (η/s) QGP (ctd.) If ε p saturates before hadronization (e.g. in PbPb@LHC (?)) v ch 2 not affected by details of hadronic rescattering below c but: v (i) 2 (p dn ), i change during hadronic phase (addl. radial flow!), and these changes depend on details of the hadronic dynamics (chemical composition dyd 2 p etc.) v 2 (p ) of a single particle species not a good starting point for extracting η/s If ε p does not saturate before hadronization (e.g. AuAu@RHIC), dissipative hadronic dynamics affects not only the distribution of ε p over hadronic species and in p, but even the final value of ε p itself (from which we want to get η/s) need hybrid code that couples viscous hydrodynamic evolution of QGP to realistic microscopic dynamics of late-stage hadron gas phase VISHNU ( Viscous Israel-Stewart Hydrodynamics n UrQMD ) (Song, Bass, UH, PRC83 (2) 2492) Note: this paper shows that UrQMD viscous hydro! U. Heinz EC*, 5/2/24 25(62)

27 Extraction of (η/s) QGP from v 2 /ε hydro (η/s)+urqmd (/S) dn ch /dy (fm -2 ) H. Song, S.A. Bass, UH,. Hirano, C. Shen, PRL6 (2) 923 η/s (fm/c) max. η/s τ dn/dy Glauber / KLN v 2 /ε MC-KLN (a) hydro (η/s) + UrQMD v 2 {2} / ε 2 part /2 KLN v 2 / ε part KLN 2 3 (/S) dn ch /dy (fm -2 ) η/s MC-Glauber (b) hydro (η/s) + UrQMD v 2 {2} / ε 2 part /2 Gl v 2 / ε part Gl η/s (/S) dn ch /dy (fm -2 ) < 4π(η/s) QGP < 2.5 All shown theoretical curves correspond to parameter sets that correctly describe centrality dependence of charged hadron production as well as p -spectra of charged hadrons, pions and protons at all centralities v2 ch /ε x vs. (/S)(dN ch /dy) is universal, i.e. depends only on η/s but (in good approximation) not on initial-state model (Glauber vs. KLN, optical vs. MC, RP vs. PP average, etc.) dominant source of uncertainty: ε Gl x vs. εkln x smaller effects: early flow increases v 2 ε by few% larger η/s bulk viscosity affects v ch 2 (p ), but not v ch 2 Zhi Qiu, UH, PRC84 (2) 249 U. Heinz EC*, 5/2/24 26(62)

28 Global description of spectra and v 2 dn/(2 dy p dp ) (GeV -2 ) %~5%* 3 5%~%* 2 %~5%* 5%~2%* 2%~3%/ 3%~4%/ 2 4%~5%/ 3 5%~6%/ 4 6%~7%/ 5 7%~8%/ 6 PHENIX SAR MC-KLN MC-Glauber p (GeV) VISHNU (H. Song, S.A. Bass, UH,. Hirano, C. Shen, PRC83 (2) 549) + (a) dn/(2 dy p dp ) (GeV -2 ) %~%* 2 %~2%* 2%~4% 4%~6%/ 6%~8%/ 2 /s =. (ideal hydro) /s =.8 /s =.6 /s =.24 (b) p (GeV) p v 2 / MC-Glauber initialization MC-KLN initialization 2 A GeV Au+Au charged hadrons s =.8 s = p (GeV) s =.6 s = p (GeV) VISHNU PHENIX v 2 {EP} (-5%)+.2 (5-%)+. (-2%)+.8 (2-3%)+.6 (3-4%)+.4 (4-5%)+.2 (5-6%) (η/s) QGP =.8 for MC-Glauber and (η/s) QGP =.6 for MC-KLN work well for charged hadron, pion and proton spectra and v 2 (p ) at all collision centralities U. Heinz EC*, 5/2/24 27(62)

29 Successful prediction of v 2 (p ) for identified hadrons in PbPb@LHC Data: ALICE, Quark Matter 2 Prediction: Shen et al., PRC84 (2) 4493 (VISH2+) Perfect fit in semi-peripheral collisions! he problem with insufficient proton radial flow exists only in more central collisions Adding the hadronic cascade (VISHNU) helps: U. Heinz EC*, 5/2/24 28(62)

30 v 2 (p ) in PbPb@LHC: ALICE vs. VISHNU v2 Data: ALICE, preliminary (Snellings, Krzewicki, Quark Matter 2) Dashed lines: Shen et al., PRC84 (2) 4493 (VISH2+, MC-KLN, (η/s) QGP =.2) Solid lines: Song, Shen, UH 2 (VISHNU, MC-KLN, (η/s) QGP =.6).3.2. π + K + p ALICE Preliminary VISHNU (η/s) QGP =.6 VISH2+ η/s =.2 5 % 2% 2 3%.2 v2. 3 4% 4 5% 5 6% p (GeV) p (GeV) p (GeV) VISHNU yields correct magnitude and centrality dependence of v 2 (p ) for pions, kaons and protons! Same (η/s) QGP =.6 (for MC-KLN) at RHIC and LHC! U. Heinz EC*, 5/2/24 29(62)

31 Successful prediction of v 2 (p ) for identified hadrons in PbPb@LHC (II) Data: ALICE, Quark Matter 22 Prediction: Shen et al., PRC84 (2) 4493 (VISH2+) Radial flow pushes v 2 for heavier hadrons to larger p heory curves are true predictions, without any parameter adjustment U. Heinz EC*, 5/2/24 3(62)

