Correlations at Intermediate p T

Correlations at Intermediate pT

Rudolph C. HwaUniversity of Oregon

Correlations and Fluctuations in Relativistic Nuclear Collisions

MIT, April 2005

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Work done in collaboration with

Chunbin Yang (Hua-Zhong Normal University, Wuhan)

Ziguang Tan (Hua-Zhong Normal University, Wuhan)

Charles Chiu (University of Texas, Austin)

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Physics at Intermediate pT

pT0 2 4 6 8 10

hardsoft semi-hard

thermal-thermal

thermal-shower

shower-shower

low intermediate high

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pdNπ

dp=

dq1

q1∫

dq2

q2

Fjj'(q1,q2)Rπ (q1,q2,p)

Fjj' =TT +TS+SS

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)

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

Transverse plane

dNπ

pTdpT

(log scale)

pT2

No small parameter (rh/RA) in the problem.

6

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)

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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 >

10



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



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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

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Correlations

2. Correlation in jets: trigger, associated particle, background subtraction, etc. (e.g., Fuqiang Wang’s talk)

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

(data to be presented tomorrow)

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Correlation function

C2(1,2) =ρ2(1,2)−ρ1(1)ρ1(2)

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

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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 correlation

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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)



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

ρ1(1)ρ1(2)[ ]1/ 2



along the diagonal

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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:

(a) central: (ST)(ST) dominates

S-S correlation weakened by separate recombination with uncorrelated (T)(T)

(b) peripheral: (SS)(SS) dominates

SS correlation strengthened by double fragmentation

The dip occurs at low pT because at higher

pT power-law suppression of 1(1) 1(2)

results in C2(1,2) ~ 2(1,2) > 0

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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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27Hwa & Tan, nucl-th/0503052



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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



Hwa & Tan, nucl-th/0503052

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Hwa & Tan, nucl-th/0503060

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Very little dependence on centrality in dAu

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and distributions (from Fuqiang Wang’s talk)

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

work done with C. Chiu

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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

Pedestal peak in &

F4

' =ξ dkkfi∫i

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

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

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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)

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∫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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z

1

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)]

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kq2

z

hard parton

shower partonShower parton

angular distribution in jet cone

Cone width

σ(x) =σ 0(1−x)

another parameter ~ 0.22

G(ψ,q2 /k) =exp−(2tan−1g(η1 +Δη,η1))

2

2σ 2(q2 / k)

⎡

⎣ ⎢ ⎤

⎦ ⎥

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Associated particle distribution in

Chiu & Hwa (2005)

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Associated particle distribution in

Chiu & Hwa (2005)

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We have not put in any (short- or long-range) correlation by hand.

The pedestal arises from the enhanced thermal medium.

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

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

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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.

For unbiased study without deciding on bkgd, we suggest the measure, G2(1,2).

Is there a hole in ?RCPG2

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recombination

Comments to stimulate discussion

• 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.

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Correlations at Intermediate p T