1. Y.Kitadono (Hiroshima ),
The analysis of the virtual photonic parton distribution functions at ILC
Y.Kitadono (Hiroshima ),
T.Ueda（Tsukuba),
K.Sasaki(Yokohama),
T.Uematsu(Kyoto)
2010/02/１9　KEK-PH10 （KEK）

2. Talk Plan
1: Introduction (photonic PDFs) 2: Heavy Quark Effect (arXiv: ) 3: Results (Gluon, Top quark PDFs in photon) 4: Summary & Future Works
KEK-PH 10 （KEK）

3. 1. Introduction What is the photon structure ? Analogy
---Nucleon structure---
probe = photon
target = Nucleon
Structure func = F2(x,Q2),FL(x,Q2),…
---Photon Structure---
probe = photon
target = photon
Structure func = F2γ(x,Q2), FLγ(x,Q2),…
‘Resolutions’
‘Target Mass’ 3
Bjorken variable
KEK-PH 10 （KEK）

4. The relation between cross section and Structure functions ,
Two photon cross section
Photon flux in e-
Relation between Structure functions and photonic PDFs,
Structure functions
Photonic PDFs(moment)
Coefficients functions
Photonic PDFs can calculate by pQCD (No free parameter),
Good tools to study QCD at high energies
also for property of parton distributions
Note: Impossible to calculate PDFs in proton by pQCD
KEK-PH 10 （KEK）

5. e- e- q2= - Q2 γ* xk k=(E, k) γ* p2= - P2 e+ e+
Physical interpretation of the Photonic PDFs (= Probability distribution)
in two photon process
( Dominates over
one photon process ) e- e-
q2= - Q2 γ*
qi (jets)
Evolution scale: Q2← P2
PDFs for parton qi xk
Momentum fraction: x
Renormalization scale: P2
qi (jets)
k=(E, k) γ*
p2= - P2 e+ e+
KEK-PH 10 （KEK）

6. - Massless LO, NLO, NNLO QCD correction,
Old results,
- Massless LO, NLO, NNLO QCD correction,
- Target mass effect (TME: kinematical effect)
- Heavy quark effect in structure functions
Next targets ,
- Heavy quark effect in PDFs (HQE: NLO)
- Phenomenological study of the virtual photon structure
functions including quark mass effect
Future (I don’t discuss today),
- SUSY QCD (many massive colored particles) effect/component in virtual photon structure functions
The formalism of QCD(OPE+RGE)
with the heavy quark effect is important
KEK-PH 10 （KEK）

7. LS, LNS = linear combination of massless quark
2. Heavy Quark Effect
We can divide the nf quarks into two groups,
one massive quark
nf quarks (one massive)
(nf -1) massless quarks
Type of PDFs,
LS, LNS = linear combination of massless quark
Focusing on this talk
H = Heavy Quark PDF
G = Gluon PDFs
KEK-PH 10 （KEK）

8. Main formalism is QCD (OPE + RGE),
Operator Product Expansion (OPE) +
Renormalization Group Equation (RGE)
Moment
Analytic Calculations
Relations (Inverse Mellin transformation)
qi(n, Q2, P2)
qi(x, Q2, P2)
Moment (n-space)
Photonic PDFs (x-space)
Numerical Calculations
KEK-PH 10 （KEK）

9. = Related Diagrams (OPE + quark mass) DGLAP eq. (RGE) Divergent term +
Finite term
Finite term contribute the absolute values of PDFs
Operator Matrix Elements
DGLAP eq. (RGE)
Divergent term contribute
the RG flow of PDFs
Heavy Quark PDF
KEK-PH 10 （KEK）

10. 3. Results Gluon PDF Top quark PDF
In this talk, I report the results of
Gluon PDF
Top quark PDF
including the heavy quark mass(now top) effect
at ILC kinematical region.
sILC /10 scale
we study the case
(√sILC= 1 TeV)
KEK-PH 10 （KEK）

11. Gluon PDF in virtual photon
Top Quark Mass Effect
KEK-PH 10 （KEK）

12. Top quark PDF in virtual photon
Large Mass Effect
KEK-PH 10 （KEK）

13. 4. Summary and Future work
Top quark effect for virtual photonic PDFs
at ILC region is investigated up to NLO in pQCD .
Renormalization scheme dependence is also studied.
(very strong !)
Threshold effect should be included in our method (OPE + RGE) for structure functions /PDFs at ILC.
Our formalism (OPE + RGE) enables us to study the heavy quark QCD and massive colored particles in beyond SM (like SUSY QCD).
KEK-PH 10 （KEK）

14. Extra Slide
KEK-PH 10 （KEK）

15. Higher Order Calculations 1/2
QCD β function (3-loop O(αs3))
Larin-Vermaseren(93)
Anomalous dimension for Hadronic Operator (3-loop O(αs3))
Moch-Vermaseren-Vogt(04)
Mixing anomalous dimension (3-loop O(αs2))
Vogt-Moch-Vermaseren(06)
KEK-PH 10 （KEK）

16. Higher Order Calculations 2/2
Hadronic Coefficient function(2-loop O(αs2))
Van-Neerven-Zijlstra(91,92)
Photon Coefficient function(2-loop O(αs2))
Changing color factor in gluonic coeff
Photon matrix element(2-loop O(αs2))
Matiounine-Smith-van-Neerven(98)
Perturbatively calculable when Λ2 << P2
KEK-PH 10 （KEK）

17. Running mass & coupling (NLO)
・ Running coupling
KEK-PH 10 （KEK）

18. - Massless LO, NLO, NNLO QCD correction,
Old results,
- Massless LO, NLO, NNLO QCD correction,
- Target mass effect (TME: kinematical effect)
- Heavy quark effect in structure functions (last workshop)
PLUTO L3
→ consistent with old experimental data
(2 groups; PLUTO, L3)
KEK-PH 10 （KEK）

19. Virtual photon structure functions
Top Quark Threshold
KEK-PH 10 （KEK）