Hadronization of Partons by Recombination

Rudolph C. HwaUniversity of Oregon

Summer School on RHIC Physics

Wuhan, China, June 2005

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Outline

An overview of the recombination modelSome questions and answers on the basicsShower partons initiated by hard partonsHadronization in heavy-ion collisions

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Parton Recombination

First studied for low-pT production in pp collision Das & Hwa, Phys. Lett. 68B, 459 (1977)

E dNπ

dpL≡H(x) = dx1

x1∫ dx2

x2Fqq (x1,x2)Rπ (x1,x2,x)

p p Fu x1( )Fd x2( )

x

H(x)

Ochs observation: H(x) is very similar to the valence quark distribution in a proton.

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Valon model -- to get the proton wave function Hwa, PRD (1980a)

Valon-recombination model -- better formulation of recombination Hwa, PRD (1980b)

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p+A collisions Hwa & CB Yang (2002a)

We studied the centrality dependence (or the number of collisions) in the valon-recombination model

good data from NA49

Hadronic collisions Hwa & CB Yang (2002b)

h + p h’ +Xh h’p

K+ + K

m

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Hadron production at high pT pp collision: mainly by fragmentation

AA collision: there were puzzles according to fragmentation

Recombination solved those puzzlesHwa & Yang, PRC 67, 034902 (2003); 70, 024905 (2004)Greco, Ko, Levai, PRL 90, 202302 (2003); PRC 68, 034904 (2003)Fries, Muller, Nonaka, Bass, PRL 90,202303(03); PRC 68, 044902 (03)

More recent developments -- 2004, 2005Correlations in jets

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Closer examination of the recombination formulas

pdNd=dq1q1∫

dq2q2Fqq (q1,q2 )R (q1,q2 , )Pion :

Proton :

pdN p

dp=dq1q1∫

dq2q2

dq3q3Fuud (q1,q2 ,q3)R(q1,q2 ,q3, )

Fqq (q1,q2 )

Questions:

1. What is the two-parton distribution ?

2. What are the recombination functions ? R (q1,q2 , )

Especially in heavy-ion collisions

Rp (q1,q2 ,q3, p)

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3. What about the gluons?4. Does entropy decrease?5. What about the spatial considerations?6. Isn’t the pion a Goldstone boson?7. Recombination versus fragmentation:

Which is more important?8. What is wrong with string

fragmentation?

More questions :

Answer in reverse order.

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recombination

8. String fragmentation • 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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High pT physics in pp collisions is well understood.What was a discovery yesterday is now used for calibration today.

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7. Recombination versus Fragmentation

Parton distribution (log scale)

pp

hadron momentum

q1+q2(recombine)

higher yield

q

(fragment)

heavy penalty

suppressed by power-law

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6. Pion is a Goldstone boson

Is it a boson due to spontaneous symmetry breaking? Or a bound state of quark-antiquark?

Both are aspects of the pion. No theory exists that can continuously transform one to the other.

In Drell-Yan process in -p collisions, the quark contents of pion and proton are probed.

p

+

-

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5. Spatial considerationsWe have formulated recombination in momentum space only so far.Shouldn’t the spatial coordinates be important also? Isn’t hadron size relevant?

In heavy-ion collisions there are two sizes: 1. nuclear transverse size

RA2. hadron transverse size rh

• If partons are parallel, but far apart, they cannot recombine

• If parton trajectories intersect, they must cross at the same space-time region --- relative momentum suppress recombination.

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Soft partons are restricted to the small spatial spread around the point where hard

parton emerges from the nuclear medium.

Groups at Duke University and Texas A&M University have Monte Carlo codes to implement space & momentum constraints on recombination.

Our approach:

We consider only collinear partons. Hard parton defines the direction of the hadron.

We do not use Monte Carlo code to generate the soft partons throughout the expanding medium. We infer from the soft pion spectrum at low pT what the soft parton distribution is.Momentum space consideration is

sufficient, and that is where observation is made.

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4. Entropyu + d → +

color: 3 X 3 1spin: 2 X 2 1

degrees of freedom decreased

R (q1,q2 , )= f(q1,q2 , )d(q1 + q2 −)

depends on wave function momentum conservation

g

q1 + q2 =d

+u Soft gluon radiation: mutates color & carries

away spin without changing RM cannot account for low momentum partons

Entropy: a global quantity that should take into account expanding volume.

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3. How do gluons hadronize?In pp collisions the parton

distributions are

Gluons carry ~1/2 momentum of proton but cannot hadronize directly.

