Hydrodynamic response to initial state fluctuations

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Malcolm Gaines
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1 University of Jyväskylä, Department of Physics POETIC Jyväskylä

2 AA-collisions Initial particle/energy production, followed by Hydrodynamic evolution, followed by Freeze-out/Hadron cascade Goal is to determine QGP properties: EoS, transport coefficients... Large(st) uncertainty in determining e.g. shear viscosity is initial state for hydrodynamic evolution Complicated (and necessary) problem is to determine initial conditions simultaneously with QGP properties

3 Hydrodynamics Conservation laws µt µν = 0 µn µ i = 0 T µν = eu µ u ν (p + Π) µν + π µν IS equations for viscous parts of T µν e.g. shear viscosity: τ π d dτ π µν + π µν = 2η µ u ν + To solve this set of equations we need Equation of state p = p(e) and T = T (e) Initial condition T µν (τ 0, x) Transport coefficients, e.g. shear viscosity η(t ), relaxation time τ π(t ),...

4 Hydrodynamics particles (Freeze-out) Cooper-Frye freeze-out for particle i = calculate number of particles crossing the freeze-out hypersurface E dn d 3 p = g i (2π) 3 dσ µ p µf i (p, x) Hadron spectra + decays dn h dydp 2 T dφ

5 Characterizing p T -spectra Fourier decomposition dn dydp 2 T dφ = dn dydpt 2 [ + 2v (p T ) cos (φ ψ ) + 2v 2 (p T ) cos [2 (φ ψ 2 )] + ] ψ n = (/n) arctan ( p T sin nφ / p T cos nφ ) v n(p T ), ψ n(p T ), dn/dy,... characterize single event Ensemble of events: Full characterization Averages: v n, ψ n,... Probability distributions: P(v n), P(ψ n),... Correlations: v n, v m, ψ n, ψ m,...

6 Characterizing initial state Eccentricity (for energy density ε) dxdy r m cos [n (φ Ψ m,n)] ε (x, y, τ 0 ) ɛ m,n = dxdy r m ε (x, y, τ 0 ) Ψ m,n = n arctan dxdy r m sin (nφ) ε (x, y, τ 0 ) dxdy r m cos (nφ) ε (x, y, τ 0 ) + π/n e m,n, Ψ m,n characterize single event (initial energy density) Ensemble of events (initial conditions): Full characterization Averages: e m,n, Ψ m,n,... Probability distributions: P(e m,n), P(Ψ m,n) Correlations: e m,n, e m,n, Ψm,n, Ψ m,n,...

Introduction Hydrodynamics Initial state Correlation between en and vn vn distributions Correlations Bessel-Fourier expansion: S. Floerchinger, U. Wiedemann: arxiv:307.

7 Introduction Hydrodynamics Initial state Correlation between en and vn vn distributions Correlations Bessel-Fourier expansion: S. Floerchinger, U. Wiedemann: arxiv: [hep-ph] Z R 2 (m) (m) dr r w (m) (r ) Jm kl r wl = i2 h (m) 0 R 2 Jm+ (kl R) wreco(nm,nl ) (r, φ) = Nl X (m=0) wl (0) J0 zl r /R l= +2 Nl Nm X X (m) wl h i (m) (m) cos m φ ϕl Jm zl r /R m= l= Conclusions

8 Connecting initial condition to hadron spectra Ensemble of events (initial conditions): Full characterization Averages: e m,n, Ψ m,n,... Probability distributions: P(e m,n), P(Ψ m,n) Correlations: e m,n, e m,n, Ψm,n, Ψ m,n,... Hydrodynamic response (EoS, η/s, T dec,...) Ensemble of events (spectra): Full characterization Averages: v n, ψ n,... Probability distributions: P(v n), P(ψ n),... Correlations: v n, v m, ψ n, ψ m Problem is that typically the connections depend on the full details of the space-time evolution...

9 Correlation coefficient How are v n s and ɛ n s related? v n ɛ n v n ɛ n Measure by linear correlation coefficient: c (a, b) = c = 0 no (linear) correlation c = ( ) fully (anti-)correlated ( ) ( ) a a ev b b ev σ aσ b ev

10 e n v n correlation F. G. Gardim, F. Grassi, M. Luzum and J. -Y. Ollitrault, Nucl. Phys. A , 503c (203) Quality v 2 fromε 2 v 3 fromε 3 v 4 fromε 4 v 5 fromε centrality

11 Monte-Carlo Glauber s (x, y) = W N part,bin i= { exp [(x x i ) 2 + (y y i ) 2] / ( 2σ 2)} (x i, y i ) position of wounded nucleon or binary collision W normalization constant Centrality selection accoring to N bin or N part p T -spectra for each event dn ev dydp T 2 dφ

