HI:C 07 - Montreal C. Pruneau, Wayne State 1 New Perspectives on Measurements of 2- and 3- Particle Correlations Claude A. Pruneau Wayne State University.

HI:C 07 - Montreal C. Pruneau, Wayne State 1

New Perspectives on Measurements of 2- and 3- Particle CorrelationsNew Perspectives on Measurements of 2- and 3- Particle Correlations

Claude A. Pruneau

Wayne State University

Detroit, MI, USA


Beyond the disappearance of Away-Side JetsBeyond the disappearance of Away-Side Jets

3<pt,trigger<4 GeV

pt,assoc.>2 GeV

Jörn Putschke, et al., STAR, Quark Matter 2006 Mark Horner , et al., STAR, Quark Matter 2006

pT,trig = 3.0-4.0 GeV/c;

pT,asso = 1.0-2.5 GeV/c

Au+Au 0-10% STAR preliminary

Near-Side Ridge Away-Side Dip


“Reappearance” of the away side jet“Reappearance” of the away side jet

STAR Phys. Rev. Lett. 97 (2006) 162301

• “Progressive” re-appearance of the away-side jet with increasing trigger pt in central Au+Au.

• Away-side yield vary with “system” size or collision centrality •Yield dramatically suppressed rel. to d+Au.

• Associated yield on the near side is independent of centrality.

8 < pt(trig)<15


Theoretical Scenarios - RidgeTheoretical Scenarios - Ridge

Parton radiates energy before fragmenting and couples to the longitudinal flow Gluon bremsstrahlung of hard-scattered parton

Parton shifted to lower pt

Radiated gluon contributes to broadening

near-side jet also looses energy (finite pathlength)!

Medium heating + Parton recombination Chiu & Hwa Phys. Rev. C72:034903,2005)

Recombination of thermal partons only indirectly affected by hard scattering not part of the jet

Radial flow + trigger bias Voloshin nucl-th/0312065, S. A. Voloshin, Nucl. Phys. A749, 287 (2005)

Armesto et al, PRL 93 (2004), nucl-ex/0405301


Theoretical Scenarios - Away Side DipTheoretical Scenarios - Away Side Dip

Mach Cone Concept/Calculations

Stoecker, Casalderry-Solana et al, Muller et al.; Ruppert et al., …

Velocity Field Mach Cone

Other Scenarios• Cherenkov Radiation

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


Talk OutlineTalk Outline

• Can the ridge and the dip be caused by jet - medium interactions, I.e. jet energy loss ?

• Is there a Mach Cone?• Explore the role of radial flow.

– Could radial flow “explain” both the ridge and dip structures?


T. Affolder, et al (CDF) PRD 65 (2002) 092002.

Average number of particles vs Jet pt (for particles with pt>0.5 GeV/c, ||<1, R=0.7)

Estimates: Jet p ~ 7 GeVYield in 3-4 GeV/c: ~0.45Yield in 1-2 GeV/c: ~1Jet p ~ 10 GeVYield in 3-4 GeV/c: ~0.35Yield in 1-2 GeV/c: ~1.5

Some Key Features of Charged Particle JetsSome Key Features of Charged Particle Jets


Some Key Features of the Near-Side RidgeSome Key Features of the Near-Side Ridge

Au+Au 0-10%

STAR preliminarypt,assoc. > 2 GeV

ridge

yield

STAR preliminary“jet” sloperidge slopeinclusive slope

“wide” “stronger in central coll.”4 < pt,trigger < 6 GeV

6 < pt,trigger < 10 GeV

•Ridge persists up to highest trigger pt

correlated or collocated to jet production and

~ independent of trigger pt.

•Ridge Spectrum ~ “bulk-like”, NOT “jet-like”.

•Ridge energy quite large - roughly a few GeV.

•Ridge comparable in Au+Au and Cu+Cu at same

Npart.

“large energy”

Jörn Putschke et al., STAR , Quark Matter 2006, Shanghai


Mark Horner , et al., STAR, Quark Matter 2006

More Key Features of the Near- and Away-Side Structures

Near-sideJet+Ridge

Near-sideJet Only

Away-side


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


QM06 - STAR - Analysis TechniquesQM06 - STAR - 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. Star uses two approaches to extract the truly correlated 3-particles component

ρ1(ϕ i ) ≡d 2N

dϕ i

ρ3(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, PRC See J. Ulery & nucl-ex/0609017/0609016


Azimuthal FlowAzimuthal 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• vn

2 suppressed.


Two Illustrative Models :Two Illustrative Models :

1= 2= 3=10o; =0o

No deflection Random Gaussian Away-Side Deflection1= 2= 3=10o; =30o

Di-Jets:

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Mach Cone

θmach

(a)

12

13

θ mach

(b)

θ mach


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.


Conical Emission Sensitivity/Efficiency CorrectionConical Emission Sensitivity/Efficiency Correction

• Jet + Mach Cone Model– On average, the jet includes 1 high pt particle, 2 low pt particles– On average, the “cone” includes 2 low pt particles,– Cone angle fixed at 70 degrees and width of 0.2 radians.

