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Internal structure of f0(980) mesonby fragmentation functions

M. Hirai (TUS)

Collaborators: S. Kumano(KEK), M. Oka(TiTech), K. Sudoh (Nishogakusha)

Phys. Rev D77 017504 (2008);arXiv:0708.1816  [hep-ph]

2008, Dec. 7 @Nagoya

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Contents

• Introduction

• Fragmentation functions– Global analysis

– Possible f0(980) configurations

• Results & discussion– Behavior of obtained FFs– 2nd moments of FFs

• Summary

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Introduction

• Internal structure of exotic hadrons– Parton distribution functions (PDFs)

• Measured by deeply inelastic scattering (DIS ) experiments

• Need stable targets (ex. proton)

– Exotic hadron ?• Decay

– Short life time, no stable target

• Produced hadrons in e+e- scattering– , k, p, a0, f0, et al.

• Fragmentation functions (FFs)– Information about internal structure of produced hadrons

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Favored FDisfavored F

Z=Eh/Eq

Fragmentation Functions (FFs)

• Information of hadronization – Containing no perturbative object– Determined by a global analysis

with experimental data in e++e h+X

• Similar behavior to PDFs– Favored FF: like valence quarks

• A constituent of produced hadrons• Peak position at medium and large-z

– Disfavored FF: like sea quarks• Peak position at small-z

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Global analysis of FF

• Cross sections: observable

• DGLAP equation:–

– Scale Q=F=R =s: center of mass energy

2

0

( )1, 1

hq s

totqtot

Qd

dz

e e h X

1 22 2,( , , ) ,

h

i F Ri

Fz

hi

zD

d dC Q

dz

ijP : j →i splitting function 2

12 22

( )( , ) , , ,

ln 2hs

j ij s izi

Q d zD z Q P D Q

Q

Coefficient Function calculable in pQCD

Fragmentation Function extracted from experiments

2

2 2h hP q Ez

Q Q

: scaling variable

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Fragmentation functions of f0(980)

• Functional form of initial distributions at Q02

• Constraint condition– 2nd moment should be finite and less than 1

•

•

1

02, 0, 0 ( ) 12nd h

i i i iM zD z dz

1

0

( 3), ( ) ,

( 2) ( 1)

1 (hh

N M M zD z dz

M

2nd 2nd

2nd momentum conservation)

2 2 20 2

02 2 2

2 2 2

2 2

3,1

4,, 1.43

5,4.3

6,

c

c bf c

b tb

t

Q Q mQ

m Q mn m

m Q mm

m Q

2 GeV

GeV

GeV

g ,* Fixed parameters: ,

lack of accuracy of experimental data

u s

1 100 00 0

0 0

0 0

0 0

0 0

20

2 20 0

2 20 0

2 20 0

2 2

( , ) (1 ) ,

( , ) ( , ),

( , ) ( , ),

( , ), ( , ),

( , ), ( , ).

f fq qf f

q q

f fq q

f fu d

f fs g

f fc c b b

D z Q N z z

D z Q D z Q

D z Q D z Q

D z Q D z Q

D z m D z m

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Possible f0(980) configurations

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Results

• 2/d.o.f. = 0.907– Total # of data: 23

• Tetra quark configuration– Becoming favored FFs for light

and strange quarks– Peak position at large-z

(z~0.85)– zu

max ~ zsmax

• SS configuration – Mu/Ms=0.43 6.73– Mu < Ms

• Large uncertainty– Need precise measurements to

reduce their huge uncertainties

• 2nd moments– Mu=0.0012 0.0107

– Ms=0.0027 0.0183

– Mg=0.0090 0.0046

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Summary

• Internal structure of the exotic hadron from fragmentation functions

– Global analysis of FFs with the data of the f0(980) production – Favored & disfavored FFs: valence & sea quarks

• Peak position: zmax

– Relation between the 2nd moments of FFs• Intuitive estimation of order counting for the f0(980) production in

various quark configurations

• Indicating tetra-quark and ss configuration– Tetra-quark: peak position of the up(down) & strange quark FFs – SS : relation of the 2nd moments– Huge uncertainties of the FFs and 2nd moments

• Low accuracy of experimental data• Need precise measurements from Bell, BaBar experiments

– Model predictions for the FFs• Relation between behavior of FFs and quark configurations

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Global analysis of FFs

• Data sets of f0(980) – HRS:4 (s=29.0 GeV, 0.165< z <0.554)

• PRL 57, 1990 (1986)

– OPAL:8 (s=91.2 GeV, 0.092< z < 0.75)• EPJ C4,19 (1998)

– DELPHI:10 (s=91.2 GeV, 0.078< z < 0.7)

• PL 449B,364 (1999),ZP C65, 587 (1995)

• Total # of data: 23

• NLO analysis– MS scheme– Reduction of theoretical uncertainty– 2/d.o.f. = 0.907

• Q2 dependence of pol-PDFs– DGLAP eq– Q0

2 = 1GeV2

• Minimizing 2

• Error estimation– Hessian method

– 2 (N=12) = 13.74

exp theo

exp

exp

( , ) ( , ),

i i

i

sys sta

A z Q A z Q

22 2

22

2 2 2

2 2 1

,

( ) ( ) ( ) ij

i j i j

D x D xD x H

a a

2

0( , ) 0.683K N s ds

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Comparison of data with theory

• Large uncertainty• Consistent with deviation of

experimental data with their errors

• Rather large uncertainty at low Q2

• Reducing the uncertainties– Needs of precise data– Wide rage of Q2

• Determination of gluon FF• Belle experiment s~10 GeV

– Heavy quark tagged data• Charm, bottom quark → f0(980)

-2

-1

0

1

2HRS

-2

-1

0

1

2OPAL

-2

-1

0

1

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

z

DELPHI