1. Wigner distributions and quark orbital angular momentum Cédric Lorcé and May 14 2012, JLab, Newport News, VA, USA

2. Outline Parton distribution zoo  PDFs, FFs, GPDs, TMDs, GTMDs  Partonic interpretation Proton spin puzzle  Decompositions  Relations with observables Model results  Overlap representation  Wigner distributions

3. Outline Parton distribution zoo  PDFs, FFs, GPDs, TMDs, GTMDs  Partonic interpretation Proton spin puzzle  Decompositions  Relations with observables Model results  Overlap representation  Wigner distributions

4. Parton distribution zoo Charges Partonic interpretation (twist-2) Vector Tensor Axial 0D Picture

5. Parton distribution zoo Partonic interpretation (twist-2) DIS Parton Distribution Functions Charges PDFs 1D Picture

6. Parton distribution zoo Partonic interpretation (twist-2) ES Form Factors PDFs Charges FFs 2D Picture Impact parameter Transverse center of momentum

7. DVCS Parton distribution zoo Partonic interpretation (twist-2) Generalized PDFs GPDs Charges PDFsFFs 3D Picture [Soper (1977)] [Burkardt (2000,2003)] [Diehl, Hägler (2005)]

8. Parton distribution zoo Partonic interpretation (twist-2) SIDIS Transverse-Momentum dependent PDFs No direct connection TMDs Position space Momentum space Mean momentum Displacement Mean position Momentum transfer gauge Charges PDFsFFs GPDs 3D Picture

9. Parton distribution zoo Partonic interpretation (twist-2) Generalized TMDs GTMDs TMDs [Wigner (1932)] [Belitsky, Ji, Yuan (2004)] [C.L., Pasquini (2011)] Quasi-probabilistic interpretation ??? Charges PDFs GPDs FFs 5D Picture

10. Parton distribution zoo Complete set GTMDs TMDs [C.L., Pasquini, Vanderhaeghen (2011)] Charges PDFs GPDs FFsTMCs TMFFs TMCs 4D Picture2D Picture Partonic interpretation (twist-2)

11. Twist-2 structure GPDsTMDs Nucleon polarization Quark polarization Nucleon polarization MonopoleDipoleQuadrupole Naive T-odd-odd

12. Twist-2 structure GTMDs Nucleon polarization Quark polarization MonopoleDipoleQuadrupole Naive OAM [Meißner, Metz, Schlegel (2009)]

13. Twist-3 structure GPDsTMDs Nucleon polarization Quark polarization Nucleon polarization Naive T-odd-odd MonopoleDipoleQuadrupole

14. Outline Parton distribution zoo  PDFs, FFs, GPDs, TMDs, GTMDs  Partonic interpretation Proton spin puzzle  Decompositions  Relations with observables Model results  Overlap representation  Wigner distributions

15. Does not satisfy canonical relations Incomplete decomposition Proton spin puzzle JiJaffe-Manohar [Ji (1997)][Jaffe, Manohar (1990)] Gauge-invariant decomposition Accessible in DIS and DVCS KineticCanonical Pros: Cons: Pros: Cons: Satisfies canonical relations Complete decomposition Gauge-variant decomposition Missing observables for the OAM Improvements: [Wakamatsu (2009,2010)] Complete decomposition Improvements: [Chen et al. (2008)] Gauge-invariant extension OAM accessible via Wigner distributions [C.L., Pasquini (2011)] [C.L., Pasquini, Xiong, Yuan(2011)] [Hatta (2011)]

16. Quark spin and OAM GPDsTMDsGTMDs Quark spin Quark kinetic OAM A LL A UU +A UT [Penttinen et al. (2000)] Pure twist-3! [Ji (1997)] A UL A LL Quark spin Quark canonical OAM A TT Model-dependent Not intrinsic! [C.L., Pasquini (2011)] [Hatta (2011)] [C.L. et al. (2012)] [Burkardt (2007)] [Efremov et al. (2008,2010)] [She, Zhu, Ma (2009)] [Avakian et al. (2010)] [C.L., Pasquini (2011)] A UL Twist-2 Twist-3

