Kazutaka Sudoh (KEK) SPIN2007, Vancouver July 30, 2007

1. Determination of Fragmentation Functions and Their Uncertainties Kazutaka Sudoh (KEK) SPIN2007, Vancouver July 30, 2007 In collaboration with M. Hirai (TITech), S. Kumano (KEK), and T.-H. Nagai (The Grad. Univ.) Reference: Phys. Rev. D 75, 094009 (2007) http://research.kek.jp/people/kumanos/ffs.html • Introduction • Global Analysis of Fragmentation Functions • Uncertainty Estimation • Summary

2. h : scaling variable q : CMS energy Coefficient Function calculable in pQCD Fragmentation Function extracted from experiments q Fragmentation Function in • Fragmentation Functions (FFs): • Fragmentation process occurs from quarks, anti-quarks, and gluons, so that Fh is expressed in terms of their contributions: energy fraction of hadron and primary quark

3. : j→i splitting function Favored FFs: Disfavored FFs: • Momentum (Energy) Sum Rule • Energy conservation should be hold for each flavor • DGLAP equation: controls the energy dependence of FFs • Favored and Disfavored Fragmentation Functions from a quark which exists in a naïve quark model from a quark which does not exist in a naïve quark model

4. Purpose for investigating FFs Fragmentation Functions (FFs) are key issue in high energy hadron production processes. • Origin of proton spin • Properties of quark-gluon matter Quark, anti-quark, gluon contribution to proton spin (gluon polarization, flavor separation) Nuclear modification (recombination, energy loss,,,)

5. Relevant Diagrams • LO: • NLO: • Advantage • No ambiguity of PDF error • Sensitive to low z behavior • Disadvantage • Insensitive to gluon FFs • Difficult to separate quark flavor (only sum of charged hadron) • Insensitive to high z behavior No initial gluon function!!

6. Present Status of FFs • There are two widely used FFs by Kretzer and KKP. • AKK is an updated version of KKP. KretzerS. Kretzer PRD62, 054001 (2000) KKPB.A. Kniehl, G. Kramer, B. Potter, NPB582, 514 (2000) AKKS. Albino, B.A. Kniehl, G. Kramer, NPB725, 181 (2005) But, these functions are very different. Large differences indicate that the current FFs have much ambiguities. In this work • Independent global analysis of FFs including new data • Estimate their uncertainties (It’s new!!!) Just after our analysis, a new parametrization including ep and pp reactions is proposedby D. de Florian, R. Sassot, M. Stratmann (PRD75, 114010 (2007), arXiv:0707.1506)

7. Comparison of Setup • Pion FFs from e+e- annihilation

8. Determination of FFs • Determination of fragmentation functions and their uncertainties in LO and NLO • Discuss NLO improvement in comparison with LO • Role of higher order corrections in the determination • Comparison with other parametrizations • SLD 2004 data (accurate) are included. New aspects in our analysis

9. Ansatz (for p±) • Function form (simplest form) • Constraint condition • 2nd moment should be finite and less than 1

10. Experimental Data: • the number of Data： 264 Kinematical coverage

11. c2Analysis • Input parameters and results • Uncertainty estimation: Hessian method • N=14, Dc2=15.94: [K(N.s): c2 distribution] MRST: EPJC23, 73; PLB531, 216 (2002)

12. Comparison with pion Data Our fit is successful to reproduce the pion data. The DELPHI data deviate from our fit at large z. Our NLO fit with uncertainties Rational deference between data and theory (Data-Theory)/Theory

13. Comparison with pion Data (2) • (Data-Theory)/Theory

14. FFs with Uncertainties for pion • Gluon and light quark FFs have large uncertainties. • Uncertainties bands become smaller in NLO compared with LO. (The data are sensitive to NLO effects.) • The NLO improvement is clear especially in gluon and disfavored FFs. • Heavy quark functions are relatively well determined.

15. Determined Functions for Kaon • Gluon and light quark FFs have large uncertainties. • Uncertainties bands become smaller in NLO compared with LO. • Heavy quark functions are relatively well determined. Two favored functions for kaon The situation is similar to the pion funcions

16. FFs with Uncertainties(p/p) • Gluon 2nd moment is fixed. • Gluon uncertainty at the peak position is underestimated. • NLO improvement is not significant for uncertainties. : average of favored and disfavored functions

17. Relevant to RHIC p0 Xsection Comparison with Other Parametrizations for pion HKNS(Hirai, Kumano, Nagai, Sudoh) Kretzer KKP(Kniehl, Kramer, Potter) AKK(Albino, Kniehl, Kramer) DSS(deFlorian, Sassot, Stratmann) • evolved to Q2=2, 10, 100 GeV2 • All functions are different, but consistent within uncertainties bands.

18. ---- DSS Comparison for Kaon and Proton kaon proton

19. Summary • Global analysis of FFs was done for independent parametrization • Determine function forms forp, K, pin LO, NLO analyses • Uncertainties of FFs were estimated • Large uncertainties in gluon and disfavored functions • Heavy quark functions are well determined. • Uncertainties could be reduced by performing NLO analysis • Importance of accurate FFs • The uncertainties at low Q2 are very important for discussing Nucleon’s spin and/or heavy ion physics. (e.g. hadron production at smallpTat RHIC) • Need for accurate “low-energy” data by Belle & BaBar • Program code for calculating our FFs is now available at http://research.kek.jp/people/kumanos/ffs.html