1. Cédric Lorcé IFPA Liège ECT* Colloquium: Introduction to quark and gluon angular momentum August 25, 2014, ECT*, Trento, Italy Spin and Orbital Angular Momentum of Quarks and Gluons in the Nucleon

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. Structure of matter 10 -14 m10 -15 m10 -18 m10 -10 m d u AtomNucleusNucleonsQuarks Atomic physics Nuclear physics Hadronic physics Particle physics Proton Neutron Up Down

5. Structure of nucleons Our picture/understanding of the nucleon evolves ! But many questions remain unanswered … Where does the proton spin come from ? How are quarks and gluons distributed inside the nucleon ? What is the proton size ? Why are quarks and gluons confined ? How are constituent quarks related to QCD ? …

6. Angular momentum decomposition SqSq SgSg LgLg LqLq SqSq SgSg LgLg LqLq SqSq JgJg LqLq Many questions/issues : Frame dependence ? Gauge invariance ? Uniqueness ? Measurability ? … Review: Dark spin Quark spin ? ~ 30 % ? ? ? [Leader, C.L. (2014)]

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

8. 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

9. 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

10. 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

11. Back to basics AM decomposition is a complicated story Let’s have a glimpse …

12. Back to basics Classical mechanics Free pointlike particle Total AM is conserved but not unique !

13. Back to basics Classical mechanics Free composite particle CM motion can be separated

14. Back to basics Classical mechanics Internal AM Conventional choice : Option 2 with Boost invariance Uniqueness Option 1 : Option 2 : Boost invariance Uniqueness The quantity is boost-invariant BUT its physical interpretation is simple only in the CM frame !

15. Frame Frame-dependent quantity (e.g. ) Boost-invariant extension (BIE) Back to basics Classical mechanics

16. Frame BIE1 Frame-dependent quantity (e.g. ) «Natural» frames Boost-invariant extension (BIE) Back to basics Classical mechanics CM (e.g. )

17. Frame BIE1 BIE2 Frame-dependent quantity (e.g. ) «Natural» frames Boost-invariant extension (BIE) Back to basics Classical mechanics CM (e.g. )

18. Back to basics Classical electrodynamics Charged pointlike particle in external magnetic field AM conservation ???

19. Back to basics Charged pointlike particle in external magnetic field Kinetic and canonical AM are different «Hidden» kinetic AM Conserved canonical AM System = matter + radiation Ambiguous ! Classical electrodynamics

20. Back to basics Quantum mechanics Pointlike particle at rest has intrinsic AM (spin) In general, only is conserved AM is quantized All components cannot be simultaneously measured

21. Back to basics Composite particle at rest Quantum average Expectation values are in general not quantized Quantum mechanics

22. Back to basics Special relativity Lorentz boosts do not commute Spin uniquely defined in the rest frame only ! Rest frame Moving frame «Standard» boost

23. Back to basics Special relativity Relativistic mass is frame-dependent No (complete) separation of CM coordinates from internal coordinates ! Lorentz contraction Relativity of simultaneity

24. Frame-dependent quantity (e.g. ) Frame LIE1 LIE2 «Natural» frames Lorentz-invariant extension (LIE) Back to basics Rest (e.g. ) Special relativity

25. 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

26. Back to basics Gauge theory Analogy with integration «Gauge» 1«Gauge» 2 RiemannLebesgue Which one is «physical» ? Some would say : Others would say: None! Only the total area under the curve makes sense Both! Choosing one or another is a matter of convenience

27. Back to basics 3 strategies : 1)Consider only simple (local) gauge-invariant quantities 2)Relate these quantities to observables 3)Try to find an interpretation (optional) Gauge theory 1)Fix the gauge 2)Consider quantities with simple interpretation 3)Try to find the corresponding observables 1)Define new complicated (non-local) gauge-invariant quantities 2)Consider quantities with simple interpretation 3)Try to find the corresponding observables

28. Gauge non-invariant quantity (e.g. ) Gauge GIE1 GIE2 «Natural» gauges Gauge-invariant extension (GIE) Back to basics Coulomb (e.g. ) Gauge theory [Dirac (1955)]