32 Back to the elephant in the room : How to eliminate the large model uncertainty in the initial eccentricity? U. Heinz EC*, 5/2/24 3(62)

33 wo observations: I. Shear viscosity suppresses higher flow harmonics more strongly Alver et al., PRC82 (2) 3493 (averaged initial conditions) Schenke et al., arxiv: (event-by-event hydro) v n /ε n v 2 /ε 2 v 3 /ε 3 v 4 /ε 4 v 5 /ε 5 v n (viscous)/v n (ideal) v n (η/s=.8)/v n (ideal) v n (η/s=.6)/v n (ideal) 2-3%.8.6 η/s n = Idea: Use simultaneous analysis of elliptic and triangular flow to constrain initial state models (see also Bhalerao, Luzum Ollitrault, PRC 84 (2) 349) U. Heinz EC*, 5/2/24 32(62)

34 wo observations: II. ε 3 appears to be less model dependent than ε 2 Zhi Qiu, UH, PRC84 (2) ε 3 (e) (MC-KLN) ε 3 (s) (MC-KLN) ε 3 (e) (MC-Glb) ε 3 (s) (MC-Glb) (b) Initial eccentricities ε n and angles ψ n : ε n e inψn rdrdφr 2 e inφ e(r,φ) = rdrdφr 2 e(r,φ) ε MC-KLN has larger ε 2 and ε 4, but similar ε 5 and almost identical ε 3 as MC-Glauber ε 4 (e) (MC-KLN) ε 4 (s) (MC-KLN) ε 4 (e) (MC-Glb) ε 4 (s) (MC-Glb) (c) b (fm) ε 5 (e) (MC-KLN) ε 5 (s) (MC-KLN) ε 5 (e) (MC-Glb) ε 5 (s) (MC-Glb) (d) Angles of ε 2 and ε 4 are correlated with reaction plane by geometry, whereas those of ε 3 and ε 5 are random (purely fluctuation-driven) ε ε b (fm) 5 5 b (fm) While v 4 and v 5 have mode-coupling contributions from ε 2, v 3 is almost pure response to ε 3 and v 3 /ε 3 const. over a wide range of centralities = Idea: Use total charged hadron v ch 3 to determine (η/s) QGP, then check v ch 2 to distinguish between MC-KLN and MC-Glauber! U. Heinz EC*, 5/2/24 33(62)

35 v2/ε2 v3/ε Combined v 2 & v 3 analysis: η/s is small! Zhi Qiu, C. Shen, UH, PLB77 (22) 5 and QM22 (e-by-e VISH2+) ALICE v 2 {2}/ε 2 {2} ALICE v 2 {4}/ε 2 {4} MC-KLN v 2 {2}/ε 2 {2} MC-KLN v 2 {4}/ε 2 {4} (a) ALICE v 3 {2}/ε 3 {2} MC-KLN v 3 {2}/ε 3 {2}. MC-KLN η/s =.2 (b) Centrality (%) v2/ε2 v3/ε ALICE v 2 {2}/ε 2 {2} ALICE v 2 {4}/ε 2 {4} MC-Glb. v 2 {2}/ε 2 {2} MC-Glb. v 2 {4}/ε 2 {4} (c) ALICE v 3 {2}/ε 3 {2} v 3 / ε 3. MC-Glb. η/s =.8 (d) Centrality (%) Both MC-KLN with η/s=.2 and MC-Glauber with η/s=.8 give very good description of v 2 /ε 2 at all centralities. Only η/s=.8 (with MC-Glauber initial conditions) describes v 3 /ε 3! PHENIX, comparing to calculations by Alver et al. (PRC82 (2) 3493), come to similar conclusions at RHIC energies (Adare et al., arxiv:5.3928, and Lacey et al., arxiv:8.457) Large v 3 measured at RHIC and LHC requires small (η/s) QGP /(4π) unless the fluctuations in these models are completely wrong and ε 3 is really 5% larger than these models predict! U. Heinz EC*, 5/2/24 34(62)

36 Sub-nucleonic fluctuations U. Heinz EC*, 5/2/24 35(62)

37 Adding sub-nucleonic quantum fluctuations Schenke, ribedy, Venugopalan, PRL8, 2523 (22) MC-Glauber U. Heinz EC*, 5/2/24 36(62)

38 Adding sub-nucleonic quantum fluctuations Schenke, ribedy, Venugopalan, PRL8, 2523 (22) MC-KLN U. Heinz EC*, 5/2/24 37(62)

39 Adding sub-nucleonic quantum fluctuations Schenke, ribedy, Venugopalan, PRL8, 2523 (22) IP-Glasma U. Heinz EC*, 5/2/24 38(62)

40 owards a Standard Model of the Little Bang B. Schenke: QM22 With inclusion of sub-nucleonic quantum fluctuations and pre-equilbrium dynamics of gluon fields: outstanding agreement between data and model Rapid convergence on a standard model of the Little Bang! Schenke, ribedy, Venugopalan, Phys.Rev.Lett. 8:2523 (22) Perfect liquidity reveals in the final state initial-state gluon field correlations of size /Q s (sub-hadronic)! U. Heinz EC*, 5/2/24 39(62) 3

