Fragmentation function is defined by

e+

e–

γ, Z

q

q

h

Fragmentation: hadron production from a quark, antiquark, or gluon

Fh (z,Q2 ) = 1σ tot

dσ (e+e− → hX)dz

σ tot = total hadronic cross section

z ≡Ehs / 2

=2EhQ

=EhEq

, s = Q2

Fh (z,Q2 ) = dyyz

1∫

i∑ Ci

zy,Q2⎛

⎝⎜⎞⎠⎟Dih (y,Q2 )

Ci (z,Q2 ) = coefficient function

Dih (z,Q2 ) = fragmentation function of hadron h from a parton i

Dih z,Q2( ) = probability to find the hadron h from a parton i

with the energy fraction z

Energy conservation: dz z0

1

∫

h∑ Di

h z,Q2( ) = 1

h = π + , π 0 , π − , K + , K 0 , K 0 , K − , p, p, n, n, ⋅ ⋅ ⋅

Simple quark model: π + (ud ), K+ (us ), p(uud), ⋅ ⋅ ⋅

Favored fragmentation: Duπ +

, Ddπ +

, ... (from a quark which exists in a naive quark model)

Disfavored fragmentation: Ddπ +

, Duπ +

, Dsπ +

, ... (from a quark which does not exist in a naive quark model)

Semi-inclusive reactions have been used for investigating ・ origin of proton spin

e + p→ ′e + h + X, p + p→ h + X (RHIC-Spin)

A+ ′A → h + X (RHIC, LHC)・ properties of quark-hadron matters

Quark, antiquark, and gluon contributions to proton spin (flavor separation, gluon polarization)

Nuclear medium effects (energy loss, …)

σ = fa (xa ,Q2 )⊗ fb (xb ,Q

2 )a,b,c∑

⊗ σ̂ (ab→ cX)⊗ Dcπ (z,Q2 )

• exotic-hadron search

(KKP) Kniehl, Kramer, Pötter (AKK) Albino, Kniehl, Kramer (HKNS) Hirai, Kumano, Nagai, Sudoh (DSS) De Florian, Sassot, Stratmann

Disfavored-quark and gluon fragmentation functions have large uncertainties.

Possibilities for f0 (980): 12

(uu + dd ), ss , 12

(uuss + ddss ), KK, or gg

e.g. if f0 (980) = ss : favored s, s → f0; disfavored u, d , u, d → f0 ,⋅ ⋅ ⋅

Hirai, SK, Oka, Sudoh, PRD 77 (2008) 017504.

Code for calculating the fragmentation functions is available at http://research.kek.jp/people/kumanos/ffs.html .

Fh (z,MZ2 ) ≈ (cV

u 2 + cAu 2 ) Du+

h (z,MZ2 ) + Dc+

h (z,MZ2 )⎡⎣ ⎤⎦

+ (cVd 2 + cA

d 2 ) Dd +h (z,MZ

2 ) + Ds+h (z,MZ

2 ) + Db+h (z,MZ

2 )⎡⎣ ⎤⎦ ≈ 0.33 Dq+

h (z,MZ2 )

q∑

•  At the Z-pole (LEP/SLD)

•  Far from the Z-pole (Belle, TASSO/TPC/HRC/TOPAZ )

Dq+h ≡ Dq

h + Dqh

Fh (z,Q2 ) ≈ 49Du+h (z,Q2 ) + Dc+

h (z,MZ2 )⎡⎣ ⎤⎦ +

19Dd +h (z,Q2 ) + Ds+

h (z,Q2 ) + Db+h (z,Q2 )⎡⎣ ⎤⎦

flavor singlet combination

(1) c-quark, b-quark FFs are determined from the flavor tagged data. (2) If we have very precise data at and far from the Z-pole, we can determine 2 independent components of the quark FFs. (3) Remaining flavor decomposition & determination of the gluon FF come from the mixing through scale evolution

e+

e–

γ, Z

q

q

h

Dih (z,Q2 )

New Belle and BarBar data in 2013

Scale evolution of Dih

→ gluon fragmentation function

Scale evolution of F2 → gluon distribution Belle: M. Leitgab et al., PRL 111 (2013) 062002.

BaBar: J. P. Lees et al., PR D 88 (2013) 032011.

Duπ +(z,Q0

2 ) = Nuπ +zαu

π +(1 − z)βu

π += Dd

π +(z,Q0

2 )

Duπ +(z,Q0

2 ) = Nuπ +zαu

π +(1 − z)βu

π += Dd

π +(z,Q0

2 ) = Dsπ +(z,Q0

2 ) = Dsπ +(z,Q0

2 )

Dcπ +(z,mc

2 ) = Ncπ +zα c

π +(1 − z)βc

π += Dc

π +(z,mc

2 )

Dbπ +(z,mb

2 ) = Nbπ +zαb

π +

(1 − z)βbπ +

= Dbπ +(z,mb

2 )

Dgπ +(z,Q0

2 )= Ngπ +zα g

π +

(1 − z)βgπ +

Dqπ −

= Dqπ +

N = MB(α + 2,β +1)

, M ≡ zD(z)dz (2nd moment)0

1

∫ , B(α + 2,β +1) = beta function

Constraint: 2nd moment should be finite and less than 1

0 < Mih < 1 because of the sum rule

h∑ Mi

h = 1

Note: constituent-quark composition π + = ud , π − = ud

At the stage of 2012 (with only preliminary Belle data)

Note: The preliminary Belle data were obtained by personal communications, so that they should not be used after the Belle final data are published in PRL.

