recombination in nuclear collisions rudolph c. hwa university of oregon critical examination of rhic...

Recombination in Nuclear Collisions

Rudolph C. HwaUniversity of Oregon

Critical Examination of RHIC Paradigms

University of Texas at Austin

April 14-17, 2010

Outline

1. Introduction

2. Earlier evidences for recombination

3. Recent development

A. Azimuthal dependence --- ridges

B. High pT jets --- scaling behavior

4. Future possibilities and common ground

1. Introduction

pQCDReCoHydro

Fragmentation

kT > pT

Hadronization

Cooper-Frye

k1+k2=pT

lower ki higher density

TT TS SS

low highintermediate

2 6

Usual domains in pT

pT

GeV/c

Regions in time

(fm/c)

1 8

hadronization

0.6rapid thermalization

hydro

Cronin effect: --- initial-state transverse broadening

What about Cronin effect for proton, larger than for ?

Early-time physics: CGC, P violation, …

Pay nearly no attention to hadronization at late times.

In ReCo: Final-state effect, not hard-scattering+Frag, not hydro.

What about semihard scattering (kT<3GeV/c) at <0.6 fm/c?

2. Earlier evidences for Recombination

A. pT distribution at mid-rapidity

Recombination function

R (k1,k2 , pT ) =k1k2

pT2

δ(k1 +k2

pT

−1)

q and qbar momenta, k1, k2, add to give pion pT

It doesn’t work with transverse rapidity yt

TT F(ki ) =Cki exp(−ki /T )dN

pTdpT

=C2

6exp(−pT /T )

TTTdN p

pT dpT

=NppT

2

mT

exp(−pT /T ) same T for partons, , p

empirical evidence

At low pT

phase space factor in RF for proton formation

Pion at y=0 p0 dN

dpT

=dk1

k1∫

dk2

k2

Fqq(k1,k2 )R (k1,k2 , pT )

Proton at y=0

p0 dN p

dpT

=dk1

k1∫

dk2

k2

dk3

k3

Fuud(k1,k2 ,k3)Rp(k1,k2 ,k3, pT )

QuickTime™ and a decompressor

are needed to see this picture.

PHENIX, PRC 69, 034909 (04)

went on to mT plot



Hwa-Zhu (preliminary)

dN p

pT dpT

=NppT

2

mT

exp(−pT /T )

Proton production from reco

Same T for , K, p --- a direct consequence of ReCo.

Slight dependence on centrality --- to revisit later

QuickTime™ and aTIFF (LZW) decompressor


B. p/ ratio

At higher pT shower partons enter the problem; TS recombination enters first for pion, and lowers the ratio.

It is hard to get large p/ ratio from fragmentation of hard partons.

Rp / (pT ) =dNp / pTdpT

dN / pTdpT

dominated by thermal partons at low pT

= pT2

mT (pT )

ReCo

C. Revisit very early formulation of recombination

[at the suggestion of organizers: Hwa, PRD22,1593(1980)]

The notion of valon needs to be introduced.

q

q

For p+pp+X we need

Rp (x1, x2 , x3, x)uud

p

Consider the time-reversed processu

ud

p puu

d

p+p+X Feynman x distribution at low pT

xdN

dx=

dx1

x1∫

dx2

x2

Fqq(x1,x2 )R (x1,x2 ,x)

Deep inelastic scattering

ee

p

Fq

We need a model to relate to the wave function of the proton

Fq

Valon modelp

U

U

Dvalons

A valence quark carries its own cloud of gluons and sea quarks --- valon

p

U

U

D

Basic assumptions

• valon distribution is independent

of probe

• parton distribution in a valon is independent of the hadron

xuv (x,Q2 ) = dy2GUx

1

∫ (y)KNS(xy,Q2 )

xdv (x,Q2 ) = dyGDx

1

∫ (y)KNS(xy,Q2 )

valence quark distr in proton

valon distr in proton, independent of Q

valance quark distribution in valon, whether in proton or in pion



Rp (x1, x2 , x3, x) =g(x1x2

x2 )2.76 (x3

x)2.05δ(

x1

x+

x2

x+

x3

x−1)

R (x1,x2 ,x) =x1x2

x2 δ(x1

x+

x2

x−1) initiated

DY process

p + p h + X in multiparticle production at low pT

p

U

U

Dvalon distribution collisio

n process

partons

chiral-symmetry breaking quarks gain masses momenta persist

U

D

RF

+


are needed to see this picture.QuickTime™ and a

decompressorare needed to see this picture.