41 What We Don!t Know B. Schenke: QM22 Model doesn t distinguish between a constant η/s of.2 or a temperature dependent η/s with a minimum of /4π Need both RHIC and LHC to sort this out! U. Heinz EC*, 5/2/24 4(62) 4

42 Other successes of the Little Bang Standard Model v n 2 /2 v n 2 / v v 2 v 3 v 4 v 5 v v 2 v 3 v 4 v 5 Gale, Jeon, Schenke, ribedy, Venugopalan, arxiv: (PRL 22) RHIC 2GeV, 3-4% filled: SAR prelim. open: PHENIX RHIC 2GeV, 3-4% filled: SAR prelim. open: PHENIX η/s =.2 η/s() p [GeV] P(v 2 / v 2 ), P(ε 2 / ε 2 ) P(v 3 / v 3 ), P(ε 3 / ε 3 ) P(v 4 / v 4 ), P(ε 4 / ε 4 ) 2-25% ε 2 IP-Glasma v 2 IP-Glasma+MUSIC v 2 ALAS. p >.5 GeV η < v 2 / v 2, ε 2 / ε % ε 3 IP-Glasma v 3 IP-Glasma+MUSIC v 3 ALAS. p >.5 GeV η < v 3 / v 3, ε 3 / ε % ε 4 IP-Glasma v 4 IP-Glasma+MUSIC v 4 ALAS. p >.5 GeV η < v 4 / v 4, ε 4 / ε 4 Model describes RHIC data with lower effective specific shear viscosity η/s =.2 In contrast to MC-Glauber and MC-KLN, IP-Sat initial conditions correctly reproduce the final flow fluctuation spectrum, generated from initial shape fluctuations by viscous hydrodynamics U. Heinz EC*, 5/2/24 4(62)

43 he Little Bang fluctuation power spectrum: initial vs. final.6.5 Little Bang density power spectra -.2% -5% 2-3% 5-6% Solid: IP Glasma Dash dotted: MC Glauber Dashed: MC KLN Flow power spectrum for ultracentral PbPb Little Bangs (Data: CMS, Quark Matter 22; heory: OSU 23).3.25 CMS poster QM22 (Wei Li) MCGlb. η/s =.8 MCKLN η/s =.2 εn v n %@lhc.3 <p <3 GeV n n Higher flow harmonics get suppressed by shear viscosity Neither MC-Glb nor MC-KLN gives the correct initial power spectrum! R.I.P. A detailed study of fluctuations is a powerful discriminator between models! U. Heinz EC*, 5/2/24 42(62)

44 Problems in d+au and p+pb.35 Ideal hydro with multi. flucts. & pre-equilibrium flow (Qin and Müller, PRC89 (24) 4492 (24)) % v 2-5% v 3-2% v 2-2% v 3 PHENIX d-au -5% v % v 2 -% v 3-2% v 2-2% v 3-5% v 2-5% v 3-2% v 2-2% v 3 ALAS p-pb -2% v 2 CMS p-pb -2% v 3 ALICE p-pb -2% v 2 ALICE p-pb -2% v p (GeV/c) p (GeV/c) Data are well described by ideal hydro, so viscous hydro will overpredict v 3 U. Heinz EC*, 5/2/24 43(62)

45 FOURIER HARMONICS IN p+a CMS COLLABORAION, PHYS.LE. B724 (23) v 2 2 / CMS Pb+Pb v 2 CMS p+pb v 2 Pb+Pb v 2 p+pb v 2 /s=.8 v 3 2 / CMS Pb+Pb v 3 CMS p+pb v 3 Pb+Pb v 3 p+pb v 3 CMS.3 GeV < p < 3GeV /s= N trk offline N trk offline Open symbols: Pb+Pb Filled symbols: p+pb Red points: IP-Glasma + MUSIC 26

46 Single event anisotropic flow coefficients In a single event, the specific initial density profile results in a set of complex, y- and p -dependent flow coefficients (we ll suppress the y-dependence): V n = v n e inψn := p dp dφe inφ p dp dφ dn dyp dp dφ dn dyp dp dφ {e inφ }, V n (p ) = v n (p )e inψ n(p ) := dφe inφ dφ dn dyp dp dφ dn dyp dp dφ {e inφ } p. ogether with the azimuthally averaged spectrum, these completely characterize the measurable singleparticle information for that event: dn dydφ = dn 2π dy dn dyp dp dφ = 2π dn ( dyp dp + 2 ( ) v n cos[n(φ Ψ n )], n= + 2 ) v n (p ) cos[n(φ Ψ n (p ))]. n= Both the magnitude v n and the direction Ψ n ( flow angle ) depend on p. v n, Ψ n, v n (p ), Ψ n (p ) all fluctuate from event to event. Ψ n (p ) Ψ n fluctuates from event to event. U. Heinz EC*, 5/2/24 44(62)

47 Higher order event plane correlations in cos(6(ψ 2 Ψ 6 )) Data: ALAS Coll., J. Jia et al., Hard Probes 22 Event-by-event hydrodynamics: Zhi Qiu, UH, PLB 77 (22) 26 (VISH2+) cos(4(ψ 2 Ψ 4 )) cos(8(ψ 2 Ψ 4 )).6 cos(6(ψ 3 Ψ 6 )) cos(2(ψ 2 Ψ 4 )).2 cos(2(ψ 3 Ψ 4 )) cos(6(ψ 2 Ψ 3 )) MC-Glb., η/s =.8 MC-KLN, η/s =.2 N part ALAS data cos((ψ 2 Ψ 5 )) VISH2+ reproduces qualitatively the centrality dependence of all measured event-plane correlations U. Heinz EC*, 5/2/24 45(62)

