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报告人:钟涛
高能所理论室
2012.10.28
Puzzle on the Pion DA from the Pion-Photon
TFF with the Belle and BaBar Data By X-G Wu, T. Huang and T. Zhong
arXiv:1206.0466
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MOTIVATION
Pion distribution amplitude (DA)
Physical meaning:
the momentun distribution of the valence quarks in the pion.
Application:
the important component for the QCD LCSR and the QCD factorization theory in dealing with the exclusive processes: pion electromagnetic form factor, pion-photon transition form factor, B to pion transition form factor…….
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MOTIVATION
Pion distribution amplitude (DA) Research methods:
solve DA’s evolution equation(the asymptotic form);
SVZ SRs and Lattice;
models(BHL LC harmonic oscillator model, CZ(Chernyak-Zhitnitsky)-model, ……).
Existent controversy:
Where the DA’s asymptotic form is applicable? How about the DA’s form in the low and middle region?
Information from the pion-photon transition form factor(TFF)
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PION-PHOTON TFF
Experimentally CELLO
(1991, <3GeV2 )
CLEO
(1998, 1.5-9.2GeV2)
• Experimentally – BABAR
(2009, 4-40GeV2 )
– Belle
(2012, 4-40GeV2 )
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PION-PHOTON TFF
• Theoretical approach – Light-Cone Sum
Rules(LCSRs)
– Light-cone pQCD approach
……
• Definition
2ie F p q
The amplitude
of eπ →eγ
Pion-photon TFF
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PION-PHOTON TFF(LC PQCD)
The pion-photon TFF can be written as the convolution of the hard-scattering amplitude and the pion wave function:
• The hard-scattering amplitude can be calculated perturbativel:
12
1 2 1 20
( ) (1 ) ( , ) ( , )H i iF Q dx dx x x T x Q x Q
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PION-PHOTON TFF(LC PQCD)
1980, Lepage and Brodsky neglect the transverse momentum of quark to predict:
21
2
22 01 2
2( ) ( ) 1 ,
3s
dx mF Q x O
x x QQ
CZ-Model
AS wave function
2 215 2(2 1) (1 (2 1) )CZ x x
6 (1 )as x x
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PION-PHOTON TFF(LC PQCD)
1996, Fu-Guang Cao, Tao Huang and Bo-Qiang Ma, with the transverse momentum of quark dependence:
2
12 2 2
1 230( ) 2 ( ) ( , ) ( , , )
16
kk kc u d i H
dF Q n e e dx x T x x
CZ-Model
BHL model
2 2
2( , ) exp
8 (1 )
kk
qBHLm
x Ax x
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PION-PHOTON TFF(LC PQCD)
2007, Tao Huang and Xing-Gang Wu, with the non-valence-quark part contribution:
2 ( ) 2 ( ) 2( ) ( ) ( )V NVF Q F Q F Q
The non-valence-
quark contribution
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Considering the k⊥-correction, non-valence-quark contribution and the NLO correction to the valence-quark contribution, the pion-photon TFF would be:
2 2
2 ( ) 2 ( ) 2
2 21
( ) 2
2 2 22 0 0
22 2
( ) 2
2 2
( ) ( ) ( ),
( )1 ( ) 1 ln
34 3
2ln 3 ( , ) ,3
( ) .(1 )
V NV
x QV s
NV
F Q F Q F Q
Qdx QF Q
xQ xQ k
x x k dk
F QQ
Pion-Photon TFF(LC pQCD)
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PION DA
The model of the pion WF can be written as:
1 2
1 2
( , ) ( , ) ( , ),k k kR
qq qqx x x
2 2
2
( , )
exp ,8 (1 )
k
k
R
q
x
mA
x x
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PION DA
The model of the pion WF can be written as:
1 2
1 2
( , ) ( , ) ( , ),k k kR
qq qqx x x
2 2
2
( , )
exp ,8 (1 )
k
k
R
q
x
mA
x x
3 2
2
( )
1 (2 1).
x
B C x
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PION DA
How to determine the parameters of the WF:
2 20
21
30( , )
16 2 3k
kkqq
fddx x
1
0
3( , 0)kqqdx x
f
The wave-function
normalization condition
The constraint derived
from π0→γγ decay
amplitude
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PION DA
How to determine the parameters of the WF:
2 20
21
30( , )
16 2 3k
kkqq
fddx x
1
0
3( , 0)kqqdx x
f
The wave-function
normalization condition
The constraint derived
from π0→γγ decay
amplitude
0.00 0.60B
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PION DA
The pion DA can be obtained as following:
2 2
0
22
0 3
3 2
23 2
2 2 2
0
2 2
2 3( , ) ( , )
16
3(1 ) (1 (2 1))
2 2
8 (1 ) 8 (1 )
dx x
f
Amx x BC x
f
m merf erf
x x x x
k
kk
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PION DA
Comparison of the pion DA model with the asymptotic-DA and the CZ-DA:
6 (1 )as x x
2 215(2 1) (1 (2 1) )
2
CZ x x
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PION DA
The pion DA Gegenbauer moments can be calculated by the following way:
12 3 2
02 00 1 2
3 2
0
( , ) (2 1)( )
6 (1 ) (2 1)
n
n
n
dx x C xa
dx x x C x
• The soft part contribution of the
pion electromagnetic form factor can be written as:
1 2
22
1 23( ) ( , , ) ( , , )
16
kk k
s
qq qq
dxdF Q x x
• The probability for finding the lowest valence quark state:
2
2
0( ) |s
qq QP F Q
• The charged mean square radius:
2
2
2
2 0
( )6 |
sqq
Q
F Qr
Q
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NUMERICAL ANALYSIS
• Q2Fπγ(Q2) with the model WF by taking mq=0.30 GeV
and by varying B within the region of [0.00,0.60]:
B=0.60
B=0.30
B=0.00
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NUMERICAL ANALYSIS
• Q2Fπγ(Q2)
with the model WF by fixing B=0.00 (Asymptotic-like DA) and by varying mq within the region [0.20,0.30] GeV:
0.30qm GeV
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NUMERICAL ANALYSIS
• Q2Fπγ(Q2)
with the model WF by fixing B=0.30 and by varying mq within the region of [0.20,0.40] GeV:
0.30qm GeV
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NUMERICAL ANALYSIS
• Q2Fπγ(Q2)
with the model WF by fixing B=0.60 (CZ-like DA) and by varying mq within the region [0.30,0.50] GeV:
0.40qm GeV
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SUMMARY
1. If we know the pion-photon TFF well, we can conveniently derive the pion DA’s correct behavior.
2. In large Q2 region, the new Belle data agrees with the asymptotic DA estimation, while to be consistent with the BABAR data,we need a much broader DA.
3. If the BABAR collaboration still insists on their neasurements, then there may indicate new physics in these form factors from pQCD.
4. Because of the effective end-point suppression due to the BHL-transverse-momentum dependence, our present model of the pion WF/DA will present a basis for deriving more reliable pQCD estimates.
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Thank you!