Flow Harmonic Probability Distribution in Heavy Ion Collision

1/28 Flow Harmonic Probability Distribution in Heavy Ion Collision Seyed Farid Taghavi Institute For Research in Fundamental Sciences (IPM), Tehran, Iran Second Iran & Turkey Joint Conference on LHC Physics October 23-26, 217 (1-4 Aban 1396)

2/28 Outline Standard Model of Heavy Ion Collision Event-By-Event Fluctuations and Flow Harmonics Standardized Cumulants of Flow Harmonic Fluctuations Flow Harmonic Probability Distributions Conclusions

Standard Model of Heavy Ion Collision Figure from: Sorensen, arxiv: 95.174 3/28

4/28 Why We Believe the Quark-Gluon Plasma Is There? Quark-Gluon Plasma Diagnostics Flow Harmonics, v 2, v 3,... Strangeness Enhancement Jet Quenching J/Ψ Suppression

Standard Model of Heavy Ion Collision Event-By-Event Fluctuations Before studying the hydro evolution... 27 #1 Pb, SNN = 2.76 TeV, #2 b = 4.5 fm #3 5/28

Standard Model of Heavy Ion Collision From Initial states to Final Hadrons Initial State Freeze Out Hydrodynamic Evolution Observation In Detector The initial state evolves by hydrodynamic equations, where µ T µν = T µν = (ɛ + P)u µ u ν + Pη µν + τ µν τ µν = η s µα νβ ( αu β + β u α) (ζ 23 ) ηs µν αu α µν = η µν + u µ u ν, η µν = diag( 1, 1, 1, 1). 6/28

Event-By-Event Fluctuations and Flow Harmonics How to Quantify the Initial State Systematically [PHOBOS Collaboration, 27], [Teaney, Yan, PRC, 211] Dipole Asymmetry, ε1 Eccentricity, ε2 Triangularity, ε3... R εn,x +iεn,y ( n = ε2,x = Ψ2 rdrdϕ ρ(r, ϕ) rn einϕ R, rdrdϕ ρ(r, ϕ) rn Ψ3 3 if n = 1. 1 if n 2 hx 2 Y 2 i, hx 2 + Y 2 i ε2,y = 2hX Yi. hx 2 + Y 2 i 7/28

8/28 Event-By-Event Fluctuations and Flow Harmonics Event-By-Event Fluctuations [Borghini, Dinh, Ollitrault, 2] Fourier analysis of azimuthal distribution of emitted particles. 2π N dn dφ = 1 + 2v n cos [n(φ ψ n )], n=1 v n e inψn e inφ dφ dn dφ einφ. In different events Ψ 2 Ψ 2 Ψ 2 Ψ 2

Event-By-Event Fluctuations and Flow Harmonics Flow Harmonics, n-particle Correlation Function [Borghini, Dinh, Ollitrault, 2] Define 2-particle correlation function c n {2} = e in(φ 1 φ 2 ) single then many events Instead of using e inφ we use c n {2}, dφ 1 dφ 2 ( dn dφ 1 dφ 2 ) ( e in(φ 1 φ 2 ) ( ) ) ( dn dφ 1 e inφ 1 dφ 1 ( ) ) dn dφ 2 e inφ 2 dφ 2 v 2 n As a result v 2 n{2} = c n {2} 9/28

Event-By-Event Fluctuations and Flow Harmonics Flow Harmonics, n-particle Correlation Function [Borghini, Dinh, Ollitrault, 2], [Borghini, Dinh, Ollitrault, 21] Generalizing the 2-Particle to n-particle Correlation Function c n {2} = e in(φ 1 φ 2 ), c n {4} = e in(φ 1+φ 2 φ 3 φ 4 ) 2 e in(φ 1 φ 2 ) 2, It is shown that v 2 n{2} = c n {2}, v 4 n{4} = c n {4}, v 6 n{6} = c n {6}/4, Spliting in v n {2k} CMS Preliminary PbPb 5.2 TeV v 2 {2k}.12.1.8.6.4.2.3 < p T η < 1. STAR < 3. GeV/c v 2 {2k} k = 1 k = 2 k = 3 k = 4. 1 2 3 4 5 6 Centrality % v 3.1.5 ATLAS Pb+Pb s NN = 2.76 TeV 1-1 L int = 7 µb, η < 2.5 v 3 {EP} v 3 {2} v 3 {4} -25% 5 1 15 2 25-6% 5 1 15 2 p [GeV] T 1/28

11/28 Standardized Cumulants of Flow Harmonic Fluctuations Heavy Ion Collision Event Generator, iebe-vishnu [Shen, Qiu, Song, Bernhard, Bass, Heinz, 214] The full process is too complicated and numerical calculations are needed. supermc Initial condition generator M initial conditions VISHNew Hydrodynamics Hydrodynamic! simulator freeze-out surface info iss Particle emission sampler no Particle space-time and momentum info (multiple times) Finished all events? yes binutilities Spectra and Particle spacetime and flow calculator momentum info osc2u prepare UrQMD ICs UrQMD Hadron rescattering simulator EbeCollector Collect data into SQLite database zip zip results and store in to results folder