48 Higher order event plane correlations in Zhi Qiu, UH, PLB 77 (22) 26 cos(4(φ2 Φ 4 )) cos(8(φ 2 Φ 4 )). cos(2(φ2 Φ 4 ))..5 cos(6(φ 2 Φ 3 )) cos(6(φ 2 Φ 6 )).. cos(6(φ 3 Φ 6 )) cos(2(φ 3 Φ 4 )).8 cos((φ2 Φ 5 )) MC-Glb., η/s =.8 MC-KLN, η/s =.2 N part Initial-state participant plane correlations disagree with final-state flow-plane correlations = Nonlinear mode coupling through hydrodynamic evolution essential to describe the data! U. Heinz EC*, 5/2/24 46(62)

49 Higher order event plane correlations in Data: ALAS Coll., J. Jia et al., Hard Probes 22 Event-by-event hydrodynamics: Zhi Qiu, UH, PLB 77 (22) 26 (VISH2+) cos(2ψ 2 6Ψ 3 + 4Ψ 4 ) cos(2ψ 2 + 3Ψ 3 5Ψ 5 ).2 cos(2ψ 2 + 4Ψ 4 6Ψ 6 ).5. cos( 8Ψ 2 + 3Ψ 3 + 5Ψ 5 ) cos( Ψ 2 + 4Ψ 4 + 6Ψ 6 ) cos( Ψ 2 + 6Ψ 3 + 4Ψ 4 ) MC-Glb., η/s =.8 MC-KLN, η/s =.2 N part ALAS data VISH2+ reproduces qualitatively the centrality dependence of all measured event-plane correlations U. Heinz EC*, 5/2/24 47(62)

50 Higher order event plane correlations in Zhi Qiu, UH, PLB 77 (22) 26. cos(2φ 2 + 3Φ 3 5Φ 5 )..2 cos(2φ 2 6Φ 3 + 4Φ 4 ) cos( 8Φ 2 + 3Φ 3 + 5Φ 5 ) MC-Glb., η/s = cos(2φ 2 + 4Φ 4 6Φ 6 ) cos( Φ Φ 4 + 6Φ 6 ).4 cos( Φ 2 + 6Φ 3 + 4Φ 4 ) MC-KLN, η/s =.2 N part Initial-state participant plane correlations disagree with final-state flow-plane correlations = Nonlinear mode coupling through hydrodynamic evolution essential to describe the data! U. Heinz EC*, 5/2/24 48(62)

51 est of factorization of two-particle spectra Factorization V n (p,p 2 ) := {cos[n(φ φ 2 )]} p p 2 vn (p ) v n (p 2 ) was checked experimentally as a test of hydrodynamic behavior, and found to hold to good approximation. Gardim et al. (PRC87(23) 39) pointed out that event-by-event fluctuations break this factorization even if 2-particle correlations are exclusively due to flow. hey proposed to study the following ratio: r n (p,p 2 ) := V n (p,p 2 ) Vn (p,p )V n (p 2,p 2 ) = v n(p )v n (p 2 )cos[n(ψ n (p ) Ψ n (p 2 ))]. v n [2](p )v n [2](p 2 ) Even in the absence of flow angle fluctuations, this ratio is < due to v n fluctuations (Schwarz inequality), except for p =p 2. But it additionally depends on flow angle fluctuations. o assess what share of the deviation from is due to flow angle fluctuations, we can compare with r n (p,p 2 ) := v n(p )v n (p 2 )cos[n(ψ n (p ) Ψ n (p 2 ))] v n (p )v n (p 2 ) which deviates from only due to flow angle fluctuations. Again, this ratio approaches for p =p 2. Gardim et al. studied r n for ideal hydro; we have studied r n and r n for viscous hydro (UH, Z. Qiu, C. Shen, PRC87 (23) 3493) and found that about half of the factorization breaking effects are due to flow angle fluctuations. U. Heinz EC*, 5/2/24 49(62)

52 Breaking of factorization by e-by-e fluctuations: v 2 ),p assoc r 2 (p ) Data: Wei Li (CMS), Hard Probes 23; heory (prediction!): UH, Z. Qiu, C. Shen, PRC87 (23) 3493,p assoc r 2 (p ),p assoc r 2 (p ),p assoc r 2 (p GeV/c < p CMS Preliminary PbPb - = 5 µb L int <.5 GeV/c s NN = 2.76 ev.-.2% centrality -5% -% p 4-5% - p assoc (GeV/c) GeV/c < p p < 2. GeV/c - p assoc (GeV/c) GeV/c < p p < 2.5 GeV/c.5 p - p assoc (GeV/c) p - p assoc (GeV/c) - p assoc (GeV/c) GeV/c < p < 3. GeV/c.5 p - p assoc (GeV/c) p - p assoc (GeV/c) p - p assoc (GeV/c) p - p assoc (GeV/c) p p VISH2+ Hydro Glauber, η/s=.8 MC-KLN, η/s=.2 - p assoc (GeV/c) Sizable effect for v 2 in ultra-central events -.2% 4-5% Hydro from Heinz and Shen PRC 87, 3493 (23) Factorization breaking effects appear to be suppressed by shear viscosity. U. Heinz EC*, 5/2/24 5(62)

