The Cumulant Analysis of Flow Harmonic Fluctuations in Heavy …particles.ipm.ir/conferences/2017/IWPPP/pdf/Taghavi.pdf · 2017. 2. 19. · Navid Abbasi, Davood Allahbakhshi, Ali

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The Cumulant Analysis of Flow HarmonicFluctuations in Heavy Ion Collisions

Seyed Farid Taghavi

In collaboration with:

Navid Abbasi, Davood Allahbakhshi, Ali Davodyand Mojtaba Mohammadi Najafabadi

Institute For Research in Fundamental Sciences (IPM), Tehran, Iran

arXiv: 1702.XXXXX

IPM Workshop on Particle Physics PhenomenologyBahman 1395

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Outline

Event-by-Event fluctuation in Heavy Ion Experiment

From Initial states to Final Hadrons

Flow Harmonics

The Cumulants of Flow Harmonics Distribution

Results

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Figure from: Sorensen, arXiv: 0905.0174

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Monte Carlo Glauber Model[Holopainen, et al, 2011]

I A simple and powerful model for heavy ion collision initialstate based on the Glauber model [Glauber, 1959].

I Nucleons are considered free and distributed byWoods-Saxon distribution,

ρ(r) = ρ0

(1

1 + exp[ r−r0

a

]) ,inside the nucleus. The r0 is the nuclear radius and a iscalled the skin depth.

I For 197Au: r0 = 6.38 fm; a = 0.535 fm.I For 207Pb: r0 = 6.62 fm; a = 0.546 fm.

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207Pb, r0 = 6.62 fm, a = 0.546 fm

Target

I The nucleons collide if

|~rti −~rpj | ≤√σNNinel/π. → Participants

I σNNinel = 6.4 mb at√

SNN = 2.76 TeV

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207Pb, r0 = 6.62 fm, a = 0.546 fm

Target Projectile

b = 4.5 fm

I The nucleons collide if

|~rti −~rpj | ≤√σNNinel/π. → Participants

I σNNinel = 6.4 mb at√

SNN = 2.76 TeV

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Smearing the Participants

I Using the location of participants, we can find the energydensity profile. We use a 2D Gaussian distribution as thesmearing function,

ρ(x, y) ∝ 12πσ2

Npart∑i=1

exp[−(x− xi)

2 + (y− yi)2

2σ2

]

I For σ = 0.6 fm,

I READY FOR HYDRODYNAMICS

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Smearing the Participants

I Using the location of participants, we can find the energydensity profile. We use a 2D Gaussian distribution as thesmearing function,

ρ(x, y) ∝ 12πσ2

Npart∑i=1

exp[−(x− xi)

2 + (y− yi)2

2σ2

]I For σ = 0.6 fm,

I READY FOR HYDRODYNAMICS

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Event-By-Event FluctuationsBefore studying the hydro evolution. . .

207Pb,√

SNN = 2.76 TeV, b = 4.5 fm

· · ·

· · ·

#1 #2 #3

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How to Quantify the Initial State Systematically[PHOBOS Collaboration, 2007], [Teaney, Yan, PRC, 2011]

Dipole Asymmetry, ε1Eccentricity, ε2Triangularity, ε3

...

εn,x+iεn,y ≡∫

rdrdϕρ(r, ϕ) rn′einϕ∫

rdrdϕρ(r, ϕ) rn′,

n′ =

{3 if n = 11 if n ≥ 2

.

Ψ2

Ψ3

ε2,x =〈x2 − y2〉〈x2 + y2〉

, ε2,y =2〈x y〉〈x2 + y2〉

.

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Hydrodynamic Evolution

Freeze OutInitial State

Observation InDetector

The initial state evolves by hydrodynamic equations,

∂µTµν = 0

whereTµν = (�+ P)uµuν + Pηµν + τµν

τµν = −ηs∆µα∆νβ(∂αuβ + ∂βuα)−(ζ −

23ηs

)∆µν∂αuα

∆µν = ηµν + uµuν , ηµν = diag(−1, 1, 1, 1).

