2. 1 Frascati, June 14, 2006 Aram Kotzinian Double spin azimuthal asymmetries A LT and A LL in semi-inclusive DIS Aram Kotzinian Torino University & INFN On leave in absence from YerPhI, Armenia and JINR, Russia In collaboration with B. Parsamyan, & A. Prokudin, hep-ph/0603194, PRD 2006; M. Anselmino, A. Efremov & B. Parsamyan, in preparation Introduction Positivity bounds for TMD DFs Status of Lorentz Invariance relations Predictions for azimuthal dependence of A LT Cahn effect and predictions for azimuthal dependence of A LL Conclusions

3. 2 Frascati, June 14, 2006 Aram Kotzinian TMD quark DFs of Nucleon Twist-two: 6 T-even & 2 T- odd Use this DFs and put soft and hard factors equal to 1 At twist-three: 8 T-even & 8 T- odd Collins, Soper (81,82); Ji, Ma, Yuan (03, 04); Collins, Metz (04)

4. 3 Frascati, June 14, 2006 Aram Kotzinian Positivity bounds Sivers function For d-quark this bound is reached at for the best fit parameters. But error bars of extracted parameters are large enough and allow to fulfill the general positivity bound. Naïve bound was used during fitting:

5. 4 Frascati, June 14, 2006 Aram Kotzinian Status of Lorentz invariance (LI) relations 2001: explicit model calculations of DFs by Kundu & Metz show that these relations are violated 2003: Goeke, Metz, Pobylitsa & Polyakov demonstrated that the same Wilson link in the definition of DFs which makes possible the existence of nonzero Sivers effect leads to violation of LI relations among DFs. Without Wilson link Sivers effect = 0. If observed by HERMES asymmetry is due to Sivers effect then LI relations can be strongly violated is g 1T smaller or larger than proposed by LI relation? Relations between k T -integrated twist-3 and twist-2 PDFs

6. 5 Frascati, June 14, 2006 Aram Kotzinian SIDIS and TMD DF

7. 6 Frascati, June 14, 2006 Aram Kotzinian Double Spin Asymmetry Weighted asymmetry is related to the first k T -momentum of g 1T : There exists a relation between first momentum of g 1T and g 2 (follows from Lorentz invariance, Tangerman & Mulders) : All ingredients (DFs & FFs) are known! First estimations by A.K. & P.Mulders, PRD 54, 1229 (1996)

8. 7 Frascati, June 14, 2006 Aram Kotzinian g 1T (1) GRV98+GRSV2000 LO, std DFs Kretzer FFs

9. 8 Frascati, June 14, 2006 Aram Kotzinian CUTS Low x, y, z & p T High x, y, z & p T + Low x, y, z & p T High x, y, z & p T + A TMD =

10. 9 Frascati, June 14, 2006 Aram Kotzinian P T -weighted asymmetries: COMPASS

11. 10 Frascati, June 14, 2006 Aram Kotzinian P T -weighted asymmetries: JLab

12. 11 Frascati, June 14, 2006 Aram Kotzinian Transverse momentum dependence (1) N is fixed by holds if (GeV/c) 2 From analysis of unpolarized P hT dependence and Cahn effect Predictions for

13. 12 Frascati, June 14, 2006 Aram Kotzinian Transverse momentum dependence (2) p T -unintegrated asymmetries

14. 13 Frascati, June 14, 2006 Aram Kotzinian Transverse momentum dependence (Deuteron)

15. 14 Frascati, June 14, 2006 Aram Kotzinian x,y, z dependence of asymmetries

16. 15 Frascati, June 14, 2006 Aram Kotzinian Cahn effect in A LL 1/Q kinematical correction: azimuthal modulation of lepton-quark non-coplanar hard scattering cross-section in presence of nonzero transverse momentum of quark in nucleon

17. 16 Frascati, June 14, 2006 Aram Kotzinian S T correction l’ l γ*γ* S STST x z

18. 17 Frascati, June 14, 2006 Aram Kotzinian Gaussian parameterization of DF x- and z-dependencies are known within this approach !

19. 18 Frascati, June 14, 2006 Aram Kotzinian cosφ h -weighted A LL

20. 19 Frascati, June 14, 2006 Aram Kotzinian Results P T -dependence, Proton target Influence of Δσ LT is negligible

21. 20 Frascati, June 14, 2006 Aram Kotzinian Results x,y,z-dependence at COMPASS

22. 21 Frascati, June 14, 2006 Aram Kotzinian Results x,y,z-dependence at HERMES

23. 22 Frascati, June 14, 2006 Aram Kotzinian Results x,y,z-dependence at JLab6

24. 23 Frascati, June 14, 2006 Aram Kotzinian Conclusions To check the self-consistency of LO QCD picture of polarized SIDIS with TMD DFs & FFs it is important to measure at different energies the azimuthal and P hT dependence of unpolarized SIDIS A LL & A LT These and also Sivers asymmetry measurements will allow to check the Lorentz invariance relations to perform a “global” analysis and extract the parameters describing the transverse momentum dependence of g 1T and g 1L (and x Bj dependence of Sivers function) taking into account the positivity bounds for TMF DFs. Data are already available at COMPESS, HERMES & JLab !

25. 24 Frascati, June 14, 2006 Aram Kotzinian Conclusions To check the self-consistency of LO QCD picture of polarized SIDIS with TMD DFs & FFs it is important to measure at different energies the azimuthal and P hT dependence of unpolarized SIDIS A LL & A LT These and also Sivers asymmetry measurements will allow to check the Lorentz invariance relations to perform a “global” analysis and extract the parameters describing the transverse momentum dependence of g 1T and g 1L (and x Bj dependence of Sivers function) taking into account the positivity bounds for TMF DFs. Data are already available at COMPASS, HERMES & JLab !

26. 25 Frascati, June 14, 2006 Aram Kotzinian Additional slides

27. 26 Frascati, June 14, 2006 Aram Kotzinian Spin physics “golden rule” Minimize relative error: Maximize: Constant acceptance, N

28. 27 Frascati, June 14, 2006 Aram Kotzinian Contributions from long. & tran. components of target polarization

29. 28 Frascati, June 14, 2006 Aram Kotzinian Transverse momentum dependence (Proton)

30. 29 Frascati, June 14, 2006 Aram Kotzinian x,y, z dependence of asymmetries

31. 30 Frascati, June 14, 2006 Aram Kotzinian x,y, z dependence of asymmetries

32. 31 Frascati, June 14, 2006 Aram Kotzinian P T -weighted asymmetries: HERMES

33. 32 Frascati, June 14, 2006 Aram Kotzinian Results P T -dependence, Deuteron target Influence of Δσ LT is negligible