53 Breaking of factorization by e-by-e fluctuations: v 3 ),p assoc r 3 (p ) Data: Wei Li (CMS), Hard Probes 23; heory (prediction!): UH, Z. Qiu, C. Shen, PRC87 (23) 3493,p assoc r 3 (p ),p assoc r 3 (p ),p assoc r 3 (p GeV/c < p CMS PbPb - = 5 µb L int <.5 GeV/c s NN = 2.76 ev.-.2% centrality -5% -% p 4-5% - p assoc (GeV/c) GeV/c < p p < 2. GeV/c - p assoc (GeV/c) GeV/c < p < 2.5 GeV/c p p - p assoc (GeV/c) - p assoc (GeV/c) GeV/c < p < 3. GeV/c.5 p - p assoc (GeV/c).5 p - p assoc (GeV/c) p - p assoc (GeV/c) p - p assoc (GeV/c) p - p assoc (GeV/c) p p VISH2+ Hydro Glauber, η/s=.8 MC-KLN, η/s=.2 - p assoc (GeV/c) Smaller effect for v 3 " more sensitive to η/s? -.2% 4-5% Hydro from Heinz and Shen PRC 87, 3493 (23) Factorization breaking effects appear to be larger for fluctuation dominated flow harmonics. U. Heinz EC*, 5/2/24 5(62)

54 Breaking of factorization by e-by-e fluctuations: v 4 ),p assoc r 4 (p ) Data: Wei Li (CMS), Hard Probes 23; heory (prediction!): UH, Z. Qiu, C. Shen, PRC87 (23) 3493,p assoc r 4 (p ),p assoc r 4 (p ),p assoc r 4 (p GeV/c < p CMS PbPb - = 5 µb L int <.5 GeV/c s NN = 2.76 ev.-.2% centrality -5% -% p 4-5% - p assoc (GeV/c) GeV/c < p < 2. GeV/c p - p assoc (GeV/c) p - p assoc (GeV/c) p - p assoc (GeV/c) GeV/c < p p < 2.5 GeV/c.5 p - p assoc (GeV/c) p - p assoc (GeV/c) - p assoc (GeV/c) GeV/c < p < 3. GeV/c.5 p - p assoc (GeV/c) p - p assoc (GeV/c) p p VISH2+ Hydro Glauber, η/s=.8 MC-KLN, η/s=.2 - p assoc (GeV/c) Smaller effect for v 4 " more sensitive to η/s? -.2% 4-5% Hydro from Heinz and Shen PRC 87, 3493 (23) Hydrodynamics qualitatively, and even semi-quantitatively, predicts all observed factorization breaking effects below p 2.5 3GeV. U. Heinz EC*, 5/2/24 52(62)

55 Pre-equilibrium dynamics (I) Match pre-equilibrium µν to viscous hydrodynamic form, at varying matching times τ match. Extreme case: pre-equilibrium = free-streaming = large τ match slow thermalization; short τ match fast thermalization. Studydependenceoffinalobservablesonτ match andcomparewithpurehydrocalculation that assumes no evolution at all between τ = and τ therm =.7fm/c. he following study by Jia Liu uses MC-KLN initial conditions for the gluon phase-space distribution. Viscous hydro evolution with η/s =.2. U. Heinz EC*, 5/2/24 53(62)

56 Pre-equilibrium dynamics (II) v 2 /ε 2, v 3 /ε 3 for thermal pions (left) and thermal protons (right) (Jia Liu, 23): v n {hermal Pion+}/ǫ n for Bumpy Events v n {hermal Proton}/ǫ n for Bumpy Events vn/ǫn(τ) τ s (fm/c) ǫ 2 v 2 ǫ 3 v 3 v 2 /ǫ 2 (τ ) with variance v 3 /ǫ 3 (τ ) with variance Pb Pb@276GeV MC KLN Centrality: %~2% # of events: 4 η/s=.2 vn/ǫn(τ) τ s (fm/c) ǫ 2 v 2 ǫ 3 v 3 v 2 /ǫ 2 (τ ) with variance v 3 /ǫ 3 (τ ) with variance Pb Pb@276GeV MC KLN Centrality: %~2% # of events: 4 η/s= Matching ime τ s (fm/c) Matching ime τ s (fm/c) U. Heinz EC*, 5/2/24 54(62)

57 Pre-equilibrium dynamics (III) p -spectra for thermal pions (left) and thermal protons (right) (Jia Liu, 23): (/2π)(dN/dypdp) (GeV 2 ) 3 2 τ s =.4fm/c τ s =.7fm/c τ s =.fm/c τ s =.3fm/c τ s =.6fm/c pure hydro from.7 fm/c p (GeV/c) Pb-Pb@276GeV MC-KLN Centrality: % 2% Number of events: 4 η/s=.2 (/2π)(dN/dypdp) (GeV 2 ) τ s =.4fm/c τ s =.7fm/c τ s =.fm/c τ s =.3fm/c τ s =.6fm/c pure hydro from.7 fm/c p (GeV/c) Pb-Pb@276GeV MC-KLN Centrality: % 2% # of events: 4 η/s=.2 U. Heinz EC*, 5/2/24 55(62)

