Fragmentation FunctionsFragmentation FunctionsFragmentation Functions Fragmentation Functions and Fragmentation Processesand Fragmentation Processes

Stefan KretzerBrookhaven National Laboratory &

RIKEN-BNL

XXXIV International Symposium on Multiparticle DynamicsJuly 26 - August 1, 2004

Sonoma State University, Sonoma County, California, USA

http://www.pv.infn.it/~radici/FFdatabase/maintained by M. Radici (Pavia) and R. Jakob

(Wuppertal)

To be updated

Outline:

Status / overview of global analysis of (unpolarized) fragmentation functions (incl. a brief conceptual introduction)

Fragmentation processes [2 examples of hadroproduction at (not-so) high pT]:

1. The double spin asymmetry

2. Rho mass shift in at high pT

π π

p

p

p

p

a

b

c c

a

b

Factorization and universalityFactorization and universality

Global analysisof

Fragmentation Functions

(largely avoiding advertisement plots)

The The Field & FeynmanField & Feynman picture of cascade fragmentation picture of cascade fragmentation

quark/gluon

hadron

Bilocal operatorBilocal operator

P+ = z k+

k+

D(z)

Collinear factorization:

e+e- annihilation (1h inclusive)

FFragmentation (or “ragmentation (or “DDecay”) ecay”) FFunctionsunctions

Scale dependence from renormalization or mass factorization: DGLAP

2 2 Analysis of e Analysis of e++ee--→hX→hX Data Data

Alternative model approaches:

Indumathi et al.

Bourrely & Soffer

Kniehl & Kramer & PötterKretzer

Bourhis & Fontannaz & Guillet & Werlen

How well are Fragmentation Functions determined from How well are Fragmentation Functions determined from ee++ee--??

Sum over all flavours (singlet combination)

u,d,s flavours and gluons

Semi-Inclusive Deep Inelastic Semi-Inclusive Deep Inelastic ScatteringScattering

Flavour Separation

E. Christova, SK, E. Leader

“valence”“favoured”“rank 1”

“sea”“unfavoured”“rank 2”

“strange”“rank 2”

favoured > unfavouredfavoured » unfavoured

Comparison with previous leading particle guess:

As seen in the HERMES pion multiplicities

Leading particle ansatz works well.

Fractional contributions from initial/final state partons

Central Rapidity Forward Rapidity

Dg

Dq

Dg

Dq

initial

final

P? [GeV]

gq

ggqqqg

E [GeV]

qg+gqqq

gg

Hadroproduction: pp→Hadroproduction: pp→ X at 200 GeV X at 200 GeV cmscms

Average Scaling VariablesAverage Scaling Variables

Symmetric / asymmetric kinematics for central / forward rapidityLarge z fragmentation is probed.

CentralRapidity

ForwardRapidity

P? [GeV]

E [GeV]

Factorized NLO pQCD and RHIC pp dataFactorized NLO pQCD and RHIC pp data

PHENIX central rapidity

STAR forward rapidity

Gluon FF and large-z constraints from hadroproduction.Gluon FF and large-z constraints from hadroproduction.

The gluon fragmentation function has been measured.

Hasn’t it?

d σ(3 jet) * fragmentation * tagging-

function

Laenen & Keller

LO NLO

ppTT

softhard

T. Hira

no @ Q

M04

(1/

(1/ ppTT)()(dNdN// dpdpTT))

??? GeV

Onset of pQCD in hadronic collisionsOnset of pQCD in hadronic collisions

The double-spin asymmetry

for .

can be shown to be (basically) positive definite in the few GeV range (at leading power accuracy).

Taking Moments, e.g.turns the non-local (xa ≠ xb) convolution into a local (in N) product

The minimum [by variation δ(Δσ)/δ(Δg)=0] is at

Inverted (from N to x)bounds Δσ from below:

AALLLL is (perturbatively) bounded by:is (perturbatively) bounded by:

Positivity

Underlying parton dynamics

The upper bound holds up to dependence on the scale where positivity is saturated. The lower bound is obtained under low p? approximations. The order of magnitude must be correct in both cases if the dynamics are:

Jäger, SK, Stratmann, Vogelsang (PRL 2004)

Perturbative high pT pions are produced in parton scattering and are decoupled (at leading twist) from the remnant. A statistical ensemble can realize J=1 either through angular momentum of spinless (Goldstone) bosons or through the spin of massive baryons. This must be expected to be disfavoured over J=0, i.e.A nonperturbative asymmetry of O(1%), even smaller than 1%, is enough to produce a characteristic transition from negative to positive asymmetry with increasing pT into the perturbative regime around 1-2(?) GeV.

