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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