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−→Hydrodynamic Evolution

Σµ

I After cooling down the produced plasma, it hadronises.I The hadron spectrum is obtained by Copper-Frye equation:

EdNd3p

=

∫pµdΣµ

(2π)3f (x, p).

The f (x, p) could be for example Boltzmann distribution,

f (x, p) ∝ exp[

pµuµ

T

]

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Final Particle Distribution is A Picture of The Initial State

A naive estimation of Cooper-Frye result

Σµ

pµ

uµ

Fluid Patch

Freeze Out Hyper Surf. I pµdΣµ → Sum the outgoingand subtract the ingoingparticles.

I Consider uµ ∼ (1− δ, vx, vy, 0)

I exp[

pµuµ

T

]= exp

[−u0√|~p|2+m2+~p·~v

T

]I The exp[· · · ] maximize for ~p ‖ ~v

and larger |~v|.

The Faster Fluid Patch, The More Particle Emission

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Final Particle Distribution is A Picture of The Initial State

Faster

Faster

Faster

Faster

Faster

More Particles

More Particles

More Particles

More Particles

More Particles

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The Picture Of Final State on the Detector

=# + # + · · ·

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Flow Harmonics

Flow Harmonics, n-Particle Correlation Functions[Borghini, Dinh, Ollitrault, 2000]

I Fourier analysis of azimuthal distribution of emittedparticles.

1N

dNdφ

= 1 +∞∑

n=1

2vn cos [n(φ− ψn)] ,

vneinψn ∝ 〈einφ〉 ∝∫

dφ dNdφ einφ.

I In a single event, ψn is fixed. Therefore if we shiftφ→ φ+ ψn then vn ∝

∫dφdNdφ e

inφ.I We will use also the following notation,

vn,x = vn cos(ψn), vn,y = vn sin(ψn)

I The number of particles are not enough to find ψn. The ψncan not be observed experimentally.

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Flow Harmonics

Flow Harmonics, n-Particle Correlation Function[Borghini, Dinh, Ollitrault, 2000]

I In different events

Ψ2

Ψ2

Ψ2Ψ2

I The quantity ein(φ1−φ2) is invariant under shift φi → φi + ψnwhere φ1 and φ2 are the azimuthal angle of two particles ina single event.

I We also havedN

dφ1dφ2=

(dNdφ1

)(dNdφ2

)+ fc(φ1, φ2).

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Flow Harmonics

Flow Harmonics, n-Particle Correlation Function[Borghini, Dinh, Ollitrault, 2000]

I The correlations between particles produced from thedecays such as ρ→ ππ contributes to the fc(φ1, φ2) whichis a non-flow correlation.

I If fc(φ1, φ2) is negligible then∫dφ1dφ2

(dN

dφ1dφ2

)ein(φ1−φ2) ≈

(∫dφ1

(dNdφ1

)einφ1

)(∫dφ2

(dNdφ2

)e−inφ2

)

∝ v2nI Define 2-particle correlation function

cn{2} = 〈ein(φ1−φ2)〉single then many events

I Thenv2n{2} = cn{2}

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Flow Harmonics

Flow Harmonics, n-Particle Correlation Function[Borghini, Dinh, Ollitrault, 2000], [Borghini, Dinh, Ollitrault, 2001]

I It is shown that

v2n{2} = cn{2}, v4n{4} = −cn{4}, v6n{6} = cn{6}/4, · · ·where

cn{2} = 〈ein(φ1−φ2)〉, cn{4} = 〈ein(φ1+φ2−φ3−φ4)〉−2〈ein(φ1−φ2)〉2, · · ·I At each order, the non-flow effects are more suppressed.I What is the fine structure in v2{2k} and v3{2k}.

