1. Cédric Lorcé IPN Orsay - LPT Orsay Orbital Angular Momentum in QCD June 27 2013, Dipartimento di Fisica, Universita’ di Pavia, Italy

2. The outline Dark spin Quark spin ? ~ 30 % The decompositions in a nutshell Canonical formalism and Chen et al. approach Geometrical interpretation of gauge symmetry Path-dependence and measurability Conclusions Basic question

3. Jaffe-Manohar (1990) The decompositions in a nutshell SqSq SgSg LgLg LqLq Noether’s theorem

4. Ji (1997) Jaffe-Manohar (1990) The decompositions in a nutshell SqSq SgSg LgLg LqLq SqSq JgJg LqLq Noether’s theorem

5. Ji (1997) Jaffe-Manohar (1990) Chen et al. (2008) The decompositions in a nutshell SqSq SgSg LgLg LqLq SqSq SgSg LgLg LqLq SqSq JgJg LqLq Gauge-invariant extension (GIE) Noether’s theorem

6. Wakamatsu (2010) Ji (1997) Jaffe-Manohar (1990) Chen et al. (2008) The decompositions in a nutshell SqSq SgSg LgLg LqLq SqSq SgSg LgLg LqLq SqSq SgSg LgLg LqLq SqSq JgJg LqLq Gauge-invariant extension (GIE) Noether’s theorem

7. Wakamatsu (2010) Ji (1997) Jaffe-Manohar (1990) Chen et al. (2008) CanonicalKinetic The decompositions in a nutshell SqSq SgSg LgLg LqLq SqSq SgSg LgLg LqLq SqSq SgSg LgLg LqLq SqSq JgJg LqLq Gauge-invariant extension (GIE) Noether’s theorem

8. Wakamatsu (2010) Ji (1997) Jaffe-Manohar (1990) Chen et al. (2008) CanonicalKinetic The decompositions in a nutshell SqSq SgSg LgLg LqLq SqSq SgSg LgLg LqLq SqSq SgSg LgLg LqLq SqSq JgJg LqLq Gauge-invariant extension (GIE) Noether’s theorem

9. The Chen et al. approach [Chen et al. (2008,2009)] [Wakamatsu (2010,2011)]

10. The Chen et al. approach Gauge transformation (assumed) [Chen et al. (2008,2009)] [Wakamatsu (2010,2011)]

11. The Chen et al. approach Gauge transformation (assumed) Pure-gauge covariant derivatives [Chen et al. (2008,2009)] [Wakamatsu (2010,2011)]

12. The Chen et al. approach Gauge transformation (assumed) Field strength Pure-gauge covariant derivatives [Chen et al. (2008,2009)] [Wakamatsu (2010,2011)]

13. The canonical formalism Textbook Dynamical variables Lagrangian [C.L. (2013)]

14. The canonical formalism Textbook Gauge covariant Dynamical variables Lagrangian [C.L. (2013)]

15. The canonical formalism Textbook Gauge covariant Gauge invariant Dynamical variables Lagrangian Dirac variables Dressing fieldGauge transformation [Dirac (1955)] [Mandelstam (1962)] [C.L. (2013)]

16. The analogy with General Relativity [C.L. (2012,2013)] Dual role

17. Pure gauge Physical polarizations The analogy with General Relativity Degrees of freedom [C.L. (2012,2013)] Dual role

18. Pure gauge Physical polarizations The analogy with General Relativity Geometrical interpretation Parallelism Curvature Degrees of freedom [C.L. (2012,2013)] Dual role

19. Pure gauge Physical polarizations Analogy with General Relativity The analogy with General Relativity Geometrical interpretation Parallelism Curvature Inertial forces Gravitational forces Degrees of freedom [C.L. (2012,2013)] Dual role

20. [Wakamatsu (2010)][Chen et al. (2008)] The Stueckelberg symmetry Ambiguous! [Stoilov (2010)] [C.L. (2013)] SqSq SgSg LgLg LqLq SqSq SgSg LgLg LqLq Coulomb GIE [Hatta (2011)] [C.L. (2013)] SqSq SgSg LgLg LqLq Light-front GIE L pot SqSq SgSg LgLg LqLq Infinitely many possibilities!

21. Gauge GIE1 GIE2 Gauge-variant operator « Natural » gauges Lorentz-invariant extensions ~ Rest Center-of-mass Infinite momentum « Natural » frames The gauge-invariant extension (GIE)

22. The geometrical interpretation [Hatta (2012)] [C.L. (2012)] Parallel transport

23. The geometrical interpretation [Hatta (2012)] [C.L. (2012)] Parallel transport

24. The geometrical interpretation [Hatta (2012)] [C.L. (2012)] Parallel transport Non-local !

25. The geometrical interpretation [Hatta (2012)] [C.L. (2012)] Parallel transport Path dependent ! Stueckelberg symmetry Non-local !

