1

2005 International Conference on QCD and Hadronic Physics, Peking University

Pion Electromagnetic Form Factor in the KT Factorization Formulae

Xing-Gang WuXing-Gang Wu

(IHEP)(IHEP)

In colarboration with Prof. Tao Huang

2

• Higher helicity components’ Contributions to the pion Higher helicity components’ Contributions to the pion ---electromagnetic form factor ---electromagnetic form factor

• Higher twist structures’ Contributions to the pionHigher twist structures’ Contributions to the pion ---electromagnetic form factor ---electromagnetic form factor

• BackgroundBackground

• SummarySummary

3

I.Background

soft regions up to 1.6GeV2 PRL26,1713(2001) by JLAB

A 12 GeV upgrade to CEBAF will offer the possibility to measure pion form factor to good precision out to Q2= 36GeV2 for the first time. This offers the best possibility to study the transition between the dominance of `soft’ and `hard’ processes in the dynamics, and to learn where the pQCD limit may be reached, and also to check the pQCD results.

4

Pion form factor in the large momentum transfer:

LO results:

Improved to LL: G.R. Farrar and D.R. Jackson, PRL23,246,1979

Twist-2

5

2) Power corrections to the LO pion form factor:

To short the gap between the theoretical prediction and the experimental, two ways are tried:

1) Non-perturbative contributions

A.P. Bakulev etal. PHYS.REV.D70, 033014(2004); O.C. Jacob and L.S. Kisslinger, Phys.Rev.Lett. 56, 225(1986); L.S. Kisslinger etal. Phys.Rev.D63,113005(2001); V.M. Braun etal. Phys.Rev.D61,073004(2000); F. Schlumpf, Phys.Rev.D50, 6895(1994)

a) higher helicity contributions

b) higher-twist effect

c) higher order in alpha_s

6

The right power behavior of the hard contribution from the higher helicity components can only be obtained by fully keeping the kT depen-dence in the hard amplitude, and that the kT dependence in LC wave function affects the hard and soft contributions substantially.

Wigner Rotation:

Instant Form

II. Higher helicity components’ Contributions to the pion form factor

1. Background and the reason for it

7

PQCD result: PRD54, 5890(1996) S.W. Wang and L.S. Kisslinger

LC PQCD result: PRD55, 7107(1997) T.Huang etal.

Qualitative analysis

Suppression

Need a quantitative results to clarify the situation!

PQCD result: J.Phys.G21, 765(1995) B.Q. Ma and T. Huang

Enhancement

8

2. LC pQCD calculations

Six LO time-ordered Feynman Diagrams

9

Upper three

Instantaneous diagram

10

Lower three

11

To do numerical calculation:

To ensure the gluon is hard enough

BHL

The higher helicty components will contribute zero at the order of Q2.

Model dependence:

energy flow

Wigner rotation

12

After integration over the azimuth angle, one may find that contributions from the higher helicity states will also provide a negative sign!

separation

13

Typical Q4- behavior

The right power behavior of the hard contribution from the higher helicity components can only be obtained by fully keeping the kT depen-dence in the hard amplitude, and that the kT dependence in LC wave function affects the hard and soft contributions substantially.

3. Conclusion

14

Higher helicity components

4. A discussion on the soft part

15

Combined twist-2

contributionNLO, Twist-3 ?

Higher Fock states ?

soft

hard

16

III. pionic twist-3 contributions

1) The pionic wave function1) The pionic wave function ：：

T. Huang etal. PRD70,014013(2004), QCD sum rule for the wave function of pion

17

The present obtained twist-3 contribution to the pion form factor is comparable or even larger than that of the leading twist in a wide intermediate energy region, Q2 (2-40) GeV2 So, and improved form for the twist-3 wave functions must be introduced to suppress the end-point singularity.

Gegenbauer polynomial expansion BHL

New Mode

l

18

DAs

our present

V.M. Braun, I.E. Filyanov and P. Ball

T. Huang etal.

19

For usual helicity component

2) pion electro-magnetic form factor in the b-space2) pion electro-magnetic form factor in the b-space PRD70,093013(2004)

20

For all the helicity components

21

Twsit-3

Only for usual helicity components

3) considering the model dependence of the twist-3 contributions3) considering the model dependence of the twist-3 contributions

22

Moments

mp0=1.3GeV2(0.320,0.370)

23

F.G.Cao etal. Z.T.Wei and M.Z.Yang

4) comparison among different calculations4) comparison among different calculations

Conclusion:Conclusion:

The transverse momentum dependence has been

included. A new form of twist-3 wave function has been adopted. The present obtained twist-3 indeed is

power suppressed.

EPJC11,501(1998) PRD67,094013(2003)

Twist-2

NLO

Twist-3

24

IV.Summary

1) Higher helicity structure and twist-3 contributions to the pion electro-magnetic form factor have been discussed. Q2-suppressed

2) The kT factorization approach, where the transverse momentum dependence in the wave function and the hard scattering kernel have been kept, ensures these two are really power suppressed.

3) Both two can give sizeable contributions in the intermediate energy regions. Taking both into consideration, the present experimental data can be roughly explained.

4) The coming CEBAF results shall test the PQCD calculations more precisely.

25

Thanks !