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）

Talk Plan 1: Introduction (photonic PDFs) 2: Heavy Quark Effect (arXiv:0912.4829) 3: Results (Gluon, Top quark PDFs in photon) 4: Summary & Future Works KEK-PH 10 （KEK）

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）

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）

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）

- 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）

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）

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）

= 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）

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）

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

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

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）

Extra Slide KEK-PH 10 （KEK）

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）

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）

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

- 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）

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