共同研究者:岡真(東工大), 熊野俊三(KEK), 破砕関数によるf0(980)の内部構造の決定 平井正紀 (順天堂大学) 共同研究者:岡真(東工大), 熊野俊三(KEK), 須藤和敬(KEK) To be published in PRD; arXiv:0708.1816 [hep-ph] 2008, Jan. 12 @KEK

Contents Introduction Fragmentation functions (FFs) Global analysis Possible f0(980) configurations Results & discussion Behavior of obtained FFs 2nd moments of FFs Summary

Internal structure of exotic hadrons Introduction Internal structure of exotic hadrons Parton distribution functions (PDFs) Measured by deeply inelastic scattering (DIS ) experiments Need stable targets (ex. proton) Exotic hadron ? Decay Short life time, no stable target Produced hadrons in e+e- scattering p, k, p, a0, f0, et al. Fragmentation functions (FFs) Information about internal structure of produced hadrons

Fragmentation Functions (FFs) Information of hadronization Probability to find the hadron h from a parton i with energy fraction z: Dhi(z) Containing no perturbative object Determined by a global analysis with experimental data in e++e h + X Similar behavior to PDFs Favored FF: like valence quarks A constituent of produced hadrons Peak position at medium and large-z Disfavored FF: like sea quarks Peak position at small-z Favored F Disfavored F Z=Eh/Eq Valence F Sea quark F XBj=pq/ph

Global analysis of FF Cross sections: observable DGLAP equation: Scale Q=mF=mR =s: center of mass energy Coefficient Function - calculable in pQCD Fragmentation Function - extracted from experiments : scaling variable : j →i splitting function

Fragmentation functions of f0(980) Functional form of initial distributions at Q02 Constraint condition 2nd moment should be finite and less than 1

Possible f0(980) configurations

Results c2/d.o.f. = 0.907 Tetra quark configuration SS configuration Total # of data: 23 Tetra quark configuration Becoming favored FFs for light and strange quarks Peak position at large-z (z~0.85) zumax ~ zsmax SS configuration Mu/Ms=0.43  6.73 Mu < Ms Large uncertainty Need precise measurements to reduce their huge uncertainties 2nd moments Mu=0.0012  0.0107 Ms=0.0027  0.0183 Mg=0.0090  0.0046

Comparison of data with theory Large uncertainty Consistent with deviation of experimental data with their errors Rather large uncertainty at low Q2 Reducing the uncertainties Needs of precise data Wide rage of Q2 Determination of gluon FF Belle experiment s~11 GeV Heavy quark tagged data Charm, bottom quarks → f0(980)

Summary Internal structure of the exotic hadron from fragmentation functions Global analysis of FFs with the data of the f0(980) production Favored & disfavored FFs: valence & sea quarks Peak position: zmax Relation between the 2nd moments of FFs Intuitive estimation of order counting for the f0(980) production in various quark configurations Indicating tetra-quark and ss configuration Tetra-quark: peak position of the up(down) & strange quark FFs SS : relation of the 2nd moments Huge uncertainties of the FFs and 2nd moments Low accuracy of experimental data Need precise measurements from the Belle experiments Model predictions for the FFs Relation between behavior of FFs and quark configurations

Global analysis of FFs Q2 dependence of FFs Data sets of f0(980) HRS:4 (s=29.0 GeV, 0.165< z <0.554) PRL 57, 1990 (1986) OPAL:8 (s=91.2 GeV, 0.092< z < 0.75) EPJ C4,19 (1998) DELPHI:10 (s=91.2 GeV, 0.078< z < 0.7) PL 449B,364 (1999),ZP C65, 587 (1995) Total # of data: 23 NLO analysis MS scheme Reduction of theoretical uncertainty c2/d.o.f. = 0.907 Q2 dependence of FFs DGLAP eq Q02 = 1GeV2 Minimizing c2 Error estimation Hessian method D2 (N=12) = 13.74