0 10 20 30 40 50 60 70 800

1

2

3

4

5

6

7

8

% Most Central

(%

)2

v

{2}2v

{4}2v

{6}2v

5 10 15 20

3v

0

0.05

0.1

{EP}3v{2}3v{4}3v

ATLAS = 2.76 TeVNNsPb+Pb

| < 2.5η, |-1bµ = 7 intL

0-25%

[GeV]T

p5 10 15 20

0

0.05

0.1

25-60%STAR

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Heavy Ion Collision Event Generator, iEBE-VISHNU[Shen, Qiu, Song, Bernhard, Bass, Heinz, 2014]

I The full process is too complicated and numericalcalculations are needed.

superMC Initial

condition generator

VISHNew Hydrodynamics

simulator

iSS Particle emission sampler

binUtilities Spectra and

flow calculator

osc2u prepare UrQMD

ICs

UrQMD Hadron

rescattering simulator

multiple

M initial conditions

freeze-out surface info

Particle space-time and

momentum info

Particle space-time and momentum info

Finished all events?

no

yes

EbeCollector Collect data into SQLite database zip results and store to results folder

Hydrodynamic!simulator

zip results and store in results folder

(multiple times)

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Event Generation

I Pb-Pb collision in√

SNN = 2.76 TeV: 7000 to 14000 forcentralities between 0 to 80% (b ∼ 0 to b ∼ r0).

I η/s = 0.08, τ0 = 0.6 fmI superMC: MC-Glauber

MC-Glauber for 50-55% centralities

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ε2,x ε3,x ε4,x

ε 2,y

ε 3,y

ε 4,y

dNevents/dεn,xdεn,y

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How to Quantify the Initial State Systematically II

Reminder of Notations From Statistics #1:Consider the distribution ρ(x) of a random variable x,

I Moment: µ′r(x) = 〈xr〉.

I Central Moment: µr(x) := 〈(x− 〈x〉)r〉.• Translational Invariance: µr(x + c) = µr(x).

I Cumulant: κr(x) obtained from the following generatingfunction

log〈eλ x〉 =∞∑

r=1

κrλr

r!

• Translational Invariance: κr(x + c) = κr(x) for n ≥ 2.• For Gaussian distribution, κr = 0 for r ≥ 3.

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I Moment: µ′r(x) = 〈xr〉.I Central Moment: µr(x) := 〈(x− 〈x〉)r〉.

• Translational Invariance: µr(x + c) = µr(x).

I Cumulant: κr(x) obtained from the following generatingfunction


r=1

κrλr

r!


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I Moment: µ′r(x) = 〈xr〉.I Central Moment: µr(x) := 〈(x− 〈x〉)r〉.

• Translational Invariance: µr(x + c) = µr(x).I Cumulant: κr(x) obtained from the following generating

function


r=1

κrλr

r!


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Cumulant Analysis of the Initial and Flow Distribution

• For Gaussian distribution, κr = 0 for r ≥ 3.

• Cumulants in terms of moments,

κ1 = 〈x〉,

κ2 = 〈x2〉 − 〈x〉2,κ3 = 〈x3〉 − 3〈x2〉〈x〉+ 2〈x〉3,

...

• Gram-Charlier A series,

p(ξ) ≈ 1√2πκ2

exp[−(ξ − κ1)2

2κ2]

×

(1 +

κ3

3!κ3/22H3(

ξ − κ1√κ2

) +κ4

4!κ22H4(

ξ − κ1√κ2

)

),

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• For Gaussian distribution, κr = 0 for r ≥ 3.• Cumulants in terms of moments,

κ1 = 〈x〉,

κ2 = 〈x2〉 − 〈x〉2,κ3 = 〈x3〉 − 3〈x2〉〈x〉+ 2〈x〉3,

...

