1. Kazutaka Sudoh (KEK) SPIN2007, Vancouver July 30, 2007 Determination of Fragmentation Functions and Their Uncertainties Determination of Fragmentation Functions and Their Uncertainties I.Introduction II.Global Analysis of Fragmentation Functions III.Uncertainty Estimation IV.Summary 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

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

3. K. Sudoh (KEK) Page-3 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 : j i splitting function Disfavored FFs: Favored FFs: from a quark which exists in a naïve quark model from a quark which does not exist in a naïve quark model

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

5. K. Sudoh (KEK) Page-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. K. Sudoh (KEK) Page-6 Present Status of FFs There are two widely used FFs by Kretzer and KKP. –AKK is an updated version of KKP. Independent global analysis of FFs including new data Estimate their uncertainties (Its new!!!) Kretzer S. Kretzer PRD62, 054001 (2000) KKP B.A. Kniehl, G. Kramer, B. Potter, NPB582, 514 (2000) AKK S. 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. Just after our analysis, a new parametrization including ep and pp reactions is proposed by D. de Florian, R. Sassot, M. Stratmann (PRD75, 114010 (2007), arXiv:0707.1506) In this work

7. K. Sudoh (KEK) Page-7 Comparison of Setup Pion FFs from e + e - annihilation HKNS (Ours)KretzerKKP(AKK) Function form # of parameters 141115(18) Mass threshold m Q 2 (m c,b =1.43, 4.3 GeV) m Q 2 (m c,b =1.4, 4.5 GeV) 4m Q 2 (2m c,b =2.98, 9.46 GeV) Initial scale 0 2 [NLO] 1.0 GeV 2 0.4 GeV 2 2.0 GeV 2 Ansatz One parameter is fixed.

8. K. Sudoh (KEK) Page-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. K. Sudoh (KEK) Page-9 Ansatz (for ± ) Function form (simplest form) Constraint condition –2 nd moment should be finite and less than 1

10. K. Sudoh (KEK) Page-10 Experimental Data: # of data TASSO TCP HRS TOPAZ SLD SLD [light quark] SLD [ c quark] SLD [ b quark] ALEPH OPAL DELPHI DELPHI [light quark] DELPHI [ b quark] 12,14,22,30,34,44 29 58 91.2 29 18 2 4 29 22 17 the number of Data 264 Kinematical coverage

11. K. Sudoh (KEK) Page-11 2 Analysis Input parameters and results – Uncertainty estimation: Hessian method –N=14, 2 =15.94: [ K(N.s) : 2 distribution] MRST: EPJC23, 73; PLB531, 216 (2002)

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

13. K. Sudoh (KEK) Page-13 Comparison with pion Data (2) (Data-Theory)/Theory

14. K. Sudoh (KEK) Page-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. K. Sudoh (KEK) Page-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. The situation is similar to the pion funcions Two favored functions for kaon

16. K. Sudoh (KEK) Page-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. K. Sudoh (KEK) Page-17 Comparison with Other Parametrizations for pion evolved to Q 2 =2, 10, 100 GeV 2 All functions are different, but consistent within uncertainties bands. HKNS (Hirai, Kumano, Nagai, Sudoh) Kretzer KKP (Kniehl, Kramer, Potter) AKK (Albino, Kniehl, Kramer) DSS (deFlorian, Sassot, Stratmann) Relevant to RHIC 0 Xsection

18. K. Sudoh (KEK) Page-18 Comparison for Kaon and Proton kaonproton ---- DSS

19. K. Sudoh (KEK) Page-19 Summary Global analysis of FFs was done for independent parametrization –Determine function forms for, K, p in 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 Q 2 are very important for discussing Nucleons spin and/or heavy ion physics. (e.g. hadron production at small p T at 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