Fragmentation Functions Fragmentation Functions and Polarized Parton Densitiesand Polarized Parton Densities

Stefan KretzerBrookhaven National

Laboratory & RIKEN-BNL

32nd International Conference on High Energy PhysicsAugust 16 - 22, 2004

Beijing, China

*** Mini-Review ***

Subset of functions from a graphical classification. R. Jakob

S. Moch NNLO

Next 20 min: (Some of) The rest of it

π π

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Factorization and universalityFactorization and universality

• Applications (“Partons in Operation”): Hard reactions involving hadrons / nuclei are ubiquitous. pQCD provides a predictive and quantitative (“Next-to-next-to-leading-order”: NNLO) field theoretic framework in terms of the quark and gluon degrees of freedom. It also “measures” the parton luminosities for hadron colliding machines.

• Investigations (“Partons under the Microscope”): pQCD is rich in structure in itself. (Some of it - which I will not minireview - is yet being investigated experimentally at the discovery level. )

Here are both aspects …

Forward high pT particle production in Forward high pT particle production in DISDIS

Daleo & Sassot:

• Inhomogeneous Evolution

• Mixing with Fracture Functions

• (Similar to n-hadron FFs: de Florian, …)

Aurenche & Basu & Fontannaz & Godbole:

• Signal for BFKL

To begin at the beginning, going back 25 years …

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)

Collins & Soper

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

What do we know about Fragmentation Functions from What do we know about Fragmentation Functions 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”

favoured > unfavouredfavoured » unfavoured

Well described by leading particle ansatz

SK

Compare:

From Guzey, Strikman, Vogelsang hep-ph/0407201

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?

OPAL hep-ex/0404026

LO NLO

LO — DGLAP

Transit to longitudinally polarized parton distributions …

Schematic example: Semi-inclusive DIS

Crucial test:Factorization!

What Factorization?

Collinear factorization:

LO

leads to the approximate factorization of x and z dependence in LO:

HERMES DIS pion multiplicities

(unpolarized hydrogen target)

Curves:

LO

NLO

(“NNLO”)

Stratmann & Vogelsang &

SK **** Under investigation by HERMES

***

Blümlein & Böttcher

ΔG is constraint by not much else than positivity:

|ΔG(x)| < g(x)

G=0.184±0.103G=0.100±0.075

Quark ModelQuark Model QQCCDD

?•Gluons

•Interaction

•Loops:

•Axial anomaly

•Renormalization

In hadronic collisions (RHIC) …

… gluons are “leaders”.

LO

The double-spin asymmetry

for .

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

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

Positivity

Underlying parton (gluon) 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)

Frank Bauer @ DIS04

PHENIX hep-ex/0404027

Summary : (with apologies for your favorite omission)

Fragmentation functions are determined from, mostly, e+e- annihilation data. Other processes, such as hadro/photo-production have provided tests of consistency / universality. Post-LEP/SLD 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

3. Resummations

Global analysis of polarized PDFs quantifies partonic decomposition of spin, with experimental inputs beyond inclusive DIS:

1. Semi-inclusive DIS asymmetries (sea decomposition)

2. High pT RHIC-spin processes (longitudinal gluon polarization)

And again, this mini-review left out many a maxi-topic.

short term

not-so-short term

***** Leftovers *****

Of particular importance, for physical (“axial anomaly”) and historical (“spin crisis”) reasons, is G :

Factorization Factorization

The Factorization is a statement in pQCD about the seperation of scales in

The LO DIS process is so simple, indeed is just a vertex / (1-x) (1-z) so that (x,z) / F(x)D(z) : The approximate (LO) factorization of x and z dependence (following from the one-particle “phase space” of LO DIS)Factorization ' Factorization for SIDIS

Every distribution is one component of a field-theoretic decomposition of nucleon structure

collinear part:

Stratmann & Vogelsang & SK

Is SIDIS ' q(x)D(z) at not-so-high Q?

Higher-twist interactions?E.g. Glück & Reya 02 suggest spin dependence of fragmentation into pionsStrictly Dq+

´ Dq-

Possible effects beyond leading twist

And if not … then what?

Comparison with previous leading particle guess:

As seen in the HERMES pion multiplicities

Leading particle ansatz works well.

Global analysisof

Fragmentation Functions

(largely avoiding advertisement plots)

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]

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:

ppTT

softhard

T. Hira

no @ Q

M04

(1/

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

??? GeV

Onset of pQCD in hadronic collisionsOnset of pQCD in hadronic collisions

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

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.

Factorization and UniversalityFactorization and Universality““Add” polarizationAdd” polarization

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