Treacherous Points in DVCS

1. B.L.G. Bakker and C.Ji, arXiv:1002.0443[hep-ph] G.R.Goldstein and S.Liuti, PRD80, 071501(R)(2009); arXiv:1006.0213[hep-ph] Treacherous Points in DVCS JLab, October 29, 2010

2. Outline • Motivation from well-known Folk-lore - Nucleon GPDs in DVCS Amplitude - Specific Frame without Large Transverse Momentum • Complete Amplitude with Lepton Current - Cancellation of Singularities - Spin Filter for DVCS Amplitudes • Benchmark DVCS Amplitudes - Three Typical Kinematics - Spin Directions in LF Helicity States • Full vs. Reduced Amplitudes • Investigation of Singularities • Angular Momentum Conservation • Conclusions

3. Basic Idea: VCS Amplitude at Tree Level Helicity Amplitude: Neglecting masses, Identity:

4. Nucleon GPDs in DVCS Amplitude X.Ji,PRL78,610(1997): Eqs.(14) and (15) Just above Eq.(14), ``To calculate the scattering amplitude, it is convenient to define a special system of coordinates.”

5. Nucleon GPDs in DVCS Amplitude A.V.Radyushkin, PRD56, 5524 (1997): Eq.(7.1) At the beginning of Section 2E (Nonforward distributions), ``Writing the momentum of the virtual photon as q=q’-ζp is equivalent to using the Sudakov decomposition in the light-cone `plus’(p) and `minus’(q’) components in a situation when there is no transverse momentum .”

6. S-Channel DVCS Amplitude Sudakov Variables ’ Keeping no transverse momentum in DVCS, we agree on

7. However, in general, such factorization doesn’t work. In the frame with large transverse momentum, e.g. we get where Thus, one cannot neglect the terms with large transverse momentum but should compute the full amplitude. In particular, longitudinal polarization of virtual photon should not be neglected in the frame with large transverse momentum.

8. Investigation of Complete Ampltude Attach the lepton current and check the spin filter for the DVCS amplitude. Singularities develop in the polarization vector as p+ 0.

9. Investigation of Complete Ampltude S.J.Brodsky, M.Diehl, and D.-S.Hwang, NPB596, 99(2001) In the well-known q+=0 frame, both lepton and hadron amplitudes are singular.

10. Complete Amplitude and Constraints There should not be any singularity in complete amplitude.

11. Three Kinematics K1,K2,K3 K1 K2 K3

12. Leptonic Amplitudes in K1,K2,K3

13. Hadronic Amplitudes in three Kinematics

14. Complete DVCS Amplitudes in three Kinematics

15. C.Carlson and C.Ji, PRD67, 116002 (2003) Spin Directions in LF Helicity States

16. Full Amp vs. Reduced Amp S-channel: U-channel:

17. Full and Reduced DVCS Amplitudes in K2

18. Investigation of Singularity in K1

19. Complete Amplitude in Tree Levelattaching Lepton Current

20. VCS at Tree Level in K1

21. Leptonic Polarized Amplitude in K1

22. Hadronic Polarized Amplitude +

23. Checking Amplitudes • Gauge invariance of each and every polarized amplitude including the longitudinal polarization for the virtual photon. • Klein-Nishina Formula in RCS. • Angular Momentum Conservation. Allowed !

24. Checking Amplitudes • Gauge invariance of each and every polarized amplitude including the longitudinal polarization for the virtual photon. • Klein-Nishina Formula in RCS. • Angular Momentum Conservation. Prohibited !

25. Hadronic Polarized Amplitude in K1

26. Full and Reduced DVCS Amplitudes in K1 & K3

27. Conclusions • Full VCS amplitude including the lepton current was computed in the tree level and compared with the corresponding amplitude reduced as in the GPD approach. • The two results (Full vs. Reduced) agree in K2, but neither in K1 nor in K3 although they appear same by swapping the real photon polarization: Reduced amplitude in K1 or K3 does not satisfy the angular momentum conservation. • It is crucial to add the longitudinal polarization as well as the transverse ones for the finite result both in the full and reduced amplitudes in K1 and K3. • Form Factor calculations are all right in any of the three kinematics exhibiting the similar swap in the helicity amplitude between K2 and K1(or K3).