Sea quark dist. Fq ~ c (1-x)7

Saturated sea quark dist. F’q ~ c’ (1-x)7

Gluon conversion to q-qbarq

q

g

Recombination of with saturated sea gives pion distribution in agreement with data.

qq

x2u(x)

x2g(x)

x [log]

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2. Recombination functionsIt depends on the wave function.q

q

uud

p

Consider the time-reversed processuud

p puud

What are the distributions of the quarks in momentum fractions in the infinite momentum frame?

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Deep inelastic scattering

e e

p

Fq

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

Fq

Valon modelp

UUD

valons

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pU

UD

Basic assumptions• valon distribution is

independent of probe• parton distribution in a valon is independent of the hadronxuv (x,Q

2 )= dy2GUx

1

∫ (y)KNS(xy,Q2 )

xdv (x,Q2 )= dyG Dx

1

∫ (y)KNS(xy,Q2 )

valence quark distr in proton

valon distr in proton, independent of Q

valance quark distr in valon, in proton or in pion

%uv (n,,Q2 )=2 %GU (n) %KNS(n,Q

2 )

%dv (n,,Q2 )= %G D (n) %KNS(n,Q

2 )

Moments by convolution theorem

known from CTEQ param

cancel in the ratio

the ratio can be determined

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3-valon exclusive distribution

Recombination function

α =1.76 β =1.05

proton Rp(x1,x2,x3,x) =x1x2x3x3 GUUD

p (x1x

, x2x

, x3x

)=g(x1x2x2 )2.76(x3

x)2.05δ(x1

x+x2

x+x3

x−1)

pion Rπ (x1,x2,x) =x1x2x2 GUD

π (x1x

,x2x

)=x1x2x2 δ(x1

x+x2

x−1)

From initiated Drell-Yan process xqv

π(x) =Ax0.64(1−x)1.11 valon model Gπ (y1,y2) =δ(y1 +y2 −1)

GU (y)= dy2∫ dy3∫ GUUD (y1, y2 , y3)

G D (y)= dy1∫ dy2∫ GUUD (y1, y2 , y3)

Single-valon inclusive distribution

Hwa & CB Yang, PRC66(2002)

GUUD(y1,y2,y3) =g(y1y2)α y3βδ(y1 +y2 +y3 −1)

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1. Two-parton distributions

Fqq (q1,q2 ) Fuud (q1,q2 ,q3)

pp collisions: low pT and large xF

Fud (x1, x2 )=Fu(x1)Fd (x2 )

Heavy-ion collisions:

Low pL (mid-rapidity), large pTThat is high pT physics.

Traditionally, hadronization at high pT is by fragmentation.However, fragmentation model has met some difficulties, most notably in p/ ratio at intermediate pT in nuclear collisions.

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Before describing what the two-parton distribution should be at high pT in heavy-ion collisions, we must first• discuss why fragmentation does not work

phenomenologically• what are the shower partons in fragmentation?• how does the nuclear medium affect

hadronization?Which parton recombines which parton is the core problem in the recombination model.

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Not possible in fragmentation model:

Dp/ q <<Dπ /q

Rp/π

Rp/π 1

Dp/q

Dπ /q

u

p/ ratio

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The black box of fragmentation

q

A QCD process from quark to pion, not calculable in pQCD

z1

Momentum fraction z < 1

Phenomenological fragmentation function

D/q

z1

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Let’s look inside the black box of fragmentation.

q

fragmentation

z1

gluon radiation

quark pair creation

Although not calculable in pQCD (especially when Q2 gets low), gluon radiation and quark-pair creation and subsequent hadronization nevertheless take place to form pions and other hadrons.

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Description of fragmentation by recombination

known from data (e+e-, p, … )

known from recombination model

can be determined

hard partonmeson

fragmentation shower partons recombination

xD(x) = dx1x1

∫ dx2x2

Fq,q (x1,x2)Rπ (x1,x2,x)

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Shower parton distributions

Fqq '(i )(x1,x2) =Si

q(x1)Siq ' x2

1−x1

⎛ ⎝ ⎜ ⎞

⎠ ⎟

Sij =

K L Ls

L K Ls

L L Ks

G G Gs

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

u

gs

s

d

du

K =KNS+L

Ks =KNS +Ls

Sud,d ,u ,u(sea) =L

valence

sea

L L DSea

KNS L DV

G G DG L

Ls DKSea

G Gs DKG

R

RK

5 SPDs are determined from 5 FFs.

assume factorizable, but constrained kinematically.

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Shower Parton Distributions

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

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BKK fragmentation functions

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If our shower parton distributions are reliable, based on the dynamical independence of the shower partons except for kinematical constraints, then we should be able to calculate the quark fragmentation function into a proton.

Data on Dup(z) not well determined. KKP parametrization has an error.

Nevertheless, there is only a discrepancy of less than a factor of 2 over 4 order of magnitude.

