Fragmentation Functions and Polarized Parton Densities Stefan Kretzer Brookhaven National Laboratory & RIKEN-BNL 32nd International Conference on High Energy Physics August , 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

ππ p p p p a b cc a b Factorization 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 DIS 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 Field & Feynman picture of cascade fragmentation

quark/gluon hadron Bilocal operator P + = z k + k+k+ D(z) Collins & Soper

Collinear factorization: e + e - annihilation (1h inclusive)

Fragmentation (or “Decay”) Functions Scale dependence from renormalization or mass factorization: DGLAP

 2 Analysis of e + e - → hX Data Alternative model approaches: Indumathi et al. Bourrely & Soffer Kniehl & Kramer & Pötter Kretzer Bourhis & Fontannaz & Guillet & Werlen

What do we know about Fragmentation Functions from e + e - ? Sum over all flavours (singlet combination) u,d,s flavours and gluons

Semi-Inclusive Deep Inelastic Scattering Flavour Separation

E. Christova, SK, E. Leader “valence” “favoured” “rank 1” “sea” “unfavoured” “rank 2” favoured > unfavoured favoured » unfavoured Well described by leading particle ansatz SK Compare:

From Guzey, Strikman, Vogelsang hep-ph/

Factorized NLO pQCD and RHIC pp data PHENIX central rapidity STAR forward rapidity Gluon FF and large-z constraints from hadroproduction.

The gluon fragmentation function has been measured. Hasn’t it?

OPAL hep-ex/

LONLO 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±0.103  G=0.100±0.075

Quark Model QCDQCDQCDQCD ? 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).

A LL  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 DIS04 PHENIX hep-ex/

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 pions Strictly D q+  ´ D q-  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 analysis of Fragmentation Functions (largely avoiding advertisement plots)

Fractional contributions from initial/final state partons Central RapidityForward Rapidity DgDg DqDq DgDg DqDq initial final P ? [GeV] gq gg qq qg E  [GeV] qg+gq qq gg Hadroproduction: pp →  X at 200 GeV cms

Average Scaling Variables Symmetric / asymmetric kinematics for central / forward rapidity Large z fragmentation is probed. Central Rapidity Forward Rapidity P ?  [GeV] E  [GeV]

Taking Moments, e.g. turns the non-local (x a ≠ x b ) 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:

pTpTpTpT soft hard T. QM04 (1/p T )(dN/dp T ) ??? GeV Onset of pQCD in hadronic collisions

Energy Conservation: Not a practical constraint. kT ordering DGLAP angular ordering MLLA ?

Some Theory … Parton Distributions: Local operator product expansion in inclusive DIS Bilocal 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 A LL  has to be positive? No: Polarized parton densities may oscillate!

Predictions for A LL  are all positive. Is this accidental or is A LL  bounded from below? The upper bound on A LL  depends on the scale at which positivity |Δg(x,μ)| ≤ g(x,μ) is saturated.

Factorization and Universality “Add” polarization ππ p p p p a b cc a b