Extraction of the x-Extraction of the x-dependence of the non-dependence of the non-

perturbative QCD b-quark perturbative QCD b-quark fragmentation distribution fragmentation distribution

componentcomponentEli Ben-Haim

LAL Orsay LPNHE Paris

HEP2003 Europhysics Conference, Aachen, July 2003

Sponsored by EEC RTN HPRN-CT-00292-2002

Eli Ben-Haim b-fragmentation HEP2003, Aachen 2

ContentsContents

Measured b-fragmentation function: summary.

Perturbative and non-perturbative QCD components of the fragmentation function.

New Method for extraction of the x-dependence of the non-perturbative QCD component.

Results of the method.

A remark concerning fitting the x-distribution and its moments.

Eli Ben-Haim b-fragmentation HEP2003, Aachen 3

Measured FragmentationMeasured Fragmentation FunctionFunction

Recent results from ALEPH (May 2001), SLD (May 2002), OPAL (October 2002, not published) and DELPHI (not published).Average < 1995 (still used): 0.7020.008Mean values:

ALEPH: 0.716 0.006 0.006DELPHI:0.71530.00070.0051OPAL: 0.71930.00160.0034SLD: 0.709 0.003 0.003

0.002(model)

New average(preliminary)

0.715 0.003

beam

B

EEx

dxdxD

tot

1)(

Definition:

Bin contents are correlated

Systematics dominate

Eli Ben-Haim b-fragmentation HEP2003, Aachen 4

Perturbative and non-perturbative componentsPerturbative and non-perturbative components

perturbative non-perturbativeTypically: gluon radiation.

Calculable.

Different approaches:

Monte Carlo.

Theoretical QCD.

Typically: hadronisation.

Non calculable.

Contains all the contributions that are not included in the perturbative

component.

Usually taken as a model.

Eli Ben-Haim b-fragmentation HEP2003, Aachen 5

=

=

folding

0.001

0.01

0.1

1

0 5 10 15 20 25 30 35 40 45N

perturbative

0.001

0.01

0.1

1

0 5 10 15 20 25 30 35 40 45N

non-perturbativeNO MODEL

1

0

1 )()(~

xDxdxND N

Mellin transform:

X-Distributions

moments

N

pert

measured

pertnon

xND

NDdNi

xD

)(~

)(~

21

)(

.

.

Non-perturbative Part: New Approach.Non-perturbative Part: New Approach.hep-ph/0302157

0

1

2

3

4

5

6

7

8

0 0.2 0.4 0.6 0.8 1x

perturbative(theoretical QCD,Monte Carlo)

-1

0

1

2

3

4

5

6

7

0 0.2 0.4 0.6 0.8 1x

ALEPH

OPAL

0

1

2

3

4

5

6

7

8

0 0.2 0.4 0.6 0.8 1x

non-perturbative(usually models)

Inverse Mellin-transform

Non perturbative function:directly extracted,not model dependent

0.001

0.01

0.1

1

0 5 10 15 20 25 30 35 40 45N

Eli Ben-Haim b-fragmentation HEP2003, Aachen 6

Results of the New Method: GeneralResults of the New Method: General

This Non perturbative functions are expected to be valid in a different environment than e+e-, only in the framework where similar perturbative assumptions are used for its extraction.

Extraction point by point.3 perturbative Approches:

Monte Carlo (JETSET 7.3).NLL QCD, Cacciari and Catani.NLL+DGE QCD, Cacciari and Gardi, hep-ph/0301047.

non-perturbative = F (perturbative)Low x region indicates gluon radiation is well accounted in the perturbative components.Higher order QCD the non perturbative part starts at larger x.The non-perturbative functions for JETSET and NLL QCD have similar shapes, translated.

