Fragmentation Functions Fragmentation Functions and Fragmentation Processes Stefan Kretzer Brookhaven National Laboratory & RIKEN-BNL XXXIV International Symposium on Multiparticle Dynamics July 26 - August 1, 2004 Sonoma State University, Sonoma County, California, USA

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

ππ p p p p a b cc a b Factorization and universality

Global analysis of Fragmentation Functions (largely avoiding advertisement plots)

The Field & Feynman picture of cascade fragmentation

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

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

How well are Fragmentation Functions determined 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” “strange” “rank 2” favoured > unfavoured favoured » 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 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]

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?

d σ(3 jet) * fragmentation * tagging-function Laenen & Keller

LONLO

pTpTpTpT soft hard T. QM04 (1/p T )(dN/dp T ) ??? GeV Onset 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 (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:

A LL  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 DIS04 PHENIX hep-ex/

Rho mass shift in pp extends to high pT STAR data Does the observation contradict ?

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

P k (Sudakov) q

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.

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(-10 3 )] 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: Motivation Status of global fits and issues for update SIDIS energy sum rule ??? Gluon Fragmentation / Jet FF measurement and their interpretation / Tagging Functions Recombination in twist expansion ??? Parton model limit g->0 ??? Low pT exponential ALL: positivity in pQCD and cross-over from statistical contribution rho-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 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

HERMES DIS  multiplicities (unpolarized hydrogen target) Curves: LO NLO (“NNLO”)

ππ p p p p a b cc a b Factorized cross section pp → π(p T ) X “Add” polarization (double-spin asymmetry)

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.

Identified Particle Production in Hard QCD Reactions as in … Fragmentation Functions Stefan Kretzer Brookhaven National Laboratory & RIKEN-BNL XXXIV International Symposium on Multiparticle Dynamics July 26 - August 1, 2004 Sonoma State University, Sonoma County, California, USA *** 2004 *** (selected issues)