12 v 2/3 vs ɛ 2/3 (0 5% centrality class) 2d histogram 2000 hydro events per case sbc η/s = 0 sbc η/s = 0.6 swn η/s = 0.6 v (a) c(ε 2,v 2 ) =0.954 C 2 =0.202 sbc η/s =0 0 5 % ε 2 v (b) c(ε 2,v 2 ) =0.985 C 2 =0.59 sbc η/s = % ε 2 v (c) c(ε 2,v 2 ) =0.99 C 2 =0.63 swn η/s = % ε c(ε 3,v 3 ) = C 3 = c(ε 3,v 3 ) = C 3 = c(ε 3,v 3 ) =0.920 C 3 =0.7 v 3 (a) sbc η/s =0 0 5 % ε 3 v (b) sbc η/s = % ε 3 v (c) Both v 2 and v 3 correlated to corresponding eccentricities (v 2 event-by-event) swn η/s = % ε 3

13 v 2/3 vs ɛ 2/3 (20 30% centrality class) sbc η/s = 0 sbc η/s = 0.6 swn η/s = 0.6 v (a) c(ε 2,v 2 ) =0.979 C 2 =0.207 sbc η/s = % ε 2 v (b) c(ε 2,v 2 ) =0.990 C 2 =0.48 sbc η/s = % ε 2 v (c) c(ε 2,v 2 ) =0.996 C 2 =0.53 swn η/s = % ε 2 v (a) c(ε 3,v 3 ) =0.893 C 3 =0.76 sbc η/s = % ε 3 Even better correlation v 3 c(ε ,v 3 ) =0.955 C 3 = (b) sbc η/s = % ε 3 v (c) c(ε 3,v 3 ) =0.973 C 3 =0.093 swn η/s = % ε 3

14 v 4 vs ɛ % v c(ε 4,v 4 ) =0.367 C 4 = (a) v c(ε 4,v 4 ) =0.5 C 4 = (b) v 4 c(ε 4,v 4 ) = C 4 =0.046 (c) sbc η/s =0 0 5 % ε sbc η/s = % ε % swn η/s = % ε 4 v (a) c(ε 4,v 4 ) =0.99 C 4 =0.9 sbc η/s = % ε 4 v (b) c(ε 4,v 4 ) =0.94 C 4 =0.033 sbc η/s = % ε 4 v (c) c(ε 4,v 4 ) =0.202 C 4 =0.030 Weak correlation in central collision, no correlation in peripheral collisions swn η/s = % ε 4

15 e m,n v n (p T ) correlation Do v n(p T ) correlate with different ɛ m,n at different p T s (ɛ m,n with different r m weight) v 2 (p T ): not really. v 3 (p T ): kind of, but rather weak difference No straightforward access to r-dependence of initial density profiles Bessel-Fourier expansion?.4.2 c( 0,2,v 2 ) c(,2,v 2 ) c( 2,2,v 2 ) c( 3,2,v 2 ) Nbin /s = % c( 0,3,v 3 ) c(,3,v 3 ) c( 2,3,v 3 ) c( 3,3,v 3 ) Nbin /s = % 2,v 2 ).0 3,v 3 ) 0.8 c( 0.8 c( p T [GeV] p T [GeV]

16 v n distributions

17 v n distributions: ATLAS measurement. G. Aad et al. [ATLAS Collaboration], arxiv: [hep-ex] G. Aad et al. [ATLAS Collaboration], arxiv: [hep-ex] ATLAS Pb+Pb s NN =2.76 TeV L int = 7 µb ATLAS Pb+Pb s NN =2.76 TeV L int = 7 µb ATLAS Pb+Pb s NN =2.76 TeV L int = 7 µb ) 2 p(v centrality: 0 % data p >0.5GeV, η <2.5 T centrality: 5 % data p >0.5GeV, η <2.5 T centrality: 20 25% data p >0.5GeV, η <2.5 T Glauber Glauber Glauber MC KLN ATLAS Pb+Pb s NN =2.76 TeV L int = 7 µb MC KLN ATLAS Pb+Pb s NN =2.76 TeV L int = 7 µb 2 MC KLN ATLAS Pb+Pb s NN =2.76 TeV L int = 7 µb ) 2 p(v centrality: 30 35% data p >0.5GeV, η <2.5 T Glauber MC KLN v 2 2 centrality: 40 45% data p >0.5GeV, η <2.5 T Glauber MC KLN v 2 2 centrality: 55 60% data p >0.5GeV, η <2.5 T Glauber MC KLN v 2