-1012345678

0 0.2 0.4 0.6 0.8 1 1.2x

JETSET 7.3

NLL QCD

NLL+DGE QCD (preliminary)

DN

P(x)

Extracted non-perturbative distributions

Eli Ben-Haim b-fragmentation HEP2003, Aachen 7

Results of the New Method: JETSETResults of the New Method: JETSET

Model parameters from fit to ALEPH’s data. Comparison: folding the x-distributions, by:

Lund and Bowler models favoured (as observed by OPAL and SLD).

Dpe

rt(x

) DN

P(x)

-1

0

1

2

3

4

5

6

7

8

0 0.2 0.4 0.6 0.8 1x

perturbative distribution (JETSET 7.3before hadronization)

Extracted non-perturbative distribution(ALEPH)

Perturbative & non-perturbative components

-1

0

1

2

3

4

5

6

7

0 0.2 0.4 0.6 0.8 1x

ALEPH

OPAL

Comparison ALEPH-OPALD

NP(

x)

Comparison to hadronisation models:Comparison to hadronisation models:

zdz

zxDzDxD

x

elpertnonpertpredicted )()()(

1mod

..

Extracted non-perturbative distribution:Extracted non-perturbative distribution:

DN

P(x)

-1

0

1

2

3

4

5

6

7

8

0.5 0.6 0.7 0.8 0.9 1x

KartvelishviliPetersonCollins-SpillerLundBowler

extracted non-perturbativedistribution

Eli Ben-Haim b-fragmentation HEP2003, Aachen 8

Results of the New Method: NLL QCDResults of the New Method: NLL QCD

Folding a physical model with the unphysical NLL QCD not meaningful.Our extracted non-pert. QCD distribution solves this problem.

Solves this

Not 0 for x>1 NLL Break down

Cacciari and Catani Nucl.Phys. B617 (2001) 253-290.

Comparison to hadronisation models:Comparison to hadronisation models:Extracted non-perturbativeExtracted non-perturbativedistribution:distribution:

DN

P(x)

-1

0

1

2

3

4

5

6

7

8

9

10

0.6 0.7 0.8 0.9 1 1.1 1.2x

Kartvelishvili

Peterson

LundBowler

extracted non-perturbativedistribution

Dpr

edic

ted(

x)

-1

0

1

2

3

4

5

6

7

8

0 0.2 0.4 0.6 0.8 1 1.2x

perturbative distribution (NLL perturbative QCD)Extracted non-perturbative distribution(ALEPH)greatest parameter dependence (QCD)

greatest scale dependence (QCD)

QCD={0.201, 0.251}

0=F0={mb/2, 2mb}

Dpe

rt(x

) DN

P(x)

new

Don’t fold NLL QCD with models !-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.2 0.4 0.6 0.8 1

x

measured distribution [ALEPH]extracted non-perturbative distributionPetersonBowlerLund

Eli Ben-Haim b-fragmentation HEP2003, Aachen 9

Fitting the x-distribution and momentsFitting the x-distribution and moments

Typically large x large N.

Each moment contains a weighted information from the bins.

The information measured in x-space is distributed among moments (not necessarily low order ones).

Moments are highly correlated.

Example for ALEPH (19 bins):

Fitting x-space data is different from fitting moments.

fraction of contribution

from each bin to the moments

moment order

fra

ctio

n 19

18

17

10

0.5

1

1.5

2

2.5

3

3.5

0 0.2 0.4 0.6 0.8 1x

1918

17

ALEPH

Eli Ben-Haim b-fragmentation HEP2003, Aachen 10

Summary and conclusionsSummary and conclusions

A model independent extraction of the x-dependence of the non-perturbative QCD component of the b-quark fragmentation function has been performed.

It depends closely on the perturbative approach.

The result differs markedly from some hadronisation models.

Perturbative=Monte-Carlo: Lund and Bowler models are favoured.

Perturbative = NLL QCD: The extracted function goes beyond x=1. Don’t use it with hadronisation models!

The non perturbative function is 0 at low x: gluon radiation is well accounted by perturbative QCD.

This Non perturbative function is expected to be valid in a different environment than e+e-, if similar perturbative assumptions are used.