Feb 2007 Big Sky, Montana Nuclear Dynamics 2007 Conference Is There A Mach Cone? For the STAR...
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Feb 2007 Big Sky, Montana
Nuclear Dynamics 2007 Conference
Is There A Mach Cone?For the STAR Collaboration
Claude Pruneau
Motivations/Goals Expectations/Models
Search + Analysis Methods Data + Results
Summary/Conclusions
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Claude Pruneau, for the STAR Collaboration, Nucl. Dyn 2007 2QuickTime™ and a
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Dip “Puzzle” Dip “Puzzle” in in 2-Particle Correlations
pTtrig = 3.0-4.0 GeV/c;
pTasso = 1.0-2.5 GeV/c
See M. Horner’s talk at QM06
Motivations Mach Cone Concept/Calculations
Stoecker, Casalderry-Solana et al, Muller et al.; Ruppert et al., …
Velocity Field Mach Cone
Other Scenarios• Cherenkov Radiation
Dremmer, Majumder, Koch, & Wang; Vitev• Jet Deflection (Flow) Fries; Armesto et al.; Hwa
vs~0.33
~1.1 rad
θM = π ± arccos(vs / c)
~ 1.9, 4.3rad
Claude Pruneau, for the STAR Collaboration, Nucl. Dyn 2007 3QuickTime™ and a
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Relative Angles Definition
1
2
3
12
13
Angular Range 0 - 360o
1: 3 < pt < 4 GeV/c (Jet Tag)2,3: 1 < pt < 2 GeV/c,
Mach Cone & Deflection Kinematical Signatures
13
12
0
Back-to-back Jets “in vacuum”Away-side broadeningAway-side deflection & flowMach Cone
Claude Pruneau, for the STAR Collaboration, Nucl. Dyn 2007 4QuickTime™ and a
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Two Analysis Techniques
ρ2 (Δϕ ij ) ≡d 2N
dΔϕ ij
Measure 1-, 2-, and 3-Particle Densities
3-particle densities = superpositions of truly correlated 3-particles, and combinatorial components. We use two approaches to extract the truly correlated 3-particles component
ρ1(ϕ i ) ≡d 2N
dϕ iρ3(Δϕ ij , Δϕ ik ) ≡
d 3N
dΔϕ ijdΔϕ ik
C3( 1 , 13) =ρ3( 1 , 13)−ρ ( 1 )ρ1(3)−ρ ( 13)ρ1()−ρ ( 13 − 1 )ρ1(1) + ρ1(1)ρ1()ρ1(3)
1) Cumulant technique: 2) Jet+Flow Subtraction Model:
J3( 1 , 13) =J 3( 1 , 13)−J ( 1 )B ( 13)
−J ( 13)B ( 1 )−B3( 1 , 13)
Simple DefinitionModel Independent.
Intuitive in conceptSimple interpretation in principle.
PROs
CONs Not positive definiteInterpretation perhaps difficult.
Model Dependentv2 and normalization factors systematics
–.
See C. Pruneau, nucl-ex/0608002 See J. Ulery & nucl-ex/0609017/0609016
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Mach Cone Search - Data set and cuts
• p+p, d+Au, = 200 GeV used as reference.
• Search For Mach Cone in Au + Au, = 200 GeV
• Minimum bias, and Central Triggers Data Samples (Run 4)
• Particle Cuts: Predicated by the observation of the “dip”• Jet tag (trigger) : 3 < pt < 4 GeV/c, ||<1• Associates: 1 < pt < 2 GeV/c, ||<1
• Collision Centrality: • Estimated based on reference multiplicity in || < 0.5.
s
s
Claude Pruneau, for the STAR Collaboration, Nucl. Dyn 2007 6QuickTime™ and a
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ρ3(Δϕ 12 ,Δϕ 13) ρ2 (12)ρ1(3) ρ2 (13)ρ1(2)
Measurement of 3-Particle Cumulant
ρ2 (23)ρ1(1) v2v2v4
• Clear evidence for finite 3-Part Correlations• Observation of flow like and jet like structures.
• Evidence for v2v2v4 contributions
C3( 1 , 13)
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3-Cumulant vs. centralityAu + Au 80-50% 30-10% 10-0%
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3-Cumulant Sensitivity to Cone Signal
Use a simple Jet + Cone toy modelJet: <N1>=1 per jet (3<pt<4 GeV/c)<N2>=2 per jet (1<pt<2 GeV/c)<Jet>/event ~ 0.27Actual data have ~1 trigger/event
Cone: <N2>=2 per jet (1<pt<2 GeV/c)
Event Mult ~ 300 to 600.
C3( 1 , 13)
Cone
Near SideJet
Claude Pruneau, for the STAR Collaboration, Nucl. Dyn 2007 9QuickTime™ and a
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C3( i , j , k)FJ ≈( )−1 J AiAj Bk
×v ( jet)v (bckg)exp(−σ )exp − ij
4σ
⎛⎝⎜
⎞⎠⎟cos k − i − j( )
3-Cumulant Background: Jet x Flow
Flowing Jet - Differential Attn. Rel. Reaction PlaneModel:Jet Emission Rel. Reaction Plane with Finite v2.2 particles from a jet 1 particle from the background
C3( 1 , 13)
(a.u.)
