The Cumulant Analysis of Flow Harmonic Fluctuations in Heavy...
Transcript of The Cumulant Analysis of Flow Harmonic Fluctuations in Heavy...
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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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Event-by-Event fluctuation in Heavy Ion Experiment
Figure from: Sorensen, arXiv: 0905.0174
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Event-by-Event fluctuation in Heavy Ion Experiment
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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Event-by-Event fluctuation in Heavy Ion Experiment
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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Event-by-Event fluctuation in Heavy Ion Experiment
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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Event-by-Event fluctuation in Heavy Ion Experiment
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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Event-by-Event fluctuation in Heavy Ion Experiment
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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Event-by-Event fluctuation in Heavy Ion Experiment
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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Event-by-Event fluctuation in Heavy Ion Experiment
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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Event-by-Event fluctuation in Heavy Ion Experiment
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 fluctuation in Heavy Ion Experiment
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 fluctuation in Heavy Ion Experiment
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 fluctuation in Heavy Ion Experiment
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 fluctuation in Heavy Ion Experiment
Event-By-Event FluctuationsBefore studying the hydro evolution. . .
207Pb,√
SNN = 2.76 TeV, b = 4.5 fm
· · ·
· · ·
#1 #2 #3
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Event-by-Event fluctuation in Heavy Ion Experiment
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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Event-by-Event fluctuation in Heavy Ion Experiment
From Initial states to Final Hadrons
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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From Initial states to Final Hadrons
From Initial states to Final Hadrons
−→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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From Initial states to Final Hadrons
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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From Initial states to Final Hadrons
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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From Initial states to Final Hadrons
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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The Cumulants of Flow Harmonics Distribution
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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The Cumulants of Flow Harmonics Distribution
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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ε 2,y
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dNevents/dεn,xdεn,y
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The Cumulants of Flow Harmonics Distribution
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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The Cumulants of Flow Harmonics Distribution
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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The Cumulants of Flow Harmonics Distribution
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 generating
function
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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The Cumulants of Flow Harmonics Distribution
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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The Cumulants of Flow Harmonics Distribution
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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The Cumulants of Flow Harmonics Distribution
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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The Cumulants of Flow Harmonics Distribution
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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The Cumulants of Flow Harmonics Distribution
Cumulant Analysis of the Initial and Flow Distribution
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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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The Cumulants of Flow Harmonics Distribution
Cumulant Analysis of the Initial and Flow Distribution
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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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The Cumulants of Flow Harmonics Distribution
Cumulant Analysis of the Initial and Flow Distribution
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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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The Cumulants of Flow Harmonics Distribution
Cumulant Analysis of the Initial and Flow Distribution
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}
○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○
■■■■■
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○ hydro◼ ATLAS
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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
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%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