1. Pion transition form factor in the light front quark model
C. C. Lih and C. Q. Geng
The 5th international Conference of Flavor Physics, Hanoi, September 24-30, 2009

2. Pion transition form factor in the light front quark model
Outline
Motivation
Light-Front Model
Form Factors
Numerical Results
Conclusion 1

3. Motivation 1. Pion transition form factor in the process πγ*→γ
The motive mainly comes from the difference between theoretical prediction and experimental data
1. Pion transition form factor in the process πγ*→γ
BaBar data － pion transition form factor at large Q2（～40 GeV2） is in contradiction with QCD prediction.
The massless QCD asymptotic limit of pion transition form factor Fπγ(Q2)
The Babar data behavior like higher-order or twist contribution in perturbative effect.
BaBar-PUB-09/006

4. Motivation 2. π0→ e＋e－ KTeV E799-II obtained
The light meson decaying into lepton pairs has provided low-energy testing of the SM.
2. π0→ e＋e－
It is a one loop process via a two-photon intermediate state.
KTeV E799-II obtained
To lowest order in QED is given by
The imaginary part of A can be found in model independent way. One can get the unitary bound for the branching ratio
Since Fπγ*γ*(0,0)=1

5. Motivation Standard model prediction
This prediction based on the use of CLEO data on transition form factor.
The result is 3.3σ below the experimental data
The possible sources of the error
－Pion form factor Fπγ*γ*
－experiment procession
－effects of new physics
a good SM test process
The Goals in this job is to test the form factors in the light front quark model.

6. Light-Front Quark Model
In the light-front approach, the wave functions for the hadronic bound states are defined in the hypersurface
LF in Drell-Yan-West
x+(=x0+x3)=0 frame
The LFQM describes very well the electromagnetic properties of the hadronic bound states.
A hadronic bound states labeled by the momentum P+ and P⊥ and the helicity λ which can be expanded in term of the Fock space, one has
Here xi is the fraction of the total longitudinal momentum and k⊥i is relative transition momentum.

7. Light-Front Quark Model
The general form of the phenomenological light front meson bound state has a structure which consists of a quark q and an anti-quark with the total momentum P and spin s （with only the Fock space sector）：
where ΦM is the amplitude of the corresponding and k1(2) is the on-mass shell light front momentum of the internal quark.
and wave function can be expressed as follows:

8. Light-Front Quark Model
Where is the wave function which constructs a state of definite spin （s,sz） out of the helicity（λ1,λ2 ） eigenstates. One have
It is a problem. It’s Not so easy to identify the light front hadronic wave function with hadronic states. The difficulty is spin structure.
with ψ being the space part wave function that depends on the dynamics. This distribution function ψ is in term of the light-front relative momentum variable （x, k⊥）. One wave function that has often been used for the meson is Gassian type
Wei-Min Zhang, Chin. J. Phys,31,717(1994).

9. Form Factors Decay constant： The decay constant of pion is defined byW p1
Axial vector current
The amplitude can be written as：
Λπ is the bound state vertex function. One could relate to the distribution functuion ψπ by,
For the decay constant, the result are
Input parameters
to fix the parameter ω.

10. Form Factors
There are two types of transition form factors Fπγ and Fπγ*γ*. γ*
Transition form factor of Fπγ p3 p2
The form factors Fpγ is defined by the Pγγ* vertex
P(p)
γ(q) p1
The amplitude can be written as：
We can extracted these form factors from the condition ‘+’ current and by comparing the definition of amplitude.

11. Form Factors Transition form factor of Fπγ*γ* γ*(q1)p3 p2
Form factors comes from a pion which couples to two photons. The hadronic matrix elements contribute to Fpγ*γ* are: p
γ*(q2) p1
The amplitude can be written as：

12. Form Factors The LF relevant quark momentum variable
The form factors of →γ*γ* can be found as：
If two photon are on mass shell, the form factor of →γγ can be written as：
In term of q1 and q2

13. Numerical Results
To numerical the meson P → γ*γ* (P＝π0, η) transition from factors within LFQM, we have to decompose into a Fock state for meson. The π0 may described as and the valence state of η can be written as：
We use the decay constant and the branching ratio of P → 2γ to specify the quark masses of mu,d,s and the meson scale parameter of ωP in ψ(x,k⊥).

14. Numerical Results The decay constants of π0 and η mesons are in MeV
The branching ratios of π0 and η mesons to 2γ are
The factor FP→2γ(0,0) can be determined via
One could extracted the parameters：
mu＝md＝0.24, ms＝0.38. and
ωπ＝0.33, ωη1＝0.26, ωη8＝0.28 in GeV

15. Numerical Results Summary of the lepton pair decays of π0 Br Exp. data
This work
ChPT
VMD QM
NLM
QED
102 B(e+e-γ)
1.198±0.032
1.18 －
105 B(e+e-e+e-)
3.14±0.3
3.29
3.28
3.46
108 B(e+e-)
7.48±0.29±0.25
6.94
7±1
6.41±0.19 6
≧4.7
It is placed inside the lowest bound of the experimental data.

16. Numerical Results Summary of the lepton pair decays of η Br Exp. data
This work
VMD
CLEO
QCD
EMT
103 B(e+e-γ)
7.8±0.5±0.7
6.95 －
9.4±0.7
6.31－6.46
6.5
104 B(μ+μ-γ)
2.14－3.01
3.0
105 B(e+e-e+e-)
±0.1
2.47
2.49－2.62
107 B(e+e-μ+μ-)
＜1.6×103
5.83
5.42－7.17
109 B(μ+μ-μ+μ-)
＜3.6×105
1.68
109 B(e+e-)
＜2.7×104
7.85
13.7
4.6±0.06
106 B(μ+μ-)
5.22
11.4
5.2±1.2
5.11±0.2

17. Numerical Results The γ*γ→π0 transition from factors multiplied by Q2
A good fit to the Babar data at high Q2

18. Conclusions
The BaBar measured transition from factors － a large uncertainty in the high Q2.
The data point in Q2＝27.31 GeV2 is consistent with QCD asymptotic limit.
－getting more precise data on
the pion transition form factor.
Further independent experimental and theoretical predictions for these decays are needed.
New physics solutions in π0→e＋e- － if the discrepancy stand between the theoretical prediction and experimental results.