• Finite Efficiency Simulation ε(ϕ ) = ε avg 1+ a4 cos(4ϕ ){ }





B : ε( ) =0.8 1+ 0.15cos(4 ){ }A : ε( ) =1



B−0.8 A




PJet ( i φ) =G( i −φ; i ) =1 i

exp −( i −φ)

i

⎛

⎝⎜⎞

⎠⎟

Jet - Flow Cross Term - A toy Model

Jet Profile:

Jet Flow: PJet−Axis(φ ψ ) =1+ vn( jet)cos(n(φ−ψ ))n∑

Background Flow: PBCKG ( k ψ ) =1+ vn(bckg)cos(n( k −ψ ))n∑

3-Cumulant:

C3( i , j , k)FJ =( )−1 J AiAj Bk

× vn( jet)vn(bckg)exp(−n i

j

ij )exp −

ij

ij

⎛

⎝⎜

⎞

⎠⎟cos n k −α i i −α j j( )( )

n∑

ij2 = σ i

2 + σ j2

α i =σ j

2

σ i2 + σ j

2

• Jet-Flow correlation arises from finite eccentricity of medium + differential absorption+quenching.

• A simple model…


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)


3-Cumulant vs. centralityAu + Au 80-50% 30-10% 10-0%


Two-Component Model Analysis

Au+Au 0-12% No Jet Flow

φ12

φ1

3

(12+13)/2-

(12-13)/2

Au+Au 0-12%

φ12

(12-13)/2

(12+13)/2-

φ1

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


Mach Cone

*

Three particle correlation

True 3PC jet correlationsDeflected Jet

*

PHENIX Preliminary

Data is consistent with the presence of a Mach Cone away-side jet but does not rule out small contributions from other topologies.

PHENIX simulation

Real dataChun Zhang, et al. PHENIX, QM06


• 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.• Finite Sensitivity: No evidence for Cone signal

• 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

Conical Search Summary


Cone and Ridge Puzzles SummaryCone and Ridge Puzzles Summary

Ridge Carries a large amount of particles and energy

(high pt particles) Not very sensitive to the trigger particle pt. Strength grows with increasing centrality.

“Cone” Seen in 2-part correlations for many pt ranges Strong yield and also carry substantial energy in 2-part. 3-Part signal still not clear. Not seen/strong in cumulant.

Medium Effect? Ridge+Cone artifacts of the way we measure correlations?

What About Radial Flow? Can radial flow affect jets?


Observations from p+p…Observations from p+p…



• Di-jets are only back-to-back in the transverse plane, non in rapidity.• In eta-phi space, this leads to a ridge-like structure at φ in p+p

PYTHIA p+p, sqrt(s)=200 GeVTrigger: 3 < pt < 20 GeV/cAssociate: 1 < pt < 2 GeV/c

M.Daugherity, et al., STAR, hep-ph/0506172

same-side

away-side – ΦΔ ~ π

p+p


Effect of Radial Flow on ResonancesEffect of Radial Flow on ResonancesC. P., PRC 74, 064910 (2006),e-Print Archive: nucl-ex/0608002

(a) 0.01 < pt(ρo) < 0.1 GeV/c, pt(1) < 0.2 GeV/c;

(b) 0.1 < pt(ρo) < 0.5 GeV/c, pt(1) > 0.3 GeV/c, pt(2) < 0.2 GeV/c

(c) 0.1 < pt(ρo) < 0.5 GeV/c, pt(1) < 0.2 GeV/c

(d) 0.6 < pt(ρo) < 1.5 GeV/c, pt(1) > 0.2 GeV/c, pt(2) < 0.2 GeV/c

(e) 1.5 < pt(ρo) < 5.5 GeV/c, pt(1) > 0.2 GeV/c, pt(2) < 0.2 GeV/c.;

(f) 5.5 < pt(ρo) < 10. GeV/c, pt(1) < 2.0 GeV/c.

Rho-decays at Finite Temperature


Effect of Transverse Radial Flow on “Clusters”Effect of Transverse Radial Flow on “Clusters”

• S. Voloshin, e.g. nucl-ex/05• Based on the blast wave model.


• Large velocities, if applicable to jets or entire pp events, or string fragmentation can lead to dramatic changes in the correlation functions.

• So… let’s try boosting pp PYTHIA events at selected radial velocities in random transverse/radial directions.

• Work Hypothesis: Maximum Coupling Between Flow and Jets. No diffusion or Attenuation.



Toy Model to Study the Effect of Radial Flow on Jet-like

Structures, C.P., S. Gavin, S. Voloshin

STAR, Phys. Rev. Lett. 92 (2004) 112301

Large Velocities !

Blastwave Fits to Spectra

• Basic Hypothesis: Matter produced in A+A @ RHIC is subject to large collective flow.


How does this work?

