Fragmentation Functions and Polarized Parton Densities

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Fragmentation Functions and Polarized Parton
Densities
Mini-Review

• Stefan Kretzer
• Brookhaven National Laboratory RIKEN-BNL

32nd International Conference on High Energy
PhysicsAugust 16 - 22, 2004Beijing, China
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Subset of functions from a graphical
classification. R. Jakob
S. Moch NNLO
Next 20 min (Some of) The rest of it
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Factorization and universality
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• 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. )

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Here are both aspects
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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

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To begin at the beginning, going back 25 years
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The Field Feynman picture of cascade
fragmentation
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Collins Soper
Bilocal operator
hadron
P z k
D(z)
k
quark/gluon
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Collinear factorization
ee- annihilation (1h inclusive)
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Fragmentation (or Decay) Functions
Scale dependence from renormalization or mass
factorization DGLAP
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?2 Analysis of ee-?hX Data
Alternative model approaches Indumathi et
al. Bourrely Soffer
Kniehl Kramer Pötter
Kretzer
Bourhis Fontannaz Guillet Werlen
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What do we know about Fragmentation Functions
from ee-?
u,d,s flavours and gluons
Sum over all flavours (singlet combination)
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Semi-Inclusive Deep Inelastic Scattering
Flavour Separation
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E. Christova, SK, E. Leader
valencefavouredrank 1
seaunfavouredrank 2
Well described by leading particle ansatz
favoured gt unfavouredfavoured unfavoured
SK
Compare
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From Guzey, Strikman, Vogelsang hep-ph/0407201
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Factorized NLO pQCD and RHIC pp data
STAR forward rapidity
PHENIX central rapidity
Gluon FF and large-z constraints from
hadroproduction.
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The gluon fragmentation function has been
measured. Hasnt it?
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OPAL hep-ex/0404026
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LO DGLAP
LO
NLO
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Transit to longitudinally polarized parton
distributions
Schematic example Semi-inclusive DIS
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Crucial testFactorization!
What Factorization?
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Collinear factorization
LO
leads to the approximate factorization of x and z
dependence in LO
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HERMES DIS pion multiplicities (unpolarized
hydrogen target)

• Curves
• LO
• NLO
• (NNLO)

Stratmann Vogelsang SK
Under investigation by HERMES
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?G is constraint by not much else than
positivity?G(x) lt g(x)
Blümlein Böttcher
?G0.1840.103?G0.1000.075
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Quark Model
QCD
?

• Gluons
• Interaction
• Loops
• Axial anomaly
• Renormalization

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In hadronic collisions (RHIC)
LO
gluons are leaders.
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The double-spin asymmetry
for .
can be shown to be (basically) positive definite
in the few GeV range (at leading twist accuracy).
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ALL? is (perturbatively) bounded by
Jäger, SK, Stratmann, Vogelsang (PRL 2004)

• 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
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PHENIX hep-ex/0404027
Frank Bauer _at_ DIS04
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• Summary (with apologies for your favorite
  omission)
• Fragmentation functions are determined from,
  mostly, ee- annihilation data. Other processes,
  such as hadro/photo-production have provided
  tests of consistency / universality. Post-LEP/SLD
  steps
• Include new data processes in the fit
• Update ee- fits (large-z data from uds
  continuum at e.g. BELLE)
• Semi-inclusive DIS (flavour)
• Hadroproduction (gluons, large-z, RHIC pp norm
  predictions for AA and spin), enabled by NLO
  Mellin moment evaluation.
• Consistency checks with jet data.
• Error analysis and coupled analysis with parton
  densities
• Resummations
• Global analysis of polarized PDFs quantifies
  partonic decomposition of spin, with experimental
  inputs beyond inclusive DIS
• Semi-inclusive DIS asymmetries (sea
  decomposition)
• 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
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Leftovers
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Of particular importance, for physical (axial
anomaly) and historical (spin crisis) reasons,
is ?G
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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

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Every distribution is one component of a
field-theoretic decomposition of nucleon structure
collinear part
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Stratmann Vogelsang SK
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Is ?SIDIS ' q(x)D(z) at not-so-high Q?
And if not then what?

• Higher-twist interactions?
• E.g. Glück Reya 02 suggest spin dependence of
  fragmentation into pions
• Strictly Dq? Dq-?
• Possible effects beyond leading twist

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Comparison with previous leading particle guess
As seen in the HERMES pion multiplicities
Leading particle ansatz works well.
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Global analysisofFragmentation
Functions (largely avoiding advertisement plots)
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Fractional contributions from initial/final state
partons
Hadroproduction pp?? X at 200 GeV cms
Central Rapidity
Forward Rapidity
gq
qggq
qq
initial
gg
qq
gg
qg
Dq
Dq
final
Dg
Dg
E? GeV
P? GeV
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Average Scaling Variables
Central Rapidity

• Symmetric / asymmetric kinematics for central /
  forward rapidity
• Large z fragmentation is probed.

P?? GeV
Forward Rapidity
E? GeV
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Taking Moments, e.g.turns the non-local (xa ?
xb) convolution into a local (in N) product
The minimum by variation d(?s)/d(?g)0 is at
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Inverted (from N to x)bounds ?s from below
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Onset of pQCD in hadronic collisions
soft
T. Hirano _at_ QM04
hard
(1/pT)(dN/dpT)
pT
??? GeV
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Energy Conservation
?
kT orderingDGLAP
angular orderingMLLA
Not a practical constraint.
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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

Just as PDFs, FFs are well defined in terms of
Scale dependence enters through renormalization
DGLAP
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2?2 channels

• Only (ii) has a negative asymmetry at parton
  level.
• (i) gtgt (ii) by about a factor 160!
• Does this mean that ALL? has to be positive?
• No Polarized parton densities may oscillate!

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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 ?g(x,µ) g(x,µ) is
saturated.
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Factorization and Universality
Add polarization
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