Correlation of Hadrons in Jets Produced at RHIC

Correlation of Hadrons in Jets Produced at RHIC

Rudolph C. HwaUniversity of Oregon

Workshop on QCD and RHIC physics

Wuhan, June 22, 2005

2

Work done in collaboration with

Chunbin Yang (Hua-Zhong Normal University, Wuhan)

Rainer Fries (University of Minnesota)

Ziguang Tan (Hua-Zhong Normal University, Wuhan)

Charles Chiu (University of Texas, Austin)

3

Regions of transverse momentum

Traditional classification in terms of scattering

pT0 2 4 6 8 10

hardsoft

pQCD + FF

A different classification in terms of hadronization

pT0 2 4 6 8 10

(low) (intermediate)

thermal-thermal thermal-shower

(high)shower-shower

Terminology used in recombination

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recombination

What about string fragmentation?

• Fragmentation is not important until pT > 9

GeV/c.• String model may be relevant for pp collisions,

• String/fragmentation has no phenomenological support in heavy-ion collisions.

but not for AA collisions.

5

pdNπ

dp=

dq1

q1∫

dq2

q2

Fjj'(q1,q2)Rπ (q1,q2,p)

Basic equations for pion production by recombination

Rπ (q1,q2,p)=

Shower parton distributions are determined from

Fragmentation function xDi

π (x) =dx1x1

∫dx2x2

Sij(x1),Si

j '(x2

1−x1

)⎧ ⎨ ⎩

⎫ ⎬ ⎭ Rπ(x1,x2,x)

q1q2

pδ(q1 +q2 −p)

Fjj ' =TT + TS +SS

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Thermal partons are determined from the final state, not from the initial state.

Transverse plane

dNπ

pTdpT

(log scale)

pT2

An event generator takes care of the spatial problem. cf. Duke and TAMU work on recombination.

We deal in momentum space only, with all partons collinear until we treat angular dependence.

k

7

thermal

fragmentation

soft

hard

TS Pion distribution (log scale)

Transverse momentum

TT

SS

Phenomenological successes of this picture

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production in AuAu central collision at 200 GeV

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Hwa & CB Yang, PRC70, 024905 (2004)

TS

fragmentation

thermal

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are needed to see this picture.All in recombination/ coalescence model

Compilation of Rp/ by R. Seto (UCR)

10

kT broadening by multiple

scattering in the initial state.

Unchallenged for 30 years.

If the medium effect is before fragmentation, then should be independent of h= or p

Cronin Effect

p

q

in pA or dA collisionsCronin et al, Phys.Rev.D (1975)h

dNdpT

(pA→ πX)∝ Aα , α >1

A

RCPp >RCP

πSTAR, PHENIX (2003)

Cronin et al, Phys.Rev.D (1975)

p >

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d+Au collisions



Hwa & CB Yang, PRL 93, 082302 (2004)

No pT broadening by multiple scattering in the initial state.Medium effect is due to thermal (soft)-shower

recombination in the final state.

soft-soft

pion



12



Hwa, Yang, Fries, PRC 71, 024902 (2005)

Forward production in d+Au collisions

Underlying physics for hadron production is not changed from backward to forward rapidity.

BRAHMS

13

Correlations

2. Correlation in jets: trigger, associated particle, background subtraction, etc.

1. Two-particle correlation with the two particles treated on equal footing.

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Correlation function

ρ2(1,2)=dNπ1π2

p1dp1p2dp2

ρ1(1) =dNπ1

p1dp1

Normalized correlation function

K2(1,2) =C2(1,2)

ρ1(1)ρ1(2)=r2(1,2)−1 r2(1,2) =

ρ2(1,2)ρ1(1)ρ1(2)

In-between correlation function

G2(1,2)=C2(1,2)

ρ1(1)ρ1(2)[ ]1/ 2

C2 (1,2) =ρ2 (1,2)−ρ1(1)ρ1(2)

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Correlation of partons in jets

A. Two shower partons in a jet in vacuum

Fixed hard parton momentum k (as in e+e- annihilation)

k

x1

x2

ρ1(1) =Sij(x1)

ρ2(1,2)= Sij(x1),Si

j'(x2

1−x1

)⎧ ⎨ ⎩

⎫ ⎬ ⎭

=12

Sij(x1)Si

j'(x2

1−x1

) +Sij (

x1

1−x2

)Sij'(x2)

⎧ ⎨ ⎩

⎫ ⎬ ⎭

r2(1,2) =ρ2(1,2)

ρ1(1)ρ1(2)

x1 +x2 ≤1

The two shower partons are correlated.