29. Infinitely many GIEs Back to basics Gauge theory […] one can generalize a gauge variant nonlocal operator […] to more than one gauge invariant expressions, raising the problem of deciding which is the “true” one. [Bashinsky, Jaffe (1998)] In other words, the gauge-invariant extension of the gluon spin in light-cone gauge can be measured. Note that one can easily find gauge-invariant extensions of the gluon spin in other gauges. But we may not always find an experimental observable which reduces to the gluon spin in these gauges. Uniqueness issue [Hoodbhoy, Ji (1999)] Some GIEs are nevertheless measurable

30. 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

31. luon Spin decompositions in a nutshell Kinetic uark luon Canonical uark luon Decomposition? uark

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

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

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

35. [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 !

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

37. Parton correlators General non-local quark correlator

38. Parton correlators Gauge transformation Gauge invariant but path dependent

39. Partonic interpretation Phase-space «density» 2+3D Longitudinal momentum Transverse momentum Transverse position [Ji (2003)] [Belitsky, Ji, Yuan (2004)] [C.L., Pasquini (2011)]

40. [C.L., Pasquini, Xiong, Yuan (2012)] [Hatta (2012)] Example : canonical OAM « Vorticity » Spatial distribution of average transverse momentum

41. Parton distribution zoo 2+3D [C.L., Pasquini, Vanderhaeghen (2011)] GTMDs Theoretical tools Phase-space (Wigner) distribution

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

43. 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

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

45. Asymmetries 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

46. 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 (2012)] [C.L., Pasquini (2011)] [C.L., Pasquini, Xiong, Yuan (2012)] [Kanazawa, C.L., Metz, Pasquini, Schlegel (2014)]

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

48. Summary We all agree on total angular momentum We disagree on its decomposition (matter of convention ?) Observables are gauge invariant but physical interpretation need not Scattering on nucleon is sensitive to AM

49. Summary Nucleon FFsPDFsTMDsGPDs GTMDs LFWFs DPDs

50. Backup slides

51. 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

52. Back to basics Quantum optics Photons have only 2 polarization (helicity) states Twisted light carry OAM

53. We measure frame-dependent quantities Then combine them in a frame-independent way And finally interpret in a special frame Back to basics Special relativity The proper length of a pencil is clearly frame independent. When we say the length of a house in the frame v = 0.9999c is the same as the proper length of the pencil, we are not saying that the length of the house is frame-independent. Rather, we are saying that the length of the house in a special frame can be known from measuring a frame-independent quantity. v [Hoodbhoy, Ji (1999)]

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

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

56. Stueckelberg symmetry Non-local ! Decomposition is path-dependent ! Path dependenceStueckelberg non-invariance ? [Hatta (2012)] [C.L. (2013)]

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

58. 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

59. Stueckelberg symmetry Degrees of freedom [C.L. (2014)] Classical Non-dynamical Quantum Dynamical plays the role of a background field ! PassiveActive

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

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

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

63. [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

64. Parametrization GTMDs TMDsGPDs Nucleon polarization Quark polarization [Meissner, Metz, Schlegel (2009)] [C.L., Pasquini (2013)] Quarks & gluons Complete parametrizations : Quarks Twist-2

65. Energy-momentum tensor A lot of interesting physics is contained in the EM tensor Energy density Momentum density Energy flux Momentum flux Shear stress Normal stress (pressure) [Polyakov, Shuvaev (2002)] [Polyakov (2003)] [Goeke et al. (2007)] [Cebulla et al. (2007)] In rest frame

66. Energy-momentum tensor In presence of spin density In rest frame No « spin » contribution ! Belinfante « improvement » Spin density gradientFour-momentum circulation

67. QCD Energy-momentum operator Matrix elements Normalization Energy-momentum tensor

68. Energy-momentum FFs Momentum sum rule Angular momentum sum rule [Ji (1997)] Vanishing gravitomagnetic moment ! Energy-momentum tensor

69. Energy-momentum FFs Momentum sum rule Angular momentum sum rule [Ji (1997)] Vanishing gravitomagnetic moment ! Non-conserved current Energy-momentum tensor

70. Leading-twist component of Link with GPDs [Ji (1997)] Accessible e.g. in DVCS ! Energy-momentum tensor