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Determination of fragmentation functions and their uncertainties

M. Hirai, S. Kumano, T. -H. Nagai, K. Sudoh

TL;DR

This work determines unpolarized fragmentation functions for π, K, and p from a global analysis of $e^+e^-$ data, explicitly estimating uncertainties with the Hessian method. The FFs are evolved using timelike DGLAP and fitted at an initial scale with a flexible parametrization that respects energy conservation via second moments. Key findings include large uncertainties at low $Q^2$, especially for gluons and light quarks, though NLO analyses substantially improve the pion and kaon determinations. The results emphasize the necessity of incorporating uncertainty estimates in hadron-production analyses at small $Q^2$ and provide a public code for computing the FFs.

Abstract

Fragmentation functions and their uncertainties are determined for pion, kaon, and proton by a global $χ^2$ analysis of charged-hadron production data in electron-positron annihilation and by the Hessian method for error estimation. It is especially important that the uncertainties of the fragmentation functions are estimated in this analysis. The results indicate that the fragmentation functions, especially gluon and light-quark fragmentation functions, have large uncertainties at small $Q^2$. There are large differences between widely-used functions by KKP (Kniehl, Kramer, and Pötter) and Kretzer; however, they are compatible with each other and also with our functions if the uncertainties are taken into account. We find that determination of the fragmentation functions is improved in next-to-leading-order (NLO) analyses for the pion and kaon in comparison with leading-order ones. Such a NLO improvement is not obvious in the proton. Since the uncertainties are large at small $Q^2$, the uncertainty estimation is very important for analyzing hadron-production data at small $Q^2$ or $p_T$ ($Q^2, p_T^2 << M_Z^2$) in lepton scattering and hadron-hadron collisions. A code is available for general users for calculating obtained fragmentation functions.

Determination of fragmentation functions and their uncertainties

TL;DR

This work determines unpolarized fragmentation functions for π, K, and p from a global analysis of data, explicitly estimating uncertainties with the Hessian method. The FFs are evolved using timelike DGLAP and fitted at an initial scale with a flexible parametrization that respects energy conservation via second moments. Key findings include large uncertainties at low , especially for gluons and light quarks, though NLO analyses substantially improve the pion and kaon determinations. The results emphasize the necessity of incorporating uncertainty estimates in hadron-production analyses at small and provide a public code for computing the FFs.

Abstract

Fragmentation functions and their uncertainties are determined for pion, kaon, and proton by a global analysis of charged-hadron production data in electron-positron annihilation and by the Hessian method for error estimation. It is especially important that the uncertainties of the fragmentation functions are estimated in this analysis. The results indicate that the fragmentation functions, especially gluon and light-quark fragmentation functions, have large uncertainties at small . There are large differences between widely-used functions by KKP (Kniehl, Kramer, and Pötter) and Kretzer; however, they are compatible with each other and also with our functions if the uncertainties are taken into account. We find that determination of the fragmentation functions is improved in next-to-leading-order (NLO) analyses for the pion and kaon in comparison with leading-order ones. Such a NLO improvement is not obvious in the proton. Since the uncertainties are large at small , the uncertainty estimation is very important for analyzing hadron-production data at small or () in lepton scattering and hadron-hadron collisions. A code is available for general users for calculating obtained fragmentation functions.

Paper Structure

This paper contains 15 sections, 35 equations, 15 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Formalism
  3. Hadron production in $e^+ e^-$ annihilations
  4. Fragmentation functions and their $Q^2$ evolution
  5. Analysis method
  6. Parametrization
  7. Experimental data
  8. $\chi^2$ analysis
  9. Uncertainties of fragmentation functions
  10. Results
  11. Comparison with experimental data
  12. Optimum fragmentation functions and their uncertainties
  13. Comparison with other parametrizations
  14. Summary
  15. Fragmentation functions for $\pi^0$, $K^0$, $\bar{K}^0$, $n$, and $\bar{n}$

Figures (15)

  • Figure 1: (Color online) Kinematical range is shown by $z$ and $Q (=\sqrt{s})$ values for the pion data.
  • Figure 2: (Color online) Comparison of our NLO results with pion-production data at $Q=M_Z$ without separation on initial partons by the ALEPH, DELPHI, OPAL, and SLD collaborations.
  • Figure 3: (Color online) Comparison with charged-pion production data by the TASSO, TPC, HRS, ALEPH, DELPHI, OPAL, and SLD collaborations. The rational differences between the data and theoretical calculations are shown as a function of $z$. The average scale $Q$=16 GeV is taken for theoretical calculations in the top figure with the TASSO data.
  • Figure 4: (Color online) Comparison with charged-pion production data by the TASSO, TOPAZ, DELPHI, and SLD collaborations. The scale is $Q$=39 GeV for theoretical calculations in the top figure.
  • Figure 5: (Color online) Comparison with charged-kaon production data by the TASSO, TPC, HRS, ALEPH, DELPHI, OPAL, and SLD collaborations. The rational differences between the data and theoretical calculations are shown. The scale is $Q$=16 GeV for theoretical calculations in the top figure.
  • ...and 10 more figures