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no correlationC2 (1,2) =[r2 (1,2)−1]ρ1(1)ρ1(2)

0

Hwa & Tan, nucl-th/0503052

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B. Two shower partons in a jet in HIC

Hard parton momentum k is not fixed.

ρ1(1) =Sj(q1) =ξ dkkfi∫

i∑ (k)Si

j(q/ k)

ρ2(1,2) = (SS) jj '(q1,q2 ) = ξ dkkfi∫

i∑ (k) Si

j (q1

k),Si

j '(q2

k − q1

)⎧⎨⎩

⎫⎬⎭

r2(1,2) =ρ2(1,2)

ρ1(1)ρ1(2)fi(k)

fi(k) fi(k)

fi(k) is small for 0-10%, smaller for 80-92%

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Correlation of pions in jets

Two-particle distribution

dNππ

p1dp1p2dp2=

1(p1p2)

2

dqi

qii∏

⎡

⎣ ⎢ ⎤

⎦ ⎥ ∫ F4(q1,q2,q3,q4)R(q1,q3,p1)R(q2,q4, p2)

F4 =(TT+ST+SS)13(TT+ST+SS)24

k

q3

q

1

q4

q2

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Correlation function of produced pions in HIC

C2(1,2) =ρ2(1,2)−ρ1(1)ρ1(2)

ρ2(1,2)=dNπ1π2

p1dp1p2dp2

ρ1(1) =dNπ1

p1dp1


Factorizable terms: (TT)13(TT)24 (ST)13(TT)24 (TT)13(ST)24

Do not contribute to C2(1,2)

Non-factorizable terms (ST+SS)13(ST+SS)24

correlated

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C2(1,2) =ρ2(1,2)−ρ1(1)ρ1(2)




22

G2(1,2)=C2(1,2)

ρ1(1)ρ1(2)[ ]1/ 2



along the diagonal

23

24



Hwa and Tan, nucl-th/0503052

RCPG2 (1,2) =

G2(0−10%)(1,2)

G2(80−92%)(1,2)

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Physical reasons for the big dip:

• competition for momenta by the shower partons in a jet

• if p1 and p2 are low, hard parton k can be

low, and the competition is severe.Recall that r2(1,2) < 1 for shower

partons.

• if p1 and p2 are high, hard parton k can be

high, but fi(k) is suppressed, so ρ1(1)ρ1(2) is

small, and C2(1,2) becomes positive.

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Correlation with trigger particle

Study the associated particle distributions

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STAR has measured: nucl-ex/0501016

Associated charged hadron distribution in pT

Background subtracted and distributions

Trigger 4 < pT < 6

GeV/c

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Associated particle pT distribution

dNππ

p1dp1p2dp2=

1(p1p2)

2

dqi

qii∏

⎡

⎣ ⎢ ⎤

⎦ ⎥ ∫ F4(q1,q2,q3,q4)R(q1,q3,p1)R(q2,q4, p2)


After background subtraction, consider only:

dNπ

p2dp2trig =

dp1p1dNππ

p1dp1p2dp24

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∫dp1p1

dNπ

p1dp14

6

∫

p1 -- trigger

p2 -- associated

(ST+SS)13(ST+SS)24

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Reasonable agreement with data




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Very little dependence on centrality in dAu

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and distributions (STAR nucl-ex/0501016)

P1

P2

pedestal

subtraction point no pedestal

short-range correlation?

long-range correlation?

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New issues to consider:

• Angular distribution (1D -> 3D)

shower partons in jet cone

• Thermal distribution enhanced due to

energy loss of hard parton

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Longitudinal

Transverse

t=0 later

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Events without jets T(q) =Cqe−q/T

Thermal medium enhanced due to energy loss of hard parton

Events with jets

T'(q) =Cqe−q/T 'in the vicinity of the jet

T’- T = T > 0new parameter

Thermal partons

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For STST recombination

enhanced thermal

trigger associated particle

Sample with trigger particles and with background subtracted

F4

' =ξ dkkfi∫i

∑ (k)T'(q3){S(q1),S(q2)}T'(q4)G(ψ,q2 /k)

Pedestal peak in &

F4tr−bg =∑∫L (ST')13 (T'T' −TT)24 +(ST')13 (ST')24 G

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kq2

z

hard parton

shower partonShower parton

angular distribution in jet cone

Cone width

σ(x) =σ 0(1−x)

G(ψ,q2 /k) =exp−(2tan−1g(η1 +Δη,η1))

2

2σ 2(q2 / k)

