Fragmentation function is defined by...which becomes one of major achievements of the Belle and...
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Transcript of Fragmentation function is defined by...which becomes one of major achievements of the Belle and...
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