Standardized Cumulants of Flow Harmonic Fluctuations Event Generation I Pb-Pb collision in SNN = 2.76 TeV: 14 for centralities between to 8% (b to b r ). I η/s =.8, τ =.6 fm I supermc: MC-Glauber #1 #2 #3 5-55% centralities dnevents /dεn,x dεn,y 2 18.6 16.6 16.4 14.4 12.2 1 12 1 8.2 12.2 ε3,y.2 1 8 ε4,y 18.4 14 ε2,y 16.6 8.2 6.2 6 4 6.4 4.4 4.4.6 2.6 2.6.6.4.2 ε2,x.2.4.6 14.6.4.2.2 ε3,x.4.6 2.6.4.2.2 ε4,x.4.6 12/28

13/28 Standardized Cumulants of Flow Harmonic Fluctuations Cumulant Analysis of the Initial and Flow Distribution We use a 2D cumulant analysis.

13/28 Standardized Cumulants of Flow Harmonic Fluctuations Cumulant Analysis of the Initial and Flow Distribution We use a 2D cumulant analysis. The generating functional is log e ξxkx+ξyky = m,n= k m x k n y m!n! A mn.

13/28 Standardized Cumulants of Flow Harmonic Fluctuations Cumulant Analysis of the Initial and Flow Distribution We use a 2D cumulant analysis. The generating functional is log e ξxkx+ξyky = m,n= k m x k n y m!n! A mn. We define the standardized cumulants as follows A mn  mn = A m 2 A n 2

Standardized Cumulants of Flow Harmonic Fluctuations Cumulant Analysis of the Initial and Flow Distribution We use a 2D cumulant analysis. The generating functional is log e ξxkx+ξyky = m,n= k m x k n y m!n! A mn. We define the standardized cumulants as follows A mn  mn = A m 2 A n 2 The cumulant of the 2D distribution of (v n,x, v n,y ) averaged over azimuthal angle c n {2k} c n {2k} and A kl are related to each other. e.g. c n {2} = A 2 1 + A 2 1 + A 2 + A 2 13/28

14/28 Standardized Cumulants of Flow Harmonic Fluctuations Cumulant Analysis of the Initial and Flow Distribution Ê (n) kl normalized cumulants from ε n ˆV(n) kl normalized cumulants from v n For n = 2, 3, The Hydrodynamic Response is Almost Linear [Teaney, Yan, PRC, 211], [Luzum, Ollitrault,...] v n α n ε n 2 2.6 18.6 18 ε2,y.4.2.2.4 16 14 12 1 8 6 4 v2,y/αn.4.2.2.4 16 14 12 1 8 6 4.6 2.6 2 ε 2,x.6.4.2.2.4.6 v 2,x /α n.6.4.2.2.4.6 Ê (n) pq ˆV (n) pq

15/28 Standardized Cumulants of Flow Harmonic Fluctuations Connection With Experimental Observation ˆV (2) Recall 3 Ê(2) 3 is skewness. It is shown that the skewness [Giacalone,Ollitrault,Yan, Noronha-Hostler, PRC,216] exp 1 γ.4.2 CMS Preliminary PbPb 5.2 TeV < 3. GeV/c T.3 < p η < 1. v 2 {4} v 2 {6} = V 3 3V 2 1 ˆV (2) 3 6 2v 2 2 {4}(v 2{4} v 2 {6}) [ v 2 2 {2} v 2 2 {4}] 3/2 and a constraint v 2 {4} = 12v 2 {6} 11v 2 {8}..2.4.6.8 γ exp 1 2.76 TeV Hydro 1. 1 2 3 4 5 6 Centrality %

16/28 Standardized Cumulants of Flow Harmonic Fluctuations Kurtosis of the Third Flow Harmonics [Abbasi,Allahbakhshi,Davody,SFT,217] For n = 3: c 3 {2} α 2 2 c 3 {4} α 4 2 = E 2 +E 2 = E 4 +2E 22 +E 4 Γ 2q 2 = cn{2q} c q n{2} Γ 2 = ( ) 4 v3 {4} v 3 {2}

17/28 Flow Harmonic Probability Distributions Different Analytical Distributions ε2,y.6.4.2.2.4.6 2 18 16 14 12 1 8 6 4 2 ε3,y.2.2 ε 2,x.2 ε 3,x.6.4.4.6.6.4.2.4.6.6.4.2.2.4.6 18 16 14 12 1 8 6 4 2 Two Dimensional Gaussian Distribution p(ε n,x, ε n,y) = [ ] 1 exp (εn,x ε ) 2 2πσ xσ y 2σx 2 ε2 n,x 2σy 2 Elliptic-Power Distribution [Yan, Ollitrault, Poskanzer, PRC, 214] p(ε n,x, ε n,y) = α π (1 ε2 )α+1/2 (1 ε2 n,x ε 2 n,y) α 1 (1 ε ε n,x) 2α+1.