18 v n distributions Full description of heavy-ion collisions should also get distributions of v n correct (not only event averaged value) 2 v 2 sbc η/s =0.0 v 2 sbc η/s =0.6 v 2 swn η/s =0.6 2 v 3 sbc η/s =0.0 v 3 sbc η/s =0.6 v 3 swn η/s =0.6 2 v 4 sbc η/s =0.0 v 4 sbc η/s =0.6 v 4 swn η/s =0.6 P(v 2 ) (a) P(v 3 ) (a) P(v 4 ) (a) v v v 4

19 v n distibutions (20 30 % centrality) Each type of initial state and η/s gives different distribution of v n s, but if we introduce scaled variable Then... δv n = vn vn ev ɛn ɛn ev, and δɛ n =. v n ev ɛ n ev v 2 sbc η/s =0.0 v 3 sbc η/s =0.0 v 4 sbc η/s =0.0 v 2 sbc η/s =0.6 v 3 sbc η/s =0.6 v 4 sbc η/s =0.6 0 v 2 sbc ǫ 2 0 v 3 sbc ǫ 3 0 v 4 sbc ǫ 4 P(δv 2 ), P(δǫ 2 ) - (a) P(δv 3 ), P(δǫ 3 ) - (b) P(δv 4 ), P(δǫ 4 ) - (c) <v 2 > =0.06 (η/s =0.0) <v 2 > =0.043 (η/s =0.6) (v 2 <v 2 >)/ <v 2 >, (ǫ 2 <ǫ 2 >)/ <ǫ 2 > <v 3 > =0.023 (η/s =0.0) <v 3 > =0.0 (η/s =0.6) (v 3 <v 3 >)/ <v 3 >, (ǫ 3 <ǫ 3 >)/ <ǫ 3 > <v 4 > =0.03 (η/s =0.0) <v 4 > =0.004 (η/s =0.6) (v 4 <v 4 >)/ <v 4 >, (ǫ 4 <ǫ 4 >)/ <ǫ 4 > Probability distributions of δv 2 and δv 3 independent of hydrodynamic evolution, and follow the corresponding δe n distribution. HN, Denicol, Holopainen and Huovinen, Phys. Rev. C 87, (203), arxiv:22.08 [nucl-th]

20 v n distibutions (20 30 % centrality) Same result if we change between the Glauber variants (sbc or swn): <v 2 > =0.039 (swn) <v 3 > =0.02 (swn) <v 4 > =0.003 (swn) 0 <v 2 > =0.043 (sbc) 0 <v 3 > =0.0 (sbc) 0 <v 4 > =0.004 (sbc) P(δv 2 ), P(δǫ 2 ) - -2 swn v 2 η/s =0.6 sbc v 2 η/s =0.6 swn ǫ 2 sbc ǫ 2 (a) P(δv 3 ), P(δǫ 3 ) - -2 swn v 3 η/s =0.6 sbc v 3 η/s =0.6 swn ǫ 3 sbc ǫ 3 (b) P(δv 4 ), P(δǫ 4 ) - -2 swn v 4 η/s =0.6 sbc v 4 η/s =0.6 swn ǫ 4 sbc ǫ 4 (c) (v 2 <v 2 >)/ <v 2 >, (ǫ 2 <ǫ 2 >)/ <ǫ 2 > (v 3 <v 3 >)/ <v 3 >, (ǫ 3 <ǫ 3 >)/ <ǫ 3 > (v 4 <v 4 >)/ <v 4 >, (ǫ 4 <ǫ 4 >)/ <ǫ 4 >

21 v n distibutions (central collisions) v 2 sbc η/s =0.0 v 3 sbc η/s =0.0 v 4 sbc η/s =0.0 v 2 sbc η/s =0.6 v 3 sbc η/s =0.6 v 4 sbc η/s =0.6 0 v 2 sbc ǫ 2 0 v 3 sbc ǫ 3 0 v 4 sbc ǫ 4 P(δv 2 ), P(δǫ 2 ) - (a) P(δv 3 ), P(δǫ 3 ) - (b) P(δv 4 ), P(δǫ 4 ) - (c) <v 2 > =0.025 (η/s =0.0) <v 2 > =0.09 (η/s =0.6) (v 2 <v 2 >)/ <v 2 >, (ǫ 2 <ǫ 2 >)/ <ǫ 2 > <v 3 > =0.06 (η/s =0.0) <v 3 > =0.0 (η/s =0.6) (v 3 <v 3 >)/ <v 3 >, (ǫ 3 <ǫ 3 >)/ <ǫ 3 > <v 4 > =0.008 (η/s =0.0) <v 4 > =0.003 (η/s =0.6) (v 4 <v 4 >)/ <v 4 >, (ǫ 4 <ǫ 4 >)/ <ǫ 4 > <v 2 > =0.03 (swn) <v 3 > =0.008 (swn) <v 4 > =0.003 (swn) 0 <v 2 > =0.09 (sbc) 0 <v 3 > =0.0 (sbc) 0 <v 4 > =0.003 (sbc) P(δv 2 ), P(δǫ 2 ) - -2 swn v 2 η/s =0.6 sbc v 2 η/s =0.6 swn ǫ 2 sbc ǫ 2 (a) P(δv 3 ), P(δǫ 3 ) - -2 swn v 3 η/s =0.6 sbc v 3 η/s =0.6 swn ǫ 3 sbc ǫ 3 (b) P(δv 4 ), P(δǫ 4 ) - -2 swn v 4 η/s =0.6 sbc v 4 η/s =0.6 swn ǫ 4 sbc ǫ 4 (c) (v 2 <v 2 >)/ <v 2 >, (ǫ 2 <ǫ 2 >)/ <ǫ 2 > (v 3 <v 3 >)/ <v 3 >, (ǫ 3 <ǫ 3 >)/ <ǫ 3 > (v 4 <v 4 >)/ <v 4 >, (ǫ 4 <ǫ 4 >)/ <ǫ 4 >