26. The path dependence [Ji, Xiong, Yuan (2012)] [Hatta (2012)] [C.L. (2013)] Canonical quark OAM operator

27. FSIISI SIDISDrell-Yan The path dependence [Ji, Xiong, Yuan (2012)] [Hatta (2012)] [C.L. (2013)] Naive T-even Canonical quark OAM operator Light-front LqLq

28. FSIISI SIDISDrell-Yan The path dependence [Ji, Xiong, Yuan (2012)] [Hatta (2012)] [C.L. (2013)] Coincides locally with kinetic quark OAM Naive T-even Canonical quark OAM operator x-based Fock-SchwingerLight-front LqLq LqLq

29. The gauge symmetry Quantum electrodynamics « Physical » [C.L. (in preparation)] « Background »

30. The gauge symmetry Quantum electrodynamics Passive « Physical » [C.L. (in preparation)] « Background »

31. The gauge symmetry Quantum electrodynamics PassiveActive « Physical » [C.L. (in preparation)] « Background »

32. The gauge symmetry Quantum electrodynamics PassiveActive « Physical » [C.L. (in preparation)] « Background » Active x (Passive) -1

33. The gauge symmetry Quantum electrodynamics PassiveActive « Physical » [C.L. (in preparation)] « Background » Active x (Passive) -1 Stueckelberg

34. The semantic ambiguity « measurable » Quid ? « physical » « gauge invariant »

35. The semantic ambiguity Observables « measurable » Quid ? « physical » « gauge invariant » Measurable, physical, gauge invariant (active and passive) E.g. cross-sections

36. The semantic ambiguity Path Stueckelberg Background Observables « measurable » Quid ? « physical » « gauge invariant » Measurable, physical, gauge invariant (active and passive) Expansion scheme E.g. cross-sections dependent E.g. collinear factorization

37. The semantic ambiguity Path Stueckelberg Background Observables Quasi-observables « measurable » Quid ? « physical » « gauge invariant » Measurable, physical, gauge invariant (active and passive) « Measurable », « physical », « gauge invariant » (only passive) Expansion scheme E.g. cross-sections E.g. parton distributions dependent E.g. collinear factorization

38. CanonicalKinetic The observability SqSq SgSg LgLg LqLq SqSq SgSg LgLg LqLq SqSq SgSg LgLg LqLq SqSq JgJg LqLq Not observableObservableQuasi-observable [Wakamatsu (2010)] [Ji (1997)] [Jaffe-Manohar (1990)] [Chen et al. (2008)]

39. The gluon spin [Jaffe-Manohar (1990)][Hatta (2011)] Light-front GIE Light-front gauge Gluon helicity distribution Local fixed-gauge interpretationNon-local gauge-invariant interpretation « Measurable », gauge invariant but non-local

40. The kinetic and canonical OAM Quark naive canonical OAM (Jaffe-Manohar) [Burkardt (2007)] [Efremov et al. (2008,2010)] [She, Zhu, Ma (2009)] [Avakian et al. (2010)] [C.L., Pasquini (2011)] Model-dependent ! Kinetic OAM (Ji) [Ji (1997)] [Penttinen et al. (2000)] [Kiptily, Polyakov (2004)] [Hatta (2012)] but No gluons and not QCD EOM! [C.L., Pasquini (2011)] Pure twist-3 Canonical OAM (Jaffe-Manohar) [C.L., Pasquini (2011)] [C.L., Pasquini, Xiong, Yuan (2012)] [Hatta (2012)]

41. Wakamatsu (2010) Ji (1997) Jaffe-Manohar (1990) Chen et al. (2008) CanonicalKinetic The conclusion SqSq SgSg LgLg LqLq SqSq SgSg LgLg LqLq SqSq SgSg LgLg LqLq SqSq JgJg LqLq Not observableObservable Quasi-observable

42. Backup slides

43. [PRD79 (2009) 014507] [Nucl. Phys. A825 (2009) 115] [PRL104 (2010) 112001] [PRD79 (2009) 113011] GTMDs TMDs Charges PDFs GPDs FFsTMCs TMFFs [PRD84 (2011) 034039] [PLB710 (2012) 486] [PRD84 (2011) 014015] [PRD85 (2012) 114006] [JHEP1105 (2011) 041] [PRD74 (2006) 054019] [PRD78 (2008) 034001] [PRD79 (2009) 074027] Phase-space densities The parton distributions

44. « Vorticity » The twist-2 OAM Quark Wigner operator [C.L., Pasquini (2011)] [C.L., Pasquini, Xiong, Yuan (2012)] [Hatta (2012)] Quark OAM operator Exact relation

45. The spin-spin-orbit correlations [C.L., Pasquini (2011)]

46. Overlap representation MomentumPolarization [PRD74 (2006) 054019] [PRD78 (2008) 034001] [PRD79 (2009) 074027] Light-front quark modelsWigner rotation The light-front wave functions

47. OAM Canonical (naive) Kinetic Canonical GTMDs TMDs GPDs Phenomenological comparison but The orbital angular momentum