1. Cédric Lorcé SLAC & IFPA Liège How to define and access quark and gluon contributions to the proton spin December 2, 2014, IIT Bombay, Bombay, India INTERNATIONAL WORKSHOP ON FRONTIERS OF QCD IIT BOMBAY DECEMBER 2-5, 2014

2. Outline What is it all about ? Why is there a controversy ? How can we measure AM ?

3. Outline What is it all about ? Why is there a controversy ? How can we measure AM ?

4. Angular momentum decomposition SqSq SgSg LgLg LqLq SqSq SgSg LgLg LqLq SqSq JgJg LqLq Many questions/issues : Frame dependence ? Gauge invariance ? Uniqueness ? Measurability ? … Reviews: Dark spin Quark spin ? ~ 30 % ? ? ? [Leader, C.L. (2014)] [Wakamatsu (2014)] ~ 20 % [de Florian et al. (2014)]

5. In short … Noether’s theorem : Continuous symmetry Translation invariance Rotation invariance Conserved quantity Total (linear) momentum Total angular momentum We all agree on the total quantities BUT … We disagree on their decomposition

6. In short … 3 viewpoints : Meaningless, unphysical discussions No unique definition ill-defined problem There is a unique «physical» decomposition Missing fundamental principle in standard approach Matter of convention and convenience Measured quantities are unique BUT physical interpretation is not unique

7. Outline What is it all about ? Why is there a controversy ? How can we measure AM ?

8. Spin decompositions in a nutshell [Jaffe, Manohar (1990)][Ji (1997)] SqSq SgSg LgLg LqLq SqSq LqLq JgJg CanonicalKinetic Gauge non-invariant ! « Incomplete »

9. Gluon spin Gluon helicity distribution [de Florian et al. (2014)] « Measurable », gauge invariant but complicated

10. Gluon spin [Jaffe-Manohar (1990)] Light-front gauge Gluon helicity distribution Simple fixed-gauge interpretation « Measurable », gauge invariant but complicated

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

12. Spin decompositions in a nutshell [Jaffe, Manohar (1990)][Ji (1997)] SqSq SgSg LgLg LqLq SqSq LqLq JgJg CanonicalKinetic Gauge non-invariant ! « Incomplete »

13. Spin decompositions in a nutshell [Chen et al. (2008)][Wakamatsu (2010)] SqSq SgSg LgLg LqLq SqSq LqLq LgLg CanonicalKinetic SgSg Gauge-invariant extension (GIE)

14. Spin decompositions in a nutshell [Chen et al. (2008)][Wakamatsu (2010)] SqSq SgSg LgLg LqLq SqSq LqLq CanonicalKinetic SgSg Gauge-invariant extension (GIE) LgLg

15. [Wakamatsu (2010)][Chen et al. (2008)] 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 !

16. Stueckelberg symmetry Geometrical interpretation [Hatta (2012)] [C.L. (2013)]

17. Stueckelberg symmetry Geometrical interpretation Fixed reference point [Hatta (2012)] [C.L. (2013)]

18. Stueckelberg symmetry Geometrical interpretation Fixed reference point Non-local ! [Hatta (2012)] [C.L. (2013)]

19. Gluon spin [Jaffe-Manohar (1990)] Light-front gauge Gluon helicity distribution Local fixed-gauge interpretation « Measurable », gauge invariant but non-local

20. 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

21. Outline What is it all about ? Why is there a controversy ? How can we measure AM ?

22. Asymmetries in QCD Example : SIDIS [Mulders, Tangermann (1996)] [Boer, Mulders (1998)] [Bacchetta et al. (2004)] [Bacchetta et al. (2007)] [Anselmino et al. (2011)] Angular modulations of the cross section are sensitive to AM

23. Parton distribution zoo [C.L., Pasquini, Vanderhaeghen (2011)] TMDs FFsPDFs Charges GPDs

24. Parton distribution zoo [C.L., Pasquini, Vanderhaeghen (2011)] GTMDs TMDs FFsPDFs Charges GPDs «Physical» objects Theoretical tools LFWFs

25. Parton distribution zoo 2+1D 2+0D 0+3D 0+1D 2+3D [C.L., Pasquini, Vanderhaeghen (2011)] GTMDs TMDs FFsPDFs Charges GPDs «Physical» objects Theoretical tools Phase-space (Wigner) distribution

26. Parton correlators Gauge invariant but path dependent 2+3D Longitudinal momentum Transverse momentum Transverse position [Ji (2003)] [Belitsky, Ji, Yuan (2004)] [C.L., Pasquini (2011)] Phase-space «density»

27. Light-front quark model results [C.L., Pasquini (2011)]

28. [C.L. et al. (2012)] [Hatta (2012)] Example : canonical OAM « Vorticity » Spatial distribution of average transverse momentum