58 vahydro: Viscous +%,-$.)$/,#*&)$&%23,#s* g"=#4p=(8b.+4.b-%:"#=(\zv%-="4-( f(τ,x,p) = f eq (p,(τ,x))+δf Q=4*+4V"#("-(:4:&-*P:(=V%#&( U-"=4*+4V"#(8B.+4.B-%:"#=(\ZV%-="4-( f(τ,x,p) = f aniso (p,λ(τ,x), ξ(τ,x) )+δ f }{{}}{{} anisotropy D. Bazow, UH, M. Strickland, 3.672!(nA4:%*=#$,&[)*+"#,'%-.o(94+:("-(SA@( ) f LRF aniso = f iso p2 +ξ(x,τ)p 2 z Λ(x,τ) V+4'%*&( 4C'%*&( ξ = p2 2 p 2 L Anisotropic hydrodynamics (ahydro) was developed by Strickland and Martinez, It treats non-equilibrium (viscous) effects arising from a spheroidal deformation of the local momentum distribution exactly (i.e. non-perturbatively), but ignores other viscous corrections. vahydro adds the remaining viscous corrections δ f perturbatively, as usually done in other viscous hydrodynamic frameworks. U. Heinz EC*, 5/2/24 56(62)

59 vahydro*c`b2;$%-* ),"VV"-;(4L&+(*$&(;4+B(.&*%"'=(*$&(X-%'(/I[:4:&-*(%VV+4Z":%J4-(+&=P'*("=( e(2%3456(7(8&"-36(%-.(!)6(3.672f( q+%-;&[c4z&.(*&+:= %+&(-&5 4*("-."#%*&=(% #4-L&#JL&(.&+"L%JL& R4:V'"#%*&.(C"*=("-('%=* *54(&P%J4-= #4++&=V4-.(*4."=="V%JL&(n94+#&=o %-.(%-"=4*+4V"# *+%-=V4+*(#4& #"&-*= In we simplified these equations for massless particles and (+)-d expansion with longitudinal boost-invariance and no transverse expansion. For this case the theory can be tested against an exact solution of the Boltzmann equation (Florkowski, Ryblewski, Strickland, PRC88 (23) 2493). U. Heinz EC*, 5/2/24 57(62)

60 4)"--B)"*52;$*7$3/2),-$%-* e(2%3456(uh, M. Strickland, 3.672f( PL P Ξ,4ΠΗ,.6GeV PL P PL P PL P Ξ,4ΠΗ 3,.6GeV Ξ,4ΠΗ,.6GeV Ξ,4ΠΗ,.6GeV Τ fm c Ξ,4ΠΗ,.6GeV Ξ,4ΠΗ 3,.6GeV Ξ,4ΠΗ,.6GeV Ξ,4ΠΗ,.6GeV Τ fm c Ξ,4ΠΗ,.6GeV Ξ,4ΠΗ 3,.6GeV Ξ,4ΠΗ,.6GeV Ξ,4ΠΗ,.6GeV Exact Solution vahydro ahydro 3 rd orderhydro Τ fm c a%-&'=(=$45(+%j4(49('4-;"*p."-%' *4(*+%-=L&+=&(V+&==P+& Y E( l(iee(!&g(_(τ E (l(ehf(9:]# S&y(*4(+";$*("=("-#+&%="-;("-"J%' :4:&-*P:[=V%#&(%-"=4*+4VB Y4V(*4(C4b4:("=("-#+&%="-;(η]S 2'%#,('"-&("=(*$&(&Z%#*(=4'PJ4- A&.(.%=$&.('"-&("=(*$&(%8B.+4 %VV+4Z":%J4-2'P&(.4*[.%=$&.('"-&("=(*$& L%8B.+4(%VV+4Z":%J4- ^+&&-(.%=$&.('"-&("=(%(*$"+.[4+.&+ R$%V:%-[\-=,4;['",&(L"=#4P= $B.+4.B-%:"#=(%VV+4Z":%J4- eu(r%"=5%'6(/defdinef( U=(5&(#%-(=&&(9+4:(*$&=&(V'4*= L%8B.+4(.4&=(P"*&(5&''("-.&&.m U. Heinz EC*, 5/2/24 58(62)

61 C%.)$/*>"%")2;$%* e(2%3456(7h, M. Strickland, 3.672]( Τf Τ n Τf n Τ Exact Solution vahydro ahydro 3 rd orderhydro 2 nd orderhydro E-*+4VB V+4.P#J4-( L%-"=$&= "-(*54( '":"*=M((".&%' $B.+4.B-%:"#s(%-. 9+&&(=*+&%:"-;( For massless particles6( -P:C&+.&-="*B("= V+4V4+J4-%'(*4 &-*+4VB(.&-="*B. 4 4ΠΗ s In the (+)-d case, which maximizes the difference between longitudinal and transverse expansion rates and thus the P /P L pressure anisotropy, vahydro outperforms all other available approaches. We expect it to improve the validity of viscous hydrodynamics for heavyion collisions at early times. U. Heinz EC*, 5/2/24 59(62)

