internal structure of f 0 (980) meson by fragmentation functions
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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
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