No adjustable parameters

1979 data (Fermilab E118)

Not sure whether anyone has done any better

Feynman’s original parton model PRL(69)

D. Shower partons in AA collisions

At higher pT Hard scattering calculable in pQCD Hadronization by fragmentation

In between hard scattering and fragmentation is jet quenching.

Fine, at very high pT (> 6GeV/c), but not reliable at intermediate pT

pT

qD

i (

pT

q)

T(q1)S(q2/q)R(q1,q2,pT)

Fragmentation: D(z) => SS recombination, but there can also be TS

recombination at lower pT

dNTS+SS

pTdpT

=1pT

2

dqq

Fi∫ (q)[TS∂i∑ (q, pT ) + SS∂ (q, pT )]pio

n

proton [TTS∑ +TSS∑ + SSS∑ ]

We need shower parton distribution.

∫dk k fi(k) G(k,q)

k

q

Description of fragmentation

known from data (e+e-, p, … )

known from recombination model

can be determined

recombination

xD(x) =dx1x1

∫dx2

x2Fq,q (x1,x2)Rπ (x1,x2,x)

shower partons

hard partonmeson

fragmentation

by recombination

Shower parton distributionsFqq '

(i )(x1,x2) =Siq(x1)Si

q ' x2

1−x1

⎛

⎝ ⎜ ⎞

⎠ ⎟

Sij =

K L Ls

L K Ls

L L Ks

G G Gs

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

u

gs

s

d

du

L L DSea

KNS L DV

GG DG

L Ls DKSea

G Gs DKG

5 SPDs are determined from 5 FFs.

assume factorizable, but constrained kinematically.

Hwa & CB Yang, PRC 70, 024904 (04)

BKK FF(mesons)Using SSS we can calculate baryon FF

DM ⇔ DB



Hwa-Yang, PRC 73, 064904 (06)

Other topics:

1. Constituent quarks, valons, chiral-symmetry breaking, f

2. Collinear recombination

3. Entropy

4. Hadronization of gluons

5. Dominance of TS over TT at pT>3 GeV/c

6. Single-particle distributions

7. RCPp(pT)> RCP

(pT)

8. Forward-backward asymmetry in dAu collisions

9. Large p/ ratio at large

10. v2 (pT) Quark-number scaling

11. Ridges

12. Correlations

earlier

later

recent

3. Recent developmentAzimuthal dependence

PHENIX 0903.4886

85<<90

30<<45

0<<15

pT

Npart

A. pT < 2 GeV/c

B. pT > 2 GeV/c

A. pT<2 GeV/c

Region where hydro claims relevance --- requires rapid thermalization

0 = 0.6 fm/c

Something else happens even more rapidly

Semi-hard scattering 1<kT<3 GeV/c

Copiously produced, but not reliably calculated in pQCD t < 0.1 fm/c

1. If they occur deep in the interior, they get absorbed and become a part of the bulk.

2. If they occur near the surface, they can get out. --- and they are pervasive.

[Tom Trainor’s minijets (?)]

On the way out of the medium, energy loss enhances the thermal partons --- but only locally.

Recombination of enhanced thermal partons ridge particles

ρ1(pT ,φ,b) = B(pT ,b) + R(pT ,φ,b)

Base, independent of , not hydro bulk

Ridge, dependent on , hadrons formed by TT reco

• Ridge can be associated with a hard parton, which can give a high pT trigger.• But a ridge can also be associated with a semihard parton, and a trigger is not necessary; then, the ridge can be a major component of

ρ1(pT ,φ,b)

Correlated part of two-particle distribution on the near side

ρ2corr (1,2) = ρ2

J (1,2) + ρ2R (1,2)

Putschketrigger

assoc part

JET RIDGE

How are these two ridges related?

BOOM



Hard parton

Ridge

without trigger

but that is a rare occurrence

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

Semihard partons, lots of them in each event

Ridges without triggers --- contribute significantly to single-particle distribution

ratatatatatata

We need an analogy

1

2

1

2

Two events: parton 1 is undetected thermal partons 2 lead to detected hadrons with the same 2

R(φ2 ) ∝ dφ1∫ ρ2R(φ1,φ2 )

Ridge is present whether or not 1 leads to a trigger.

Semihard partons drive the azimuthal asymmetry with a dependence that can be calculated from geometry. Hwa-Zhu, 0909.1542, PRC (2010)

If events are selected by trigger (e.g. Putschke QM06, Feng

QM08), the ridge yield is integrated over all associated particles 2.