62 Conclusions (I) Quark-Gluon Plasma is by far the hottest and densest form of matter ever observed in the laboratory. Its properties and interactions are controlled by QCD, not QED. It is a liquid with almost perfect fluidity. Its specific shear viscosity at RHIC and LHC energies is (η/s) QGP ( c <<2 c ) = 2 4π ±5% his is significantly below that of any other known real fluid. Precision comparison of harmonic flow coefficients at RHIC and LHC provides first serious indications for a moderate increase of the specific QGP shear viscosity between 2 c and 3 c. A more precise extraction of (η/s) QGP requires the simultaneous determination of the initial fluctuation spectrum. U. Heinz EC*, 5/2/24 6(62)

63 Conclusions (II) Viscous relativistic hydrodynamics provides a quantitative description of QGP evolution. By coupling viscous fluid dynamics for the QGP stage to microscopic evolution models of the dense early pre-equilibrium and dilute late hadronic freezeout stages, a complete dynamical description of the strongly interacting matter created in ultra-relativistic heavy-ion collisions has been achieved. his dynamical theory has made successful predictions for the first Pb+Pb collisions at the LHC that were quantitatively precise and non-trivial (in the sense that they disagreed with other predictions that were falsified by the data). he Color Glass Condensate theory (IP-Sat model) provides a mostly successful spectrum of initial-state gluon field fluctuations, but has problems in ultracentral PbPb and ppb at LHC. A large set of flow fluctuation observables, so far only partially explored, (over)constrains this initial fluctuation spectrum. = We are rapidly converging on the Standard Model for the Little Bang U. Heinz EC*, 5/2/24 6(62)

64 hanks to: Dennis Bazow, Steffen Bass, Scott Moreland, Zhi Qiu, Chun Shen, Huichao Song, Mike Strickland for their collaboration, to Gabriel Denicol, Matt Luzum, Bjoern Schenke, Jean-Yves Ollitrault for many in-depth discussions, and to Paul Sorensen for letting me use some of his beautiful slides Monday, December 2, 3

65 Supplements U. Heinz EC*, 5/2/24 62(62)

66 Flow in ultra-central PbPb collisions ~ 2M events for.-.2% centrality gered events v n vs n from two-particle correlations v n {2part, Δη > 2} CMS PbPb s NN L int.3 < p %, HF.-2.5%, HF -%, HF+NPixel.-.2%, HF+NPixel.-.2%, HF+NPixel CMS PAS HIN-2- = 2.76 ev - = 5 µb < 3. GeV/c. CMS Preliminary All orders of v n tends to saturate around.-.2% centrality è Predominantly induced by participant fluctuations Wei Li, Hard Probes 23 n 3

67 Single event anisotropic flow coefficients In a single event, the specific initial density profile results in a set of complex, y- and p -dependent flow coefficients (we ll suppress the y-dependence): V n = v n e inψn := p dp dφe inφ p dp dφ dn dyp dp dφ dn dyp dp dφ {e inφ }, V n (p ) = v n (p )e inψ n(p ) := dφe inφ dφ dn dyp dp dφ dn dyp dp dφ {e inφ } p. ogether with the azimuthally averaged spectrum, these completely characterize the measurable singleparticle information for that event: dn dydφ = dn 2π dy dn dyp dp dφ = 2π dn ( dyp dp + 2 ( ) v n cos[n(φ Ψ n )], n= + 2 ) v n (p ) cos[n(φ Ψ n (p ))]. n= Both the magnitude v n and the direction Ψ n ( flow angle ) depend on p. v n, Ψ n, v n (p ), Ψ n (p ) all fluctuate from event to event. Ψ n (p ) Ψ n fluctuates from event to event. U. Heinz EC*, 5/2/24 63(62)

68 p -dependent flow angles and their fluctuations n(ψn(p) Ψn) π π/2 π/2 π (a) n= n=2 n=3 n=4 n=5 n=6 Glb, 2-3% p (GeV) Except for directed flow (n=), Ψ n (p ) Ψ n fluctuates most strongly at low p Directed flow angle Ψ (p ) flips by 8 at p GeV for charged hadrons (pions) and at p.5gev for protons (momentum conservation) σ(cos(n(ψn(p) Ψn))) σ(cos(n(ψn(p) Ψn))).8 (b) Glb, 2-3% π n= n=2 n=3 n=4 n=5 n= p (GeV).8 (c) Glb, 2-3% p (GeV) p n= n=2 n=3 n=4 n=5 n=6 U. Heinz EC*, 5/2/24 64(62)

69 Elliptic and triangular flow comparison (I).6.4 KLN, -5% KLN, -5%.2 π + p v 2 [2] v 2 v 2 {EP} v 2 {2}.. 2. p (GeV) π + p v 3 [2] v 3 v 3 {EP} v 3 {2}.. 2. p (GeV).5 KLN, 6-7% KLN, 6-7%..5 π + p v 2 [2] v 2 v 2 {EP} v 2 {2}.. 2. p (GeV) π + p 4v 3 [2] 4 v 3 4v 3 {EP} 4v 3 {2}.. 2. p (GeV) In central collisions, angular fluctuations suppress v n {EP}(p ) and v n {2}(p ) below the mean and rms flows at low p (clearly visible for protons) his effect disappears in peripheral collisions, but a similar effect then takes over at higher p, for both pions and protons. U. Heinz EC*, 5/2/24 65(62)

