1. 1 SPIN 2006, Kyoto, October 6, 2006 Aram Kotzinian Double spin azimuthal asymmetries A LT and A LL in semi-inclusive DIS Aram Kotzinian YerPhI, Armenia and JINR, Russia and INFN, Torino In collaboration with B. Parsamyan, & A. Prokudin, hep-ph/0603194; Phys.Rev. D73 (2006) M. Anselmino, A. Efremov & B. Parsamyan, hep-ph/0608048, submitted to PRD Introduction Positivity bounds for TMD DFs Status of Lorentz Invariance relations Predictions for azimuthal and p hT - dependence of A LT Predictions for azimuthal and p hT - dependence of A LL Conclusions

2. 2 SPIN 2006, Kyoto, October 6, 2006 Aram Kotzinian TMD quark DFs of Nucleon Twist-two: 6 T-even & 2 T- odd At twist-three: 8 T-even & 8 T- odd

3. 3 SPIN 2006, Kyoto, October 6, 2006 Aram Kotzinian Generalized 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 of Sivers DF:

4. 4 SPIN 2006, Kyoto, October 6, 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

5. 5 SPIN 2006, Kyoto, October 6, 2006 Aram Kotzinian SIDIS and TMD DF

6. 6 SPIN 2006, Kyoto, October 6, 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)

7. 7 SPIN 2006, Kyoto, October 6, 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 =

8. 8 SPIN 2006, Kyoto, October 6, 2006 Aram Kotzinian P T -weighted asymmetries

9. 9 SPIN 2006, Kyoto, October 6, 2006 Aram Kotzinian Factorized Gaussian parameterization From analysis of unpolarized P hT dependence and Cahn effect All μ are constant independent of flavor, x and z

10. 10 SPIN 2006, Kyoto, October 6, 2006 Aram Kotzinian Transverse momentum dependence (Proton) Proton Deuteron

11. 11 SPIN 2006, Kyoto, October 6, 2006 Aram Kotzinian Orbital momentum and g 1L Model by Brodsky, Hwang, Ma & Schmidt, NPB 593 (2001) 311

12. 12 SPIN 2006, Kyoto, October 6, 2006 Aram Kotzinian P T -dependence of A LL With factorized Gaussian parameterization of DFs and FFs we have simple expressions

13. 13 SPIN 2006, Kyoto, October 6, 2006 Aram Kotzinian Predicted P T -dependence of A LL Proton Deuteron

14. 14 SPIN 2006, Kyoto, October 6, 2006 Aram Kotzinian Azimuthal dependence of A LL l’ l γ*γ* S STST x z In O(1/Q) approximation we have kinematical (Cahn) corrections and contribution from transverse component of target polarization

15. 15 SPIN 2006, Kyoto, October 6, 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

16. 16 SPIN 2006, Kyoto, October 6, 2006 Aram Kotzinian cosφ h -weighted A LL

17. 17 SPIN 2006, Kyoto, October 6, 2006 Aram Kotzinian Results P T -dependence Influence of Δσ LT is negligible Proton Deuteron

18. 18 SPIN 2006, Kyoto, October 6, 2006 Aram Kotzinian Results x,y,z-dependence at COMPASS

19. 19 SPIN 2006, Kyoto, October 6, 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 g 1T, g 1L and Sivers function taking into account the positivity bounds for TMF DFs. Data are already available at COMPASS, HERMES & JLab !