Extraction of the Non-Perturbative b-Quark Fragmentation Function From Delphi to CDF?

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Extraction of the Non-Perturbative b-Quark
Fragmentation FunctionFrom Delphi to CDF?

• Eli Ben-Haïm
• Patrick Roudeau
• Laboratoire de lAccélérateur Linéaire
• Orsay
• Aurore Savoy-Navarro
• LPNHE
• Paris

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Introduction

• b-quark fragmentation in ee- annihilation
• Definitions
• the variable x
• Measured fragmentation function

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• D can be split into a perturbative and a
  non-perturbative components
• Perturbative- Calculable within perturbation
  theory bb pair production, gluon radiation.
• Monte Carlo simulation.
• Theoretical calculation.
• Non-perturbative- hadronisation, all the
  contributions that are not included in the
  perturbative part.
• Main perspective using the non-perturbative
  component in CDF.

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• Recent results from Aleph (May 2001), SLD (May
  2002), Delphi (not published) and OPAL (October
  2002).
• Average lt 1995 0.702 ?0.008
• Mean values
• Aleph 0.716?0.006?0.006
• SLD 0.709 ?0.003 ?0.003 ?0.002(model)
• Delphi 0.7153?0.0007?0.0051
• OPAL 0.7193 ?0.0016 ?0.0034

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Subjects to be presented

• New method direct extraction of the
  non-perturbative QCD component of the b-quark
  fragmentation function.
• Perspectives.

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Direct Extraction of the Non-Perturbative QCD
Component

• The usual approach taking a functional form for
  the non-perturbative component (Peterson, Lund).
• We propose an alternative approach
• The perturbative and non-perturbative
  distributions of the variable x are folded by
• If applying to both perturbative and
  non-perturbative distributions the Mellin
  transformation
• (simple moments for integer N).
• Folding the distributions becomes simply

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• The last expression allows us to extract the
  non-perturbative component in the N-space (Mellin
  conjugate space).
• It is a simple ratio.
• and then, we bring it back to the x-space by
  taking the inverse Mellin transformation

Direct, model independent method to obtain the
non- perturbative component.

• This can be done for perturbative component
• From Monte Carlo.
• From Theoretical QCD calculation.

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The N-space distributions
Nth moment value (D(N), Dpert(N)
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About the Theoretical QCD Calculation
Cacciari and Catani Nucl.Phys. B617 (2001)
253-290.

• Resummation of Large logarithmic terms (up to
  NLL) to all perturbative orders in ?s.
• Generalization of previous calculations adding
  soft gluons contributions at large x.
• Calculations are done in Mellin conjugate space.
• Factorization of process dependent and process
  independent functions in the N distribution

?0 leading order cross section.
C process dependent coefficient function.
DNgluons perturbative gluon radiation. E
AP evolution operator.
DNgluons,ini initial condition.

• Calculation is reliable for N not too large.

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Further comments for the theoretical calculation
x-distribution

• Contains non-physical parts. Not under control in
  the large x region due to discrepancies at large
  N.
• Slightly distorted (even when physical) by the
  large x (large N) behavior.

Scale and parameter dependence
Nth moment value Dpert(N)
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Results
Corresponding to perturbative components from
Monte Carlo
Theoretical QCD
Dpert(x), DNP(x)

• Non-physical region at low x due to
• Unphysical regions in perturbative distribution
  from theory.
• A problem in the perturbative component from the
  Monte Carlo or in the measured distribution.

The Non perturbative function may be used in a
different environment than ee- collisions, only
in the framework of a similar perturbative
assumptions as were used for its extraction.
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If we cancel the non-physical part in the MC
non-perturbative component
The measured function becomes
The perturbative component becomes
Dpert(x)
Dmeasured(x)
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Comparing with models

• Model parameters are those who best fit Delphis
  data.

Perturbative component from Monte Carlo
Perturbative component from Theory
Dpert(x), DNP(x)

• The extracted non-perturbative component does not
  look like the common models, especially for the
  theory.

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perspectives

• Completing our measurement.
• How to implement in CDF?
• Model independent method check on other domains
  of energy for ee-.
• Check for charm fragmentation.
• Study the properties of the leading track
  accompanying the B.