⎡

⎣ ⎢ ⎤

⎦ ⎥

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1Ntrig

dNdΔη

=dη1 dp2p2 dp1p1

dNtrig−bg

p1dp1p2dp24

6

∫passocmin

4

∫−0.7

0.7

∫

dη1−0.7

0.7

∫ dp1p1dNtrig

p1dp14

6

∫

dNtrig

p1dp1=

ξp1

3 dkkfi∫i∑ (k) dq1∫ T'(p1 −q1)S

q1

k⎛ ⎝

⎞ ⎠

dNtrig−bg

p1dp1p2dp2=

ξ(p1p2)

3 dkkfi∫i∑ (k) dq1 dq2 ⋅∫∫

×

T'(p1 −q1)Sq1

k⎛ ⎝

⎞ ⎠

T'(q2)T'(p2 −q2)−T(q2)T(p2 −q2)[ ]

+T'(p1 −q1){Sq1

k⎛ ⎝

⎞ ⎠ ,S

q2

k−q1

⎛

⎝ ⎜ ⎞

⎠ ⎟ }T'(p2 −q2)G(ψ,q2 /k)

⎧

⎨ ⎪

⎩ ⎪

⎫

⎬ ⎪

⎭ ⎪

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Pedestal in

P1,2 = dp2pmin(1,2)

4

∫dN(T'T'−TT)

dp2|trig

more reliable

0.15 < p2 < 4 GeV/c, P1 = 0.4

2 < p2 < 4 GeV/c, P2 = 0.04

P1

P2

less reliableparton distribution

T'(q) =Cqe−q/T ' T ’ adjusted to fit pedestal

find T ’= 0.332 GeV/c

cf. T = 0.317 GeV/cT = 15 MeV/c

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Associated particle distribution in

Chiu & Hwa, nucl-th/0505014

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Associated particle distribution in

Chiu & Hwa, nucl-th/0505014

42

The peaks in & arise from the recombination of thermal partons with shower partons in jets with angular spread.

2. The pedestal arises from the enhanced thermal medium.

That is the feedback from the hard parton through lost energy to the soft partons. By longitudinal expansion it gives rise to the long-range correlation.

Correlation exists among the shower partons, since they belong to the same jet.

That may be regarded as the short-range correlation --- though only kinematical (sufficient so far).

1.

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Autocorrelation

Correlation function C2 (1,2) =ρ2 (1,2)−ρ1(1)ρ1(2)

1,2 on equal footing --- no trigger

Define

θ− =θ2 −θ1φ− =φ2 − φ1

Autocorrelation: Fix and , and integrate over all other variables in

θ− φ−

C2 (1,2)

The only non-trivial contribution to

near , would come from jets θ− : 0 φ− : 0

A(θ−,φ−)

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p2

p1

x

yz

θ1θ2

pion momentum

space

q2

q1

x

yz

2

1

k

parton momentum

space

A(−,φ−)

-

H (θ1,θ2 ,φ−)P()

G( 1, 2 )Gaussian in jet cone

45

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Autocorrelation

Chiu & Hwa (2005b)

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Other recent work done on recombination

Fries, Muller, Bass, PRL94, 122301(2005)

Correlation

Muller, Fries, Bass, nucl-th/0503003

Beyond the valence quark approximation

Majumder, E. Wang, X.N. Wang, LBNL-57478

Modified fragmentation function

Greco and C.M. Ko, nucl-th/0505061

Scaling of hadron elliptic flow

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Conclusion

Parton recombination provides a framework to interpret the data on jet correlations.

There seems to be no evidence for any exotic correlation outside of shower-shower correlation in a jet.

If future analysis finds no hole in , then some dynamical correlation among the shower partons may be needed.

RCPG2

Autocorrelation without subtraction is a good place to compare theory and experiment.

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Porter & Trainor, ISMD2004, APPB36, 353 (2005)


are needed to see this picture.Transverse rapidity yt

( pp collisions )

G2

STAR

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51Hwa & Tan, nucl-th/0503052



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z

θ1

θ

p1

trigger

Assoc p2kq2

z

hard parton

shower parton

ψ =θ −θ1

η−η1 =Δη

tanψ2

=g(η,η1)=e−η −e−η1

1+e−η−η1

=e−η1e−Δη −1

1+e−Δη−2η1

⎡

⎣ ⎢ ⎤

⎦ ⎥

Expt’l cut on trigger: -0.7 < 1 < +0.7k

jet cone exp[−ψ 2 /2σ 2(x)]

Documents

Correlation of Hadrons in Jets Produced at RHIC