29. Kinetic vs canonical OAM Quark naive canonical OAM (Jaffe-Manohar) Model-dependent ! Kinetic OAM (Ji) but No gluons and not QCD EOM ! Pure twist-3 Canonical OAM (Jaffe-Manohar) [C.L., Pasquini (2011)] [C.L. et al. (2012)] [Kanazawa et al. (2014)] [Mukherjee et al. (2014)] [Ji (1997)] [Penttinen et al. (2000)] [Burkardt (2007)] [Efremov et al. (2008,2010)] [She, Zhu, Ma (2009)] [Avakian et al. (2010)] [C.L., Pasquini (2011)]

30. Lattice results CIDI [Deka et al. (2013)]

31. Summary We all agree on total angular momentum but we disagree on its decomposition (matter of convention ?) Observables are gauge invariant but physical interpretation need not Information about AM is encoded in polarized parton distributions Reviews: [Leader, C.L. (2014)] [Wakamatsu (2014)]

32. Summary Nucleon FFsPDFsTMDsGPDs GTMDs LFWFs DPDs

33. Backup slides

34. Semantic ambiguity Path Stueckelberg Background Observables Quasi-observables « measurable » Quid ? « physical » « gauge invariant » Measurable, physical, gauge invariant and local « Measurable », « physical », gauge invariant and non-local Expansion scheme E.g. cross-sections E.g. parton distributions -dependent E.g. collinear factorization

35. Gauge fixing GIE1 GIE2 « Natural » gauges Lorentz-invariant extensions ~ Rest Center-of-mass Infinite momentum « Natural » frames Stueckelberg symmetry Gauge non-invariant operator Stueckelberg fixing [C.L. (2013)]

36. CanonicalKinetic 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)]

37. Two different approaches Lagrangian Hamiltonian Time Space Lorentz covariance Physical interpretation ManifestNot manifest ComplicatedSimple

38. Two different approaches Stueckelberg invariant Stueckelberg fixed Physical dofs Gauge dof Gauge invariance Physical interpretation LocalNon-local ComplicatedSimple

39. Stueckelberg symmetry Non-local color phase factor Path dependenceStueckelberg non-invariance Path-dependent Path-independent [C.L. (2013)]

40. Photon spin and OAM Should we be happy with ? Well… for a circularly polarized plane wave travelling along z Two descriptions related by a non-zero surface term !

41. Photon spin and OAM Should we be happy with ? [Ghai et al. (2009)] Single-slit experiment

42. Photon spin and OAM Should we be happy with ? [O’Neil et al. (2002)] [Garcés-Chavéz et al. (2003)] Optically trapped microscopic particle

43. Back to basics Gauge theory Gauge invariant Gauge non-invariant […] in QCD we should make clear what a quark or gluon parton is in an interacting theory. The subtlety here is in the issue of gauge invariance: a pure quark field in one gauge is a superposition of quarks and gluons in another. Different ways of gluon field gauge fixing predetermine different decompositions of the coupled quark-gluon fields into quark and gluon degrees of freedom. [Bashinsky, Jaffe (1998)] A choice of gauge is a choice of basis

44. Back to basics Time dependence and interaction Forms of dynamics Scale and scheme dependence Should Lorentz invariance be manifest ? Quantum gauge transformation Surface terms Evolution equation How are different GIEs related ? Should the energy-momentum tensor be symmetric ? Topological effects ? Longitudinal vs transverse … As promised, it is pretty complicated … Additional issues

45. Canonical formalism Dynamical variables Lagrangian [C.L. (2013)] Starting point Energy- momentum Gauge invariant ! Generalized angular momentum Conserved tensors Gauge covariant Translation invariance Lorentz invariance

46. Canonical formalism Dynamical variables Lagrangian Starting point Energy- momentum Gauge invariant ! Generalized angular momentum Conserved tensors Gauge invariant Translation invariance Lorentz invariance Dirac variables Dressing field [Dirac (1955)] [Mandelstam (1962)] [Chen (2012)] [C.L. (2013)]

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

48. Back to basics Special relativity Different foliations of space-time Instant-form dynamics Light-front form dynamics [Dirac (1949)] «Space» = 3D hypersurface «Time» = hypersurface label Light-front components Time Space Energy Momentum

49. PassiveActive « Physical » « Background » Active x (Passive) -1 Stueckelberg Stueckelberg symmetry Quantum Electrodynamics Phase in internal space

50. Light-front wave functions (LFWFs) Fock expansion of the nucleon state Probability associated with the Fock states Momentum and angular momentum conservation gauge

51. [C.L., Pasquini, Vanderhaeghen (2011)] ~ Overlap representation Light-front wave functions (LFWFs) GTMDs MomentumPolarization

52. [C.L., Pasquini, Vanderhaeghen (2011)] Light-front wave functions (LFWFs) Light-front quark models Wigner rotation Light-front helicity Canonical spin SU(6) spin-flavor wave function