70 Elliptic and triangular flow comparison (II): v n ratios.2. KLN, -5% KLN, -5% π, v 2 [2]/ v 2 π, v 2 {2}/v 2 {EP} p, v 2 [2]/ v 2 p, v 2 {2}/v 2 {EP} p (GeV) π, v 3 [2]/ v 3 π, v 3 {2}/v 3 {EP} p, v 3 [2]/ v 3 p, v 3 {2}/v 3 {EP}.. 2. p (GeV) Except for where the numerator or denominator goes through zero, for central collisions these ratios are equal to 2/ π.3, independent of p. Expected if flow angles are randomly oriented (Bessel- Gaussian distribution for v n, see Voloshin et al., PLB 659, 537 (28)). Not true in peripheral collisions, especially not for v 2 (Gardim et al., ) hat this works even for v n {2}/v n {EP} suggests an approximate factorization of angular fluctuation effects! U. Heinz EC*, 5/2/24 66(62)

71 Elliptic and triangular flow comparison (III): v n ratios (a) Central collisions: (b).8 Glb, -5% Glb, -5%.6.4 (c) π, v 2 {EP}/ v 2 π, v 2 {2}/v 2 [2] p, v 2 {EP}/ v 2 p, v 2 {2}/v 2 [2] (d) π, v 3 {EP}/ v 3 π, v 3 {2}/v 3 [2] p, v 3 {EP}/ v 3 p, v 3 {2}/v 3 [2].8 Glb, -5% Glb, -5%.6 π, v 2 (p )v 2 / v 2 2 (p ) v 2 2 π, cos(2(ψ 2 (p ) Ψ 2 )) p, v 2 (p )v 2 / v 2 2 (p ) v 2 2 p, cos(2(ψ 2 (p ) Ψ 2 )).. 2. p (GeV) π, v 3 (p )v 3 / v 2 3 (p ) v 2 3 π, cos(3(ψ 3 (p ) Ψ 3 )) p, v 3 (p )v 3 / v 2 3 (p ) v 2 3 p, cos(3(ψ 3 (p ) Ψ 3 )).. 2. p (GeV) he angular fluctuation factor cos[n(ψ n (p ) Ψ n )] completely dominates the p -dependence of these ratios! Angular fluctuations have similar effect as poor event-plane resolution: they reduce v n. Angular fluctuations are effective both at low and high p, but not at intermediate p. he window for seeing flow angle fluctuation effects at low p is smaller for pions than for protons. U. Heinz EC*, 5/2/24 67(62)

72 Elliptic and triangular flow comparison (IV): v n ratios Peripheral collisions: (a) (b) (c) Glb, 6-7% π, v 2 {EP}/ v 2 π, v 2 {2}/v 2 [2] p, v 2 {EP}/ v 2 p, v 2 {2}/v 2 [2] (d) Glb, 6-7% π, v 3 {EP}/ v 3 π, v 3 {2}/v 3 [2] p, v 3 {EP}/ v 3 p, v 3 {2}/v 3 [2].8 Glb, 6-7% Glb, 6-7%.6 π, v 2 (p )v 2 / v 2 2 (p ) v 2 2 π, cos(2(ψ 2 (p ) Ψ 2 )) p, v 2 (p )v 2 / v 2 2 (p ) v 2 2 p, cos(2(ψ 2 (p ) Ψ 2 )).. 2. p (GeV) π, v 3 (p )v 3 / v 2 3 (p ) v 2 3 π, cos(3(ψ 3 (p ) Ψ 3 )) p, v 3 (p )v 3 / v 2 3 (p ) v 2 3 p, cos(3(ψ 3 (p ) Ψ 3 )).. 2. p (GeV) he window for seeing flow angle fluctuation effects at low p closes in peripheral collisions. U. Heinz EC*, 5/2/24 68(62)

73 Flow angle fluctuation effects for higher order v n (p ) Central collisions; solid: v n (p ) ; dashed: v n {EP}(p ):.5 v 2 π v 6/3 p π p.6.2 v 3 v 7 2v 4 π π p p 2v 8 π π p p.6.2 p (GeV).6.2 3v 5 π p 3v 9 π p.6.2 As harmonic order n increases, suppression of v n {EP}(p ) (or v n {2}(p )) from flow angle fluctuations for protons gets somewhat weaker but persists to larger p. U. Heinz EC*, 5/2/24 69(62)

74 Conclusions Boththemagnitudesv n andtheflowanglesψ n dependonp andfluctuatefromeventtoevent. In each event, the p -averaged (total-event) flow angles Ψ n are identical for all particle species, but their p distribution differs from species to species. he mean v n values and their p -dependence at RHIC and LHC have already been shown to put useful constraints on the QGP shear viscosity and its temperature dependence (see next talk by B. Schenke) he effects of v n and Ψ n fluctuations can be separated experimentally by studying different V n measures based on two-particle correlations. Flow angle correlations are a powerful test of the hydrodynamic paradigm and will help to further constrain the spectrum of initial-state fluctuations and QGP transport coefficients. Studying event-by-event fluctuations of the anisotropic flows v n and their flow angles Ψ n as functions of p, as well as the correlations between different harmonic flows (both their magnitudes and angles), provides a rich data base for identifying the Standard Model of the Little Bang, by pinning down its initial fluctuation spectrum and its transport coefficients. U. Heinz EC*, 5/2/24 7(62)