Y R (φ1) ∝ dφ2∫ ρ2R(φ1,φ2 )

Enhanced thermal partons on average move mainly in the direction normal to the surface

~|2-1|<~0.33 Correlated emission model

(CEM) Chiu-Hwa, PRC 79 (09)

Geometrical consideration in Ridgeology

For every hadron normal to the surface there is a limited line segment on the surface around 2

through which the semihard parton 1 can be emitted.


are needed to see this picture.QuickTime™ and a decompressor




b normalized to RA

Ridge due to enhanced thermal partons near the surface

R(pT,,b) S(,b)nuclear density

S(,b) 2

Base

ρ1(pT ,φ,b) = B(pT ,b) + R(pT ,φ,b)

Ridge



base

ridge

inclusive


are needed to see this picture.base

ridge

inclusive

RH-L.Zhu (preliminary)

ρ1 (pT ,φ,b) = B(pT ,b) + R(pT ,φ,b) = N(pT ,b)[e− pT /T0 + e− pT /T1 (b)aD(b)S(φ,b)]



Single-particle distribution at low pT without elliptic flow, but with Ridge

T0 for base

T1(b) for ridge

a can be determined from v2, since S(,b) is the only place that has dependence.



ridge bas

e

Azimuthal dependence of ρ1(pT,,b) comes entirely from Ridge ---

In hydro, anisotropic pressure gradient drives the asymmetry

x

y

requiring no rapid thermalization, no pressure gradients.

Since there more semihard partons emerging at ~0 than at ~/2, we get in ReCo anisotropic R(pT,,b),

∝ S(φ1,b)



Hwa-Zhu, PRC (10)

Y R (φ1) ∝ dφ2∫ ρ2R(φ1,φ2 )

Ridge yield’s dependence on trigger



Feng QM08

φs ⇒ φ1

Normalization adjusted to fit, since yield depends on exp’tal cuts

Normalization is not readjusted.

s dependence is calculated

S(,b) correctly describes the dependence of correlation

Nuclear modification factor



art

Summary

dependencies in

Ridge R(pT,,b) v2(pT,b)=<cos 2 > yield YR() RAA(pT,,b)

are all inter-related --- for pT<2 GeV/c

Hwa-Zhu, 0909.1542 PRC (2010)

B. pT>2 GeV/cPHENIX 0903.4886

Need some organizational simplification. and b are obviously related by geometry.

Scaling behavior in --- a dynamical path length

5 centralities and 6 azimuthal angles () in one universal curve for each pTLines are results of calculation in Reco.

Hwa-Yang, PRC 81, 024908 (2010)





Complications to take into account:

• details in geometry

• dynamical effect of medium

• hadronization

Nuclear medium that hard parton traverses

x0,y0

k

Dynamical path length

=γl (x0 , y0 ,φ,b) γ to be determined

Geometrical path length

l (x0 , y0 ,φ,b) = dtD[x(t),y(t)]

0

t1 (x0 ,y0 ,φ,b)

∫D(x(t),y(t))

Geometrical considerations

Average dynamical path length

(φ,b) = γ dx0dy0∫ l (x0 , y0 ,φ,b)Q(x0 , y0 ,b)

Q(x0 , y0 ,b) =TA(x0 ,y0 ,−b / 2)TB(x0 ,y0 ,b / 2)

d2rsTA(rs+

rb / 2)TB(

rs−

rb / 2)∫

Probability of hard parton creation at x0,y0





Define

P(,φ,b) = dx0 dy0Q(x0 ,y0∫ ,b)δ[ −γl (x0 ,y0 ,b)] (φ,b) = dξξP(∫ ξ ,φ,b)

KNO scaling

P(,φ,b) =ψ (z) (φ,b)

z = / dzψ (z) =1∫dzzψ (z) =1∫

For every pair of and c:

• we can calculate

• PHENIX data gives

(φ,c)

RAA (φ,c)

We can plot the exp’tal data

RAA ( )

There exist a scaling behavior in the data when plotted in terms of

(φ,c)









Theoretical calculation in the recombination model Hwa-Yang, PRC 81, 024908 (2010) ( γ = 0.11 )

ρ1

TS +SS (pT ,φ,b) =dq

q∫ Fii

∑ (q,φ,b)H i (q, pT )

b

q

TS+SS recombination

G(k,q,) =qδ(q−ke− )

degradation

hadronization

dNihard

kdkdyy=0

= fi (k)

Fi (q,) = dkkfi∫ (k)G(k,q,)

k probability of hard parton creation with momentum k

geometrical factors due to medium

dNTS

pTdpT

=1pT

2

dqq∫

i∑ Fi (q)TS∂ (q, pT )