Frank Bauer @ DIS04

PHENIX hep-ex/0404027

Rho mass shift in pp extends to high pTRho mass shift in pp extends to high pT

STAR data

Does the observation contradict ?

ρ ρ

p

p

p

p

a

b

c c

a

b

Resonant (p-wave) contribution to the 2-pion fragmentation function. (Bachetta & Radici)

P

k

(Sudakov)

q

P

k

Qualitative (dual) features:

• Heavier (light) hadrons come with a harder FF. (The low scale evolution is cut-off.)

• Heavier hadrons are suppressed. (The virtual parton has to survive a multiple of its perturbative lifetime)

• Resonances will be shifted to lower mass.

P

k

Quantitative (order of magnitude) estimate:

And slowly approaching with increasing .

Summary : (with apologies for the omission of heavy quark fragmentation)

Fragmentation functions are determined from, mostly, e+e- annihilation data. Other processes, such as hadro/photo-production have provided tests of consistency and universality. Next steps:

1. Include new data & processes in the fit:

i. Update e+e- fits (large-z data from uds continuum at e.g. BELLE)

ii. Semi-inclusive DIS (flavour)

iii. Hadroproduction (gluons, large-z, RHIC pp norm predictions for AA and spin), enabled by NLO Mellin moment evaluation.

iv. Consistency checks with jet data.

2. Error analysis and coupled analysis with parton densities (à la CTEQ)

Two recent RHIC measurements – resonance production and the double spin asymmetry for pion production - exemplify the rich phenomenology of identified particle production in hard QCD processes:

1. The perturbative spin asymmetry can be bounded to be (basically) positive [>O(-103)] for pT < 4 GeV.

2. Resonance mass shifts of the observed order are to be expected at large pT from parton fragmentation into the resonance decay products.

short term

not-so-short term

***** Leftovers *****

Brain(?)storm:

MotivationStatus of global fits and issues for updateSIDISenergy sum rule ???Gluon Fragmentation / Jet FF measurement and their interpretation / Tagging FunctionsRecombination in twist expansion ???Parton model limit g->0 ???Low pT exponentialALL: positivity in pQCD and cross-over from statistical contributionrho-meson production and shifted rho mass

Collaborations with: E. Christova, E. Leader, W. Vogelsang, H. Yokoya, A. Dumitru, A. Bachetta, M. Radici, …

Energy Conservation:

Not a practical constraint.

kT orderingDGLAPangular ordering

MLLA

?

Some Theory …

Parton Distributions:Local operator product expansion in inclusive DISBilocal operator definition

Fragmentation Functions:No local OPE (no inclusive final state)Bilocal operator definition

Scale dependence enters through renormalization: DGLAP

Just as PDFs, FFs are well defined in terms of

HERMES DIS multiplicities

(unpolarized hydrogen target)

Curves:

LO

NLO

(“NNLO”)

π π

p

p

p

p

a

b

c c

a

b

Factorized cross section ppFactorized cross section pp→→ππ(p(pTT) X) X““Add” polarization (double-spin Add” polarization (double-spin asymmetry)asymmetry)

2→2 channels:

Only (ii) has a negative asymmetry at parton level.

(i) >> (ii) by about a factor 160!

Does this mean that ALL has to be positive?

No: Polarized parton densities may oscillate!

Predictions for ALL are all positive. Is this

accidental or is ALL bounded from

below?

The upper bound on ALL depends on the

scale at which positivity |Δg(x,μ)| ≤ g(x,μ) is saturated.

Identified Particle Production Identified Particle Production in Hard QCD Reactionsin Hard QCD Reactionsas in …as in …Fragmentation FunctionsFragmentation Functions

Stefan KretzerBrookhaven National Laboratory &

RIKEN-BNL

XXXIV International Symposium on Multiparticle DynamicsJuly 26 - August 1, 2004

Sonoma State University, Sonoma County, California, USA

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