Work in progress to assess the strength of this term in the cumulant and systematics.
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Jet-Flow Subtraction Method
See J. Ulery, nucl-ex/0609017/0609016
Δ12
Δ 1
3
ρ3(Δϕ 12 , Δϕ 13) / Trigger
Δ12
2-Part Correlation
Flow background
“Jetty”signal
Δ12
Δ 1
3
Estimate/Remove JetBackground Hard-Soft Term
Claude Pruneau, for the STAR Collaboration, Nucl. Dyn 2007 11QuickTime™ and a
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Estimate/Remove Trigger 2-Background Soft-soft term
Δ 1
3
Δ12
Jet-Flow Subtraction Method (cont’d)Estimate/Remove Trigger Background Flow
v2(1) v2(2) v22
v4 (1) v4 (2) + +v2(1)v2(2)v2(3) v2
4
Δ 1
3
Δ12
Δ 1
3
Δ12
v4=1.15v22
Claude Pruneau, for the STAR Collaboration, Nucl. Dyn 2007 12QuickTime™ and a
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Jet - Flow Subtraction Method - System Size Dependence (1)
(12+13)/2-
(12-13)/2
Δ12 Δ12 Δ12
Δ 1
3
Δ 1
3
Δ 1
3
pp d+AuAu+Au 50-80%
Claude Pruneau, for the STAR Collaboration, Nucl. Dyn 2007 13QuickTime™ and a
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Jet - Flow Subtraction Method - System Size Dependence (1)
Δ 1
3
Δ 1
3
Δ 1
3
Δ12Δ12
Δ12
(12+13)/2-
(12-13)/2
Au+Au 30-50% Au+Au 10-30% Au+Au 0-10%
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12
1
3Jet - Flow Subtraction Result in Au+Au - Triggered 0-12%
Diagonal and Off-diagonal structures are suggestive of conical emission at an angle of about 1.45 radians in central Au+Au.
Deflected Jet + Cone
Cone
Near Side
Elongated Away Side Jet
Claude Pruneau, for the STAR Collaboration, Nucl. Dyn 2007 15QuickTime™ and a
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Yield and SystematicsAu+Au 0-12% No Jet Flow
12
1
3
(12+13)/2-
(12-13)/2
Au+Au 0-12%
12
(12-13)/2
(12+13)/2-
1
3
Nominal Model:• Used “reaction plane” v2 estimates• Used Zero Yield at 1 rad for
normalizations
“Systematics” Estimates:• Vary v2 in range: v2{2} - v2{4}• Vary point of normalization
Turn Jet-Flow background term on/off
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• Use 3 Particle Azimuthal Correlations.• Identification of correlated 3-particle from jet and predicted Mach
cone is challenging task.• Must eliminate 2-particle correlation combinatorial terms.• Must remove flow background - including v2v2, v4v4, and v2v2v4
contributions.• Use two approaches: Cumulant & Jet - Flow Subtraction Model
• Cumulant Method• Unambiguous evidence for three particle correlations.• Clear indication of away-side elongated peak.• No evidence for Cone signal given flow backgrounds
• Jet-Flow Background Method• Model Dependent Analysis
• Cone amplitude sensitive to magnitude v2 and details of the model.
• Observe Structures Consistent with Conical emission in central collisions
Summary/Conclusions
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Additional Material
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Azimuthal FlowPF ( i |ψ ) =1+ vm
m∑ (i)cos m i −ψ( )( )
ρ,F ( i , j ) = ( )− FiFj − Fi Fj + FiFj vm(i)vm( j)cos m i − j( )( )m∑⎛
⎝⎜⎞⎠⎟
Particle Distribution Relative to Reaction Plane
2- Cumulants
ρ3,F ( i , j , k) = ( )−3
FiFjFk − FiFj Fk( ) vm(i)vm( j)cos m i − j( )( )m∑
+permutations (j,k,i) and (k,i,j) of above
FiFjFk vp(i)vm( j)vn(k)
δ p,m+n cos p i −m j −n k( )
+δm,p+n cos −p i + m j −n k( )
+δn,m+k cos −p i −m j +n k( )
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
p,m,n∑
−constant terms
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪
Reducible2nd order in v
Irreducible3rd order in v
3- Cumulants
• 3-Cumulant Flow Dependence : • Irreducible v2v2v4 contributions
• Must be modeled and manually subtracted• v2
2 suppressed but finite• v2
2 cancellation possible with modified cumulant.