TS∂ (q, pT ) =

dq2

q2∫ Si

j (q2

q) dq1∫ Ce−q1 /T R (q1,q2 , pT )

dNSS

pTdpT

=1pT

2

dqq∫

i∑ Fi (q)SS∂ (q, pT )

xDi (x) =

dx1

x1∫

dx2

x2

Sij (x1),Si

j '(x2

1−x1

)⎧⎨⎩

⎫⎬⎭R (x1,x2 ,x)

x =pT / q

Nuclear modification factor

RAA (pT ,φ,c) =

dNAA / dpTdφ

NcolldNpp / dpT

only adjustable parameter γ = 0.11

=γl (x0 , y0 ,φ,b)

4. Future Possibilities

At kT not too large, adjacent jets can be so close that shower partons from two parallel jets can recombine.

H ii '(q,q ', pT ) =1pT

2

dq1

q1∫

dq2

q2

Sij (

q1

q)Si '

j '(q2

q')R

Γ (q1,q2 , pT )

≅ΓRπ (q1,q2 , pT )

Γ - probability for overlap of two shower partons

ρAA2 j ∝ Ncoll

2

RAA2 j (pT ,φ,c) =

ρAA2 j (pT ,φ,c)

Ncollρpp1 j (pT ,c)

At LHC, the densities of hard partons is high.

A. Two-jet recombination at LHC

Two hard partons

dNAA2 j

pT dpT dφ=

dqq∫

dq'q'

Fi (qii '∑ ,φ,b)Fi '(q',φ,b)Hii '(q,q', pT )



Scaling

1jet QuickTime™ and a decompressor


Scaling badly broken

Hwa-Yang, PRC 81, 024908 (2010)

2jet

Pion production at LHC

Observation of large RAA at pT~10 GeV/c will be a clear signature of 2-jet recombination.

>1 !

Proton production due to qqq reco is even higher.

Hwa-Yang, PRL 97 (06)

B. Back-to-back dijets

C. Forward production of p and

D. Large correlation

E. Auto-correlation

F. P violation: hadronization of chirality-flipped quarks

G. CGC: hadronization problem

Common ground with the 2-component model of UW-UTA alliance

B. Two-component model





T.Trainor, 0710.4504, IJMPE17,1499(08)

Hwa-Yang, PRC70,024905(04)




Similar to our Base, B ~ exp(-pT/T0), T0 independent of b

minijets

Strong enhancement of hard component at small yt

Similar to our Ridge, R ~ exp(-pT/T1), T1 depends on b

ρAA

npart / 2= SNN (yt ) + ν H AA (yt ,ν )

SNN(yt) is independent of

Ridge due to semihard partons --- minijets?



Comparison

Recombination 2-component

semihard partons minijets

recombination of enhanced fragmentation thermal partons

Ridges --- TT reco effect of jet on medium low-yt enhancement

Jets --- TS+SS effect of medium on jet high-yt suppressionρ1 = B + R + J ρ1 = S + H

no dependence on depend on b and

B+R accounts for v2 at pT<2GeV/c some quadrupole component without hydro without hydro ρ(η Δ ,φΔ )

ρ ref

In Recombination

averaged over B(pT) R(pT,b)ρ

1

h (pT ,b) = N h (pT ,b)[e− pT /T0 + A(b)e− pT /T1 ]



In 2D autocorrelation

UW-UTA alliance

dependence

ρ(η Δ ,φΔ )

ρ ref

R(pT ,φ,b)=N(pT ,b)e−pT /T1 (b)aD(b)S(φ,b)






Scaling in variable that depends on initial-state collision parameters only

ρρref

=Δρnf

ρ ref

(η Δ ,φΔ ) + 2Δρ[m]

ρ refm=1

2

∑ cos(mφΔ )

No hydro

Trainor, Kettler, Ray, Daugherity

minijet contribution

φΔ ηΔ

from the hard comp 2<yt<4

I would like to know how it depends on at each b



cf. our ridge component

Conclusion

We should seek common grounds as well as recognize differences.

• Has common ground with minijets.

At pT<2GeV/c, ridges due to semihard scattering and TT reco account for various aspects of the data.At pT>2GeV/c, hard scattering and TS+SS reco account for the scaling behavior observed.

• Recombination can accommodate fragmentation.

• Has thermal distribution at late times, though not thermalization and hydro expansion at early times.

recombination in nuclear collisions rudolph c. hwa university of oregon critical examination of rhic...

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