Fh (z,Q2 ) = 1σ tot

dσ (e+e− → hX)dz

Orninate: Fπ (data) − Fπ (theory)

Fπ (theory)

Good agreement with the Belle data

as well as the others like SLD, LEP data

Note: Preliminary Belle data of 2012 (≠ PRL 111 (2013) 062002) are used in this presentation.

Significant improvement due to the Belle measurements

Leading order (LO) Next-to-leading order (NLO)

with Belle data

without Belle data

2nd moment: M = dz z0

1

∫ Dih z,Q2( )

Leading order (LO)

Next-to-leading order (NLO)

with Belle data without Belle data

Significant improvement due to the Belle measurements

Significant improvement due to the Belle measurements

Leading order (LO) Next-to-leading order (NLO)

with Belle data

without Belle data

So far so good, but …

At the stage of 2013 (with final Belle and BaBar data)

Belle (2013)

BaBar-conv (2013) BaBar-prompt (2013)

1σ tot

dσ (e+e− → hX)dz

s = Mz

s = 10.54 GeV

HKNS07

1σ tot

dσ (e+e− → hX)dz

s = Mzs = 10.54 GeV

HKNS07

BaBar Belle

Question: Why BaBar > Belle ?! Belle: ISR BaBar: Lifetime cut (τ < 3 ⋅10−11 sec)

How to use the Belle 2013-final data tables in our global analysis

Total fragmentation function: F = 1σ total

dσ h

dz

Belle data table: dσh

dz⎛⎝⎜

⎞⎠⎟ EISR<0.5%⋅ s /2

= dσh

dz⋅ c

c = normalization correction factor due to the kinematical cut EISR < 0.5%⋅ s / 2. Total fragmentation function F

Ffinal Belle data ≡1

σ total( )EISR<0.5%⋅ s /2

dσ h

dz⎛⎝⎜

⎞⎠⎟ EISR<0.5%⋅ s /2

= 1c ⋅ σ total( )no cut for EISR

⋅ c ⋅ dσh

dz⎛⎝⎜

⎞⎠⎟ no cut for EISR

= 1c ⋅ σ total( )no cut for EISR

dσ h

dz⎛⎝⎜

⎞⎠⎟ final Belle data

= 1σ total

dσ h

dz, if 1

σ total( )no cut for EISR

dσ h

dz⎛⎝⎜

⎞⎠⎟ no cut for EISR

= 1σ total

dσ h

dz

1σ total

dσ h

dz⎛⎝⎜

⎞⎠⎟Theory

⇔ 1c ⋅ σ total( )no cut for EISR

dσ h

dz⎛⎝⎜

⎞⎠⎟ final Belle data

= 1c ⋅σ total

dσ h

dz⎛⎝⎜

⎞⎠⎟ final Belle data

c = 0.65 (Monte Carlo simulation, Belle paper)

χ tot2 = 782 for 342 data, χ 2 / dof = 2.4

χ 2 (Belle) = 117 for 78 data

Analysis with Belle data Analysis with Belle and BaBar data

χ tot2 = 1076 for 377 data, χ 2 / dof = 3.0

χ 2 (Belle) = 74 for 78 dataχ 2 (BaBar) = 185 for 35 data

Issues i Belle and BaBar data are not very consitent with other data (e.g. SLD, LEP data) + DGLAP evolution. i Belle and BaBar data are not consitent with each other.

0.2 0.2

χ tot2 = 324 for 323 data, χ 2 / dof = 1.1

χ 2 (Belle) = 4.6 for 78 data

Analysis with Belle data Analysis with Belle and BaBar data

χ tot2 = 1357 for 368 data, χ 2 / dof = 3.9

χ 2 (Belle) = 86 for 78 dataχ 2 (BaBar) = 717 for 45 data

Issues i Belle data are consitent with other data (e.g. SLD, LEP data) + DGLAP evolution. i BaBar data are not very consitent with other data.

0.2 0.2

Issues and Comments i Kinematical cuts of Belle and BaBar data: ISR, lifetime of final hadrons ⇒ Why Belle < BaBar? i The other data are not consitent with Belle-pion, but they are consistent with Belle-kaon. ⇒ May not be an ISR correction issue of Belle. i z-dependent functions are similar in kaon between Belle and BaBar ⇒ The issue could is simply the normalizaiton?? i Belle and BaBar data are not consistent with other data (+DGLAP evolution).

0.2 0.2

→ Solving these issues, we should be able to obtain accurate fragmentation functions, which becomes one of major achievements of the Belle and BaBar experiments.

The End

The End