Effect of Radial Flow on Jet-like Structures

>0

A+A participant regionp+p collision (in vacuum)

p+p boosted by high radial flow: focusing

p+p boosted by low radial flow: focusing + deflection




PYTHIA p+p @ sqrt(s)=200 GeV; 3<pt<20 && 1<pt<2



=0.1



=0.2








=0.3 =0.4 =0.5PYTHIA p+p @ sqrt(s)=200 GeV; 3<pt<20 && 1<pt<2

“Dip” “Suppression”









=0.1 =0.23-cumulants








=0.3 =0.4 =0.5



SummarySummary

• A+A Studies of 2- and 3- Particle Correlations reveal new “unforeseen” structures.

• While they are many theoretical interpretations, and predictions based on energy loss, the strengths of the structures are quite large, and put in question the notion they are produced by energy loss.

• No Clear/Robust (Model Independent) Evidence For Mach Cone Yet!!• Explored the effect of Strong Radial Flow Using p+p events from

PYTHIA.– 2- and 3- particle correlations.– Radial Flow Induces Patterns in Azimuthal Correlations that are “similar” to

Conical Emission.– Produces a “relocation” of the pp away side ridge to the near side.– Pros:

• Explains “simply” two phenomena at once.• Explains the large particle/energy carried by the ridge.

– Cons:• Requires a strong acceleration field that otherwise leaves the intrinsic correlation

“unchanged”

– Many Open issues: Effects of Quenching, Diffusion, Momentum Conservation, Requires detailed modeling, and comparison with data.

• Handle of Early Time System Expansion?


Additional Material


Some Properties of CumulantsCumulants are not positive definiteThe number of particles in a bin varies e-by-e: ni = <ni> + εi

n1n2 = n1 + ε1( ) n + ε( ) = n1 n + ε1ε

n1n2n3 = n1 + ε1( ) n + ε( ) n3 + ε3( )

= n1 n n3 + n1 εε3 + n ε1ε3 + n3 ε1ε + ε1εε C3 = ε1εε

Cumulant for Poisson Processes (independent variables) are null

C2 = n1n − n1 n = ε1ε =0 C3 = ε1εε =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 = ε1ε

n1n2 − n1 n

n1 n

=−1M


More Properties of CumulantsConsider a Superposition of α=1,…, s processes

Number of particles in a phi bin in a given event: ni = nα,iα=1

s

∑1- Particle Density: ni = nα,i

α=1

s

∑

2- Particle Density: n1n2 = nα ,1α=1

s

∑⎛⎝⎜⎞⎠⎟

n,=1

s

∑⎛

⎝⎜⎞

⎠⎟= nα ,1nα ,

α=1

s

∑ + nα ,1n,α≠

s

∑

Product of Single Particle Densities: n1 n2 = nα ,1α=1

s

∑⎛⎝⎜⎞⎠⎟

n,=1

s

∑⎛

⎝⎜⎞

⎠⎟= nα ,1 nα ,

α=1

s

∑ + nα ,1 n,α≠

s

∑

2-Cumulant: C2 = Cα,α=1

s

∑ + COVα (1,)α≠

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 = nα ,1α=1

s

∑⎛⎝⎜⎞⎠⎟

n,=1

s

∑⎛

⎝⎜⎞

⎠⎟nγ,

γ=1

s

∑⎛

⎝⎜⎞

⎠⎟= nα ,1nα ,nα ,3

α=1

s

∑ + nα ,1n,nγ,3α≠≠γ

s

∑

3-Cumulant: C3 = Cα,3α=1

s

∑ + COVαγ (1,,3)α≠≠γ

s

∑

Enables Separation of Jet (Mach Cone) and Flow Background.


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 φij differences.








PYTHIA p+p @ sqrt(s)=200 GeV; 3<pt<20 && 0.2<pt<1

=0.1 =0.2=0. NO BOOST








=0.3 =0.4 =0.5

PYTHIA p+p @ sqrt(s)=200 GeV; 3<pt<20 && 0.2<pt<1









=0.1 =0.2








PYTHIA p+p @ sqrt(s)=200 GeV; 3<pt<20 && 2<pt<3=0.4 =0.5=0.3








PYTHIA p+p @ sqrt(s)=200 GeV; 3<pt<20 && 0.2<pt<13-cumulants =0.1 =0.2

Yield normalized per bin (72x72)






=0.3 =0.5PYTHIA p+p @ sqrt(s)=200 GeV; 3<pt<20 && 0.2<pt<1





are needed to see this picture.QuickTime™ and a

TIFF (Uncompressed) decompressorare needed to see this picture.

PYTHIA p+p @ sqrt(s)=200 GeV; 3<pt<20 && 2<pt<3=0.1 =0.2








=0.3 =0.4 =0.5


HI:C 07 - Montreal C. Pruneau, Wayne State 1 New Perspectives on Measurements of 2- and 3- Particle Correlations Claude A. Pruneau Wayne State University.

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