• Gram-Charlier A series,

p(ξ) ≈ 1√2πκ2

exp[−(ξ − κ1)2

2κ2]

×

(1 +

κ3

3!κ3/22H3(

ξ − κ1√κ2

) +κ4

4!κ22H4(

ξ − κ1√κ2

)

),

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Cumulant Analysis of the Initial and Flow Distribution• The expansion coefficients are

? Skewness:γ1 =

µ3

κ3/22

=κ3

κ3/22

γ1 < 0 γ1 = 0 γ1 > 0? Kurtosis:

K =µ4κ22

=κ4κ22

+ 3 → γ2 = K − 3

γ2 < 0 γ2 = 0 γ2 > 0

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ε2,x ε3,x ε4,x

ε 2,y

ε 3,y

ε 4,y

I We use a 2D cumulant analysis.

I The generating functional is

log〈eξxkx+ξyky〉 =∑

m,n=0

kmx kny

m!n!Amn.

I We define the normalized cumulants as follows

Âmn =Amn√Am20An02

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60

80

100

120

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160

ε2,x ε3,x ε4,x

ε 2,y

ε 3,y

ε 4,y

I We use a 2D cumulant analysis.I The generating functional is

log〈eξxkx+ξyky〉 =∑

m,n=0

kmx kny

m!n!Amn.

I We define the normalized cumulants as follows

Âmn =Amn√Am20An02

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I Similar to 2D distribution for εn,x and εn,y, there is adistribution for vn,x and vn,y.

I Ê(n)kl → normalized cumulants from εnI V̂(n)kl → normalized cumulants from vn

For n = 2, 3, The Hydrodynamic Response is Almost Linear[Teaney, Yan, PRC, 2011], [Luzum, Ollitrault, . . .]

v̂n ' κnε̂nI From Homogeneity of cumulants: V(n)pq ' κp+qn E(n)pq ,

therefore,

V̂(n)pq ' Ê (n)pq

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Results

iEBE-VISHNU Output, For n = 2

[Ollitrault,PRC,2016]

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Results

iEBE-VISHNU Output, For n = 3

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Results

iEBE-VISHNU OutputI We learned

• For n = 2Â(2)10 ∼ O(1),

Â(2)30 , Â(2)40 , Â

(2)04 ∼ O(10

−1).

• For n = 3Â(3)40 ∼ Â

(3)04 ∼ O(10

−1)

The rest are approximately zero.

The n-Particle Correlations, cn{2k} are A(n)kl where the effectof ψn is Integrated Out, Therefore,by using the simplifications studied above

The V̂(n)pq can be written in terms of vn{2k}.

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Results

Connection With Experimental Observation

For n = 2:

V̂(2)30 '−66√

2(21v2{6} − 22v2{8})2(v2{6} − v2{8})[v22{2} − (21v2{6} − 22v2{8})2

]3/2V̂(2)40 ' V̂

(2)04 '

8(21v2{6} − 22v2{8})3(v2{4} − 12v2{6}+ 11v2{8})[v22{2} − (21v2{6} − 22v2{8})2

]2If we can ignore V̂(2)40 ' V̂

(2)04 then we

find [Ollitrault,PRC,2016]

V̂(2)30 '−6√

2v22{4}(v2{4} − v2{6})[v22{2} − v

22{4}

]3/2and a constraint

v2{4} = 12v2{6} − 11v2{8}

○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○

■■■■■

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

○ hydro◼ ATLAS

0 20 40 60 80-1.0-0.50.0

0.5

centrality [%]

γ1expt

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Results

Kurtosis of the Third Flow Harmonics

For n = 3:

V̂(3)40 ' V̂(3)04 ' −2

(v3{4}v3{2}

)4

0 20 40 60 80

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0

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A04 A40

%Centrality%Centrality

HydroInitialATLAS

HydroInitialATLAS

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Results

Thank You!

Event-by-Event fluctuation in Heavy Ion ExperimentFrom Initial states to Final HadronsFlow HarmonicsThe Cumulants of Flow Harmonics DistributionResults

Documents

The Cumulant Analysis of Flow Harmonic Fluctuations in Heavy …particles.ipm.ir/conferences/2017/IWPPP/pdf/Taghavi.pdf · 2017. 2. 19. · Navid Abbasi, Davood Allahbakhshi, Ali