22 v n distibutions: MC-KLN (CGC) vs MC-Glauber Two different Glauber limits give the same distribution, but... <v 2 > =0.039 (CGC) <v 2 > =0.048 (CGC + NBD) <v 2 > =0.024 (Glauber Npart) CGC CGC + NBD Glauber N part P(δv 2 ) 0 - η/s =0.6 <ǫ 2 > =0.23 (CGC) <ǫ 2 > =0.278 (CGC + NBD) <ǫ -2 2 > =0.42 (Glauber Npart) (v 2 <v 2 >)/ <v 2 > v 2 ǫ 2 For example different variations of MC-KLN model give clearly different δv 2 distributions δv 2 distributions can distinquishe between the initial state models (without doing a single hydrodynamic calculation) Initial conditions from A. Dumitru

23 v n distibutions: Tuning MC-Glauber model Different distributions of nucleons (or quarks): CQS = constitutient quark scaling, HS = hard sphere P(ε 2 ) P(v 2 ) ATLAS data 0-5% 5-% CQS -5% 5-20% sheet 20-25% 25-30% Glauber % 35-40% δv 2, δε 2 δv 2,δε 2 Thorsten Renk and HN HS

24 v n distibutions: Tuning MC-Glauber model 0 5-% centrality P(δε 2 ), P(δv 2 ) 0. ATLAS data σ = 0.6 fm σ =.0 fm σ = 0.6 fm, KNO σ = 0.6 fm, σ = 0.3 fm σ = 0.6 fm, σ = 0.3 fm, KNO δε 2, δv 2 Thorsten Renk and HN

25 v n distibutions: IP-Glasma C. Gale, S. Jeon, B. Schenke, P. Tribedy and R. Venugopalan, Phys. Rev. Lett., (203) 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 ) % ε 2 IP-Glasma v 2 IP-Glasma+MUSIC v 2 ATLAS 0. p T > 0.5 GeV η < v 2 / v 2, ε 2 / ε % ε 3 IP-Glasma v 3 IP-Glasma+MUSIC v 3 ATLAS 0. p T > 0.5 GeV η < v 3 / v 3, ε 3 / ε % ε 4 IP-Glasma v 4 IP-Glasma+MUSIC v 4 ATLAS 0. p T > 0.5 GeV η < v 4 / v 4, ε 4 / ε 4

26 (v i, v j ) correlations % (a) sbc η/s =0.0 sbc η/s =0.6 swn η/s = % (b) sbc η/s =0.0 sbc η/s =0.6 swn η/s = % (c) sbc η/s =0.0 sbc η/s =0.6 swn η/s =0.6 c(v 2,v 3 ) c(v 3,v 4 ) c(v 2,v 4 ) p T [GeV] p T [GeV] p T [GeV] c(ɛ 2, ɛ 3 ) c(v 2, v 3 ) c(ɛ 2, ɛ 4 ) c(v 2, v 4 ) c(ɛ 3, ɛ 4 ) c(v 3, v 4 ) sbc η/s = sbc η/s = swn η/s =

27 Conclusions v 2 and v 3 linearly correlated to corresponding eccentricities (v 2 event-by-event) Correlation between v 4 and e 4 depends strongly on the details of fluid dynamics Scaled distributions P(δv n) follow directly from the initial conditions (details of the fluid dynamical evolution do not matter) Clear constraints for the initial condition (without doing a single hydro run) (v i, v j ) correlations provide additional information about the transport properties of the matter

Event-by-event distribution of azimuthal asymmetries in ultrarelativistic heavy-ion collisions

Event-by-event distribution of azimuthal asymmetries in ultrarelativistic heavy-ion collisions Hannu Holopainen Frankfurt Institute for Advanced Studies in collaboration with G. S. Denicol, P. Huovinen,