Claude Pruneau, for the STAR Collaboration, Nucl. Dyn 2007 19QuickTime™ and a
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Two Illustrative Models :
σ1= σ2= σ3=10o; σ=0o
No deflection Random Gaussian Away-Side Deflectionσ1= σ2= σ3=10o; σ=30o
Di-Jets:
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Mach Cone
θmach
(a)
12
13
θ mach
(b)
θ mach
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Some Properties of CumulantsCumulants are not positive definiteThe number of particles in a bin varies e-by-e: ni = <ni> + i
n1n2 = n1 + 1( ) n + ( ) = n1 n + 1
n1n2n3 = n1 + 1( ) n + ( ) n3 + 3( )
= n1 n n3 + n1 3 + n 13 + n3 1 + 1 C3 = 1
Cumulant for Poisson Processes (independent variables) are null
C2 = n1n − n1 n = 1 =0 C3 = 1 =0
Cumulant for Bi-/Multi-nomial Processes ~ 1/Mn-1
(independent variables, but finite multiplicity)
n1 =p1M
n =pM
Var(n1) = n1 − n1
=p1(1−p1)M
n1n =p1pM
Where M is a reference multiplicity
C2 = n1n − n1 n = 1
n1n2 − n1 n
n1 n
=−1M
Claude Pruneau, for the STAR Collaboration, Nucl. Dyn 2007 21QuickTime™ and a
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More Properties of CumulantsConsider a Superposition of =1,…, s processes
Number of particles in a phi bin in a given event: ni = n,i=1
s
∑1- Particle Density: ni = n,i
=1
s
∑
2- Particle Density: n1n2 = n ,1=1
s
∑⎛⎝⎜⎞⎠⎟
nβ,β=1
s
∑⎛
⎝⎜⎞
⎠⎟= n ,1n ,
=1
s
∑ + n ,1nβ,≠β
s
∑
Product of Single Particle Densities: n1 n2 = n ,1=1
s
∑⎛⎝⎜⎞⎠⎟
nβ,β=1
s
∑⎛
⎝⎜⎞
⎠⎟= n ,1 n ,
=1
s
∑ + n ,1 nβ,≠β
s
∑
2-Cumulant: C2 = C,=1
s
∑ + COVβ (1,)≠β
s
∑
Cumulant of a sum of processes equals sum of cumulants + sum of covariances between these processes.
• If the processes are independent, these covariances are null.• At fixed multiplicity, these covariances are of order 1/Mn-1.
3- Particle Density: n1n2n3 = n ,1=1
s
∑⎛⎝⎜⎞⎠⎟
nβ,β=1
s
∑⎛
⎝⎜⎞
⎠⎟nγ,
γ=1
s
∑⎛
⎝⎜⎞
⎠⎟= n ,1n ,n ,3
=1
s
∑ + n ,1nβ,nγ,3≠β≠γ
s
∑
3-Cumulant: C3 = C,3=1
s
∑ + COVβγ (1,,3)≠β≠γ
s
∑
Enables Separation of Jet (Mach Cone) and Flow Background.
Claude Pruneau, for the STAR Collaboration, Nucl. Dyn 2007 22QuickTime™ and a
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Example: 2-particle Decay: ρ → + + −
2-Cumulant
Maxwell Boltzman, T=0.2 GeVIsotropic Emission/Decay of rho-mesons, with pion background.
• 3-Particle Density contains 2-body decay signals.• 2-Body Signal Not Present in 3-cumulant.
Suppression of 2-part correlations with 3-cumulant
Many resonances, e.g. ρ 0
s , N*, … contribute to the soft-soft term, and likely to the hard-soft as well.
Claude Pruneau, for the STAR Collaboration, Nucl. Dyn 2007 23QuickTime™ and a
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Cumulant Method - Finite Efficiency Correction
• Use “singles” normalization to account for finite and non-uniform detection efficiencies.
• Example: ρ2 (Δϕ ij )
ρ1ρ1(Δϕ ij )=
ρ2 (Δϕ ij )
ρ1(ϕ i )ρ1(ϕ j )δ (Δϕ ij −ϕ i +ϕ j )∫Robust Observables
ρ2 (Δϕ ij )
ρ1ρ1(Δϕ ij )
Measured
=ε 2 (ϕ i ,ϕ j )ρ 2
theory (ϕ i ,ϕ j )
ε1(ϕ i )ρ 1
theory (ϕ i )ε1(ϕ j )ρ 1
theory (ϕ j )δ (Δϕ ij −ϕ i +ϕ j )dϕ idϕ j∫
=ρ
2
theory (ϕ i ,ϕ j )
ρ1
theory (ϕ i )ρ 1
theory (ϕ j )δ (Δϕ ij −ϕ i +ϕ j )dϕ idϕ j∫
provided
ε 2 (ϕ i ,ϕ j ) = ε1(ϕ i )ε1(ϕ j ) verified for sufficiently large ij differences.
Claude Pruneau, for the STAR Collaboration, Nucl. Dyn 2007 24QuickTime™ and a
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What changed since QM05
Background subtractedQM2005
Au+Au 0-10% most central
Example Acceptance Correction• Increased data sample• Two Analysis Methods• Jet-Flow Background Method:
• Improved efficiency corrections• Reduce the number of free parameters