Flow Harmonic Probability Distributions Integration Over Azimuthal Direction Bessel-Gaussian [Voloshin, Poskanzer, Tang, Wang, PLB, 28] 3 2.5 2 ε 2 (a) [ p(ε n) = εn σ 2 exp ε2 n + ] ( ) ε2 ε ε n 2σ 2 I σ 2 P(ε 2 ) 1.5 1.5 2 ε 3 (b) Elliptic-Power [Yan, Ollitrault, Poskanzer, PRC, 214] P(ε 3 ) 1.5 1.5 p(ε n) = 2αε n(1 ε 2 n )α 1 (1 ε 2 )α+1/2 1 π (1 ε ε n cos ϕ) 2α 1 dϕ π Power (for odd n) [Yan, Ollitrault, PRL, 214] 2αε n(1 ε 2 n) α 1 P(ε 4 ) 2 1.5 1.5 ε 4 (c) Power Elliptic Power Bessel-Gaussian.2.4.6.8 1 ε n 75%-8% Centrality [Yan, Ollitrault, Poskanzer, PRC, 214] 18/28

Flow Harmonic Probability Distributions A New Distribution 2D Gram-Charlier A Series P(ξ x, ξ y) 1 + H 2π e (ξx A 1 ) 2 (ξy A 1 )2 2A 2 2A 2 A 2 A 2 where H = Â mn m!n! Hen( ξx A 1 )He m( ξy A 1 ) A2 A2 m=n=1, m+n 3 Integrate Over Azimuthal Direction Radial-Gram-Charlier p(v 3 ) = [ 1 + Γ 2 L 2 ( v2 3 c 3 {2} ) + Γ 4 6 L 3 ( v2 3 c 3 {2} ) + ] [ ] 2v 3 exp c 3 v2 3 {2} c 3 {2} Γ 2q 2 = c 3 {2q}/c q 3 {2}, ( ) 4 Γ v 3 {4} 2 = v 2 3 {2} 19/28

2/28 Flow Harmonic Probability Distributions Radial-Gram-Charlier Fit p T dependence of standardized radial cumulants. Flow and non-flow effects?!

Flow Harmonic Probability Distributions Work On Progress Bessel-Gaussian [ p(v n) = vn σ 2 exp v2 n + ] ( ) v2 v v n 2σ 2 I σ 2 RGC (I n = I n(v v n/σ 2 )) p(v n ) = (Q + q 2 Q 2 + q 4 Q 4 + q 6 Q 6 + ) ( v n ) [ ] σ exp v2 2 n +v2 2σ 2 Considering 2σ 2 v 2 n {2} v2 Q = I, [ ] vn Q 2 = I + v I 1, [ ] [ ] 2 vn vn Q 4 = 2I 4 I 1 +2 I 2, v v [ ] [ ] 2 [ ] 3 vn vn vn Q 6 = 6I +18 I 1 18 I 2 +6 I 3, v v v q 2 =, q 4 = 1 4 q 6 = 1 18 [ v 4 n {4} v 4 (v 2 n {2} v2 )2 [ 2v 6 n {6} 3v 2 v4 n {4} + ] v6 ], (v 2 n {2} v2 )3 21/28

Flow Harmonic Probability Distributions Validity Checks p(vn) 1 8 6 4 2 5-55% centralities Q Q 4 Q 6 Q 8 Ellip.-Power..5.1.15.2.25.3 Fine-splitting: [Miller,Snellings,23],[Giacalone,Ollitrault,Yan, Noronha-Hostler, PRC,216] Using the approximation v v n{4} leads to q 4 = and q 6 γ 1 v n q 8 11(v 2 {6} v 2 {8}) (v 2 {4} v 2 {6}) q 1 higher fine splittings 22/28

23/28 Conclusions Conclusions The distribution of v 3 has kurtosis. Its value is obtained by (v 3 {4}/v 3 {2}) 4. Does its p T dependence has any physical significances? A new distribution (RGC) for p(v 3 ) is introduced. The standardized cumulants are appeared as coefficients in a expansion and can be found by fitting. Generalization of RGC to arbitrary n The fine splitting of v 2 {2k} are related to RGC (n = 2) expansion coefficients.

24/28 Conclusions Thank You!

BACKUP 25/28

26/28 Conclusions For n = 2 [Ollitrault,PRC,216]

27/28 Conclusions For n = 3

Conclusions 28/28