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Once the shower parton distributions are known, they can be applied to heavy-ion collisions.The recombination of thermal partons with shower partons becomes conceptually unavoidable.

D(z)

h

qA A

Conventional approach

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Once the shower parton distributions are known, they can be applied to heavy-ion collisions.The recombination of thermal partons with shower partons becomes conceptually unavoidable.

hNow, a new component

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Pion formation: qq distributionthermal

shower

soft component

soft semi-hard components

usual fragmentation(by means of recombination)

TSFqq =TT+TS+SS

Proton formation: uud distribution

Fuud =TTT +TTS +TSS +SSS

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T(p1)=p1

dNqth

dp1=Cp1exp(−p1/T)

Thermal distribution

Fit low-pT data to determine C & T.

Shower distribution in AuAu collisions

S(p2)=ξ∑i ∫dkkfi(k)Sij(p2 /k)

hard parton momentum

distribution of hard parton i in AuAu collisions

Contains hydrodynamical properties, not included in our model.

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fi (k) =dNihard

kdkdyy=0density of hard partons with pT = k

Input: parton distributions CTEQ5L nuclear shadowing EKS98 hard scattering pQCD

Srivastava, Gale, Fries, PRC 67, 034903 (2003)

dNjet

d2 pTdyy=0=K C

(1+pT / B)β

C, B, are tabulated for i=u, d, s, u, d, g K=2.5

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T(p1)=p1

dNqth

dp1=Cp1exp(−p1/T)

Thermal distribution

Fit low-pT data to determine C & T.

Shower distribution in AuAu collisions

S(p2)=ξ∑i ∫dkkfi(k)Sij(p2 /k)

hard parton momentum

distribution of hard parton i in AuAu collisions

SPD of parton j in shower of hard parton i

fraction of hard partons that get out of medium to produce shower

calculable

Contains hydrodynamical properties, not included in our model.

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thermal

fragmentation

soft

hard

TS Pion distribution (log scale)

Transverse momentum

TT

SS

Now, we go to REAL DATA, and real theoretical results.

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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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Proton production in AuAu collisions

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

TTS+TSS

TSS

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QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Proton/pion ratio

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QuickTime™ and aTIFF (LZW) decompressor

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

Compilation of Rp/ obtained by 3 groups

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Puzzle in pA or dA collisions

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

pq

hdNdpT

(pA→ πX)∝ Aα , α >1

A

RCPp >RCP

π STAR, PHENIX (2003)

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

p >

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RHIC data from dAu collisions at 200 GeV per NN pair

Ratio of central to peripheral collisions: RCP

RCPh (pT ) =

dNhdpT

1NColl

central( )dNhdpT

1NColl

peripheral( )

PHENIX and STAR experiments found (2002) RCP

p (pT )>RCPπ (pT )

Can’t be explained by fragmentation.

=geometrical factor(central) × Dh

geometrical factor(peripheral) × Dh

(in fragmentation model)

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RCPp (pT )>RCP

π (pT )STAR

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d

dcentral peripheral

more T more TS

less T less TS

RCPh (pT ) =

dNhdpT

1NColl

central( )dNhdpT

1NColl

peripheral( ) ⇒ more TS

less TS >1

d+Au collisions (to study the Cronin Effect)

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d+Au collisions

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.QuickTime™ and a

TIFF (LZW) decompressorare needed to see this picture.

Pions

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

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QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Proton

Thermal-shower recombination is negligible.

Hwa & Yang, PRC 70, 037901 (2004)

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Nuclear Modification Factor

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

RCPp >RCP

π

This is the most important result that validates parton recombination.

2q , each quark has ~1/2 of momentum

3q p, each quark has ~1/3 of p momentum

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Molnar and Voloshin, PRL 91, 092301 (2003).

Parton coalescence implies that v2(pT) scales with the number of

constituents STAR data

Azimuthal anisotropy

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Forward-backward asymmetry in d+Au collisions

Expects more forward particles at high pT than backward particles

If initial transverse broadening of parton gives hadrons at high pT, then

• backward has no broadening

• forward has more transverse broadening

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QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Backward-forward asymmetry at intermediate pT

backwardforward

in d+Au collisions

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More interesting behavior found in large pT and large pL region.

It is natural for parton recombination to result in forward-backward asymmetry

Less soft partons in forward (d) direction than backward (Au) direction.Less TS recombination in forward than in backward direction.

Forward-backward asymmetry by recombination

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QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

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 data

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SummaryWe have discussed

• some basic issues about recombination• application to intermediate and high pT

physics in heavy-ion collisions• resolved several puzzles on single-particle distributions in HIC

We have not coveredCorrelation of hadrons in jets

(Thursday)