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

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…….

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)

PION-PHOTON TFF

Experimentally CELLO

(1991, <3GeV2 )

CLEO

(1998, 1.5-9.2GeV2)

• Experimentally – BABAR

(2009, 4-40GeV2 )

– Belle

(2012, 4-40GeV2 )

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

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

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

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

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

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

qq

NV

F Q F Q F Q

Qdx QF Q

xQ xQ k

x x k dk

F QQ

Pion-Photon TFF(LC pQCD)

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

qq

q

x

mA

x x

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

qq

q

x

mA

x x

3 2

2

( )

1 (2 1).

x

B C x

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

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

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 )

qq

dx x

f

Amx x BC x

f

m merf erf

x x x x

k

kk

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

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

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

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

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

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

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.

Thank you!