1. Probing hadron structure from hard processes
Hiroyuki Kawamura (KEK)
“New Hadrons” Workshop 2010
Feb. 28 RIKEN

2. Hard Processes
Internal structure of hadrons can be probed by processes with large
momentum transfer (“hard process”).
Hard scale ↔ quarks & gluons inside hadrons.
Inclusive &
Initial
Inclusive &
Final
DIS
e+ e- → h + X h e e+ ↔ ↔ h
PDF e- FF
Exclusive &
Initial /Final
Exclusive &
Decay
Form Factor
B → l nu gamma π π ↔ B ↔
LC wavefunction
LC wavefunction

3. QCD Factorization Inclusive Exclusive
= factorization of hard and soft QCD dynamics
Inclusive
Calculable
DIS =
PDFs
Theoretically easy to handle ,,, (Proof by O.P.E. )
Exclusive
“End-point
singularity”
Information of w.f. is relevant even to show factorization.
More elaborate setup needed.
Brodsky & Lepage etc.

4. Global analysis of FFs Inclusive hadron production Factoriziation Data
In collaboration with Hirai, Kumano, Saito
Inclusive hadron production
Factoriziation
Data
Calculable

5. Fragmentation functionh
≈ probability density of a → h
with mom. fraction z a
DGLAP equation ↔ scale dep.
Time-like splitting function
─ matrix form in flavor singlet channel
HKNS07

6. HKNS07 results Mainly determined by LEP1 data at the Z mass.
Large uncertainties in gluon & sea quark components.
─ Can be reduced by mixing effects through evolution
─ High precision data from BELLE are crucial !!
Data will be released soon.

7. Small-z problems for FF
de Florian, Stratmann, Vogelsang (‘98)
(1) Time-like splitting functions
Moch, Vogt (‘07)
Double log corrections at each order of perturbation.
Negative cross section at some value of z.
log z resummation using angular ordering (color coherence)
are known up to next-to leading order (MLLA).
─ Difficult to unify the evolutions in the small-z and intermediate region.
Albino, Kniehl, Kramer, Ochs (‘04)
Albino, Kniehl, Kramer (’04, ‘06)

8. (2) Hadron mass effect Unity at production threshold
Important for Belle data for K.
“Rescaled” variable :
There are still other power corrections (higher twist corr.).
Naïve solution
Set a lower limit :
Conventionally,
For BELLE data

9. FFs for Partonic structure can be explored by the shape of FFs.
Hirai, Kumano, Oka, Sudoh (‘08)
Partonic structure can be explored by the shape of FFs.
Tetraquark picture
s-sbar picture
soft (disfavored)
hard (favored)
Peak positions & 2nd moments
Large uncertainty  no definite conclusion
Data from Belle or Belle II are desirable.

10. Summery
Internal structure of hadrons can be probed from hard processes.
─ Non-perturbative functions (PDFs, FFs, etc.) defined via
“QCD Factorization” ↔ Internal structure of hadrons
─ Well-defined quantities extracted from experimental data.
We are preparing for a new global fit of fragmemtation functions with
the first Belle data is coming soon.
─ First high precision data at low
→ Impacts on
─ Small-z problems : timelike splitting functions,hadron mass effects,
higher twist effects etc.
Towards exploring internal structure of exotic hadrons
─ Fragmentation functions, exclusive production, decays with HQET

11. Fragmentation function
Operator definition
DGLAP equation
Time-like splitting function