17. OAM and origin dependence RelativeIntrinsicNaive Transverse center of momentum Physical interpretation ? Depends on proton position Equivalence IntrinsicRelativeNaive Momentum conservation

18. Outline Parton distribution zoo  PDFs, FFs, GPDs, TMDs, GTMDs  Partonic interpretation Proton spin puzzle  Decompositions  Relations with observables Model results  Overlap representation  Wigner distributions

19. Momentum Fock expansion of the proton state Fock states Simultaneous eigenstates of Light-front helicity Overlap representation

20. Light-front wave functions Proton state Eigenstates of parton light-front helicity Eigenstates of total OAM Probability associated with the N,  Fock state Normalization Overlap representation gauge

21. Fock-state contributions Overlap representation [C.L., Pasquini (2011)] [C.L. et al. (2012)] GTMDs TMDs GPDs Kinetic OAM Naive canonical OAM Canonical OAM

22. Model calculations Light-front 3Q models [C.L., Pasquini (2011)] Models are not QCD Truncation of Fock space spoils Lorentz covariance In model calculations, one would expectbut [Carbonell, Desplanques, Karmanov, Mathiot (1998)] GTMDs TMDs GPDs

23. Model calculations Unpol d quark in unpol d nucleon [C.L., Pasquini (2011)] favored disfavored Left-right symmetryNo net quark OAM

24. Model calculations Unpol d quark in longitudinally pol d nucleon Proton spin u-quark OAM d-quark OAM [C.L., Pasquini (2011)]

25. Model calculations Proton spin u-quark OAM d-quark OAM [C.L. et al. (2012)] Unpol d quark in longitudinally pol d nucleon

26. Quark spin u-quark OAM d-quark OAM Model calculations Longitudinally pol d quark in unpol d nucleon [C.L., Pasquini (2011)]

27. Proton spin u-quark spin d-quark spin Model calculations Longitudinally pol d quark in longitudinally pol d nucleon [C.L., Pasquini (2011)]

28. Emerging picture Longitudinal [C.L., Pasquini (2011)] Transverse [Burkardt (2005)] [Barone et al. (2008)]

29. Summary Parton distribution zoo  PDFs, FFs, GPDs, TMDs, GTMDs  Partonic interpretation Twist-2 phase-space distributions Proton spin puzzle  Decompositions Kinetic versus canonical  Relations with observables Model results  Overlap representation  Wigner distributions Consistent emerging picture

30. Incoherent scattering DVCS vs. SIDIS DVCSSIDIS GPDs TMDs FFs Factorization Compton form factor Cross section process dependent perturbative « universal » non-perturbative hardsoft

31. GPDs vs. TMDs GPDsTMDs Correlator Dirac matrix Wilson line Off-forward!Forward! FSIISI e.g. SIDISe.g. DY

32. LC helicity and canonical spin LC helicityCanonical spin Nucleon polarization Quark polarization Nucleon polarization [C.L., Pasquini (2011)]

33. Interesting relations Model relations ** * * * * Flavor-dependent Flavor-independent Linear relationsQuadratic relation Bag LF  QSM LFCQM S Diquark AV Diquark Cov. Parton Quark Target [Jaffe, Ji (1991), Signal (1997), Barone & al. (2002), Avakian & al. (2008-2010)] [C.L., Pasquini, Vanderhaeghen (2011)] [Pasquini & al. (2005-2008)] [Ma & al. (1996-2009), Jakob & al. (1997), Bacchetta & al. (2008)] [Ma & al. (1996-2009), Jakob & al. (1997)] [Bacchetta & al. (2008)] [Efremov & al. (2009)] [Meißner & al. (2007)] * =SU(6) * * * * * *

34. Geometrical explanation Preliminaries Quasi-free quarks Spherical symmetry [C.L., Pasquini (2011)] Conditions: Light-front helicity Canonical spin Wigner rotation (reduces to Melosh rotation in case of FREE quarks)

35. Geometrical explanation Axial symmetry about z

36. Geometrical explanation Axial symmetry about z