1. Gluon Spin and OAM with Different Definitions INT Workshop Feb 6-17, 2012 Orbital Angular Momentum in QCD Xiang-Song Chen Huazhong University of Science & Technology 陈相松 华中科技大学 武汉

2. Nucleon spin comes from the spin and orbital motion of quarks and gluons --- Chairman Mao A universally correct statement for the nucleon spin

3. Actual practice: Challenge and Controversy Gauge Invariance! Elliot Leader (2011)

4. I.Chief theoretical framework and key issues (uniqueness, applicability) II.Leader’s criteria of separating momentum and angular momentum III.The issue of convenience and fine- tuning in actual application IV.Another complementary example: graviton (spin-2 gauge particle) V.Prospect Outline (of lecture series)

5. Related recent papers 1)Art of spin decomposition Xiang-Song Chen, Wei-Min Sun, Fan Wang, T. Goldman, Phys. Rev. D 83, 071901(R) (2011). 2) Proper identification of the gluon spin Xiang-Song Chen, Wei-Min Sun, Fan Wang, T. Goldman, Phys. Lett. B 700, 21 (2011). 3) Physical decomposition of the gauge and gravitational fields Xiang-Song Chen, Ben-Chao Zhu, Phys. Rev. D 83, 084006 (2011). 4) Spin and orbital angular momentum of the tensor gauge field. Xiang-Song Chen, Ben-Chao Zhu, Niall Ó Murchadha, arXiv:1105.6300

6.  Review of the theoretical efforts  Uniqueness of separating a gauge field into physical and pure-gauge components.  The prescription for actual application  The non-Abelian gluon field  Short summary of added contributions (compared to the familiar separation of a vector field) I. Chief theoretical framework and key issues (uniqueness, applicability)

7.  1988-1996: Dark age, no gauge-invariance  1997-2000: Two approaches towards gauge- invariance: Operator/Matrix Element  2001-2007: Another miserable stage  2008-2010: The field-separation method  2011: Revival of the naïve canonical approach by Elliot Leader  2012: Reconciliation of Leader’s Criteria with gauge-invariance at operator level History of theoretical efforts: Brief Review

8. 1988-1996: Dark age, no gauge-invariance Concentration on quark spin, the only gauge-invariant piece, from ~0% to ~30%

9. X. Ji, Phys. Rev. Lett. 78, 610 (1997) X.S. Chen, F. Wang, Commun.Theor. Phys. 27:212 (1997) 1997: Manifestly gauge-invariant decomposition of the nucleon spin

10. X.S. Chen, F. Wang, hep-ph/9802346: a path-integral proof M. Anselmino, A. Efremov, E. Leader, Phys. Rep. 261:1 (1995). 1998: A delicate and appealing possibility: gauge-invariant matrix element of gauge- dependent operators in certain states

11. Problem with the covariant derivative L K is not quantized, thus does not help to solve/label a quantum state Electron in a magnetic field

12. Questioning the path-integral proof of gauge-invariant matrix element for gauge-dependent operators Explicit counter example by perturbative calculation P. Hoodbhoy, X. Ji, W. Lu, PRD 59:074010 (1999); P. Hoodbhoy, X. Ji, PRD 60, 114042 (1999).

13. Revealing the unreliability of the utilized conventional path-integral approach X.S. Chen, W.M. Sun, F. Wang, JPG 25:2021 (1999). W.M. Sun, X.S. Chen, F. Wang, PLB483:299 (2000); PLB 503:430 (2001). Questioning the path-integral proof of gauge-invariant matrix element for gauge-dependent operators---continued The common practices can be wrong: Averaging over the gauge group; Interchange of the integration order

14. Limitation to covariant quantization in the covariant gauge! E. Leader, PRD 83:096012 (2011) The recent proof of Elliot Leader by canonical quantization

15. Mixed use of different decompositions! In both theory and experiments! 2001-2007: Another miserable stage A typical confusion: S g ~0, L g ~0, L’ q ~0, then where is the nucleon spin?!

16. Key Observation: Dual Role of the Gauge Field 2008-2010: The field-separation method

17. Physical decomposition of the gauge field and its dual role

18. Advantage (usage) of the decomposition Physical quantity = f(A phys, D pure,…)

19. Application: Consistent separation of nucleon momentum and spin van Enk, Nienhuis, J. Mod. Opt. 41:963 (1994) Chen, Sun, Lü, Wang, Goldman, PRL 103:062001 (2008)

20. The conventional gauge-invariant “quark” PDF The gauge link (Wilson line) restores gauge invariance, but also brings quark-gluon interaction, as also seen in the moment relation:

21. The modified quark PDF With a second moment:

22. The conventional gluon PDF Relates to the Poynting vector:

23. Gauge-invariant polarized gluon PDF and gauge-invariant gluon spin

24. Physical separation of the Abelian Field: Prescription

25. Physical separation of the Abelian Field: Solution

26. Physical separation of the Abelian Field: Uniqueness

27. Physically controllable boundary conditions: Vanishing at a finite surface within a certain accuracy Open surfaces: Well-defined mathematically, ill-defined physically!!!

28. Closer look at the distinct behaviors Open boundary: The field persists constantly to infinity

29. Separation of non-Abelian field

30. Perturbative solution

31. The explicit expressions

32. Short summary of the contributions added (compared to the familiar separation of a vector field)  A four-dimensional formulation including time-component  The generalization to non-Abelian field  The pure-gauge covariant derivative  Clarification on the impossibility of distinct extension

33.  The new controversies and Leader’s compelling criteria  Recalling the Poincare algebra and subalgebra for and interacting system  Generators for the physical fields: QED  The quark-gluon system II. Leader’s criteria of separating momentum and angular momentum

34. The new controversy and Leader’s Criteria

35. Interacting theory: Structure of Poincare generators

36. Interacting theory: Poincare (sub)algebra

37. Generators for the gauge-invariant physical fields - translation

38. Generators for the gauge-invariant physical fields - Rotation

39. The quark-gluon system

40. Generator for the gauge- invariant quark field

41. Generator for the gauge- invariant gluon field

42. Some detail in the proof

43.  Hint from a forgotten practice: Why photon is ignored for atomic spin?  The fortune of choosing Coulomb gauge  Quantitative differences  Fine-tuning for the gluon spin and OAM III. The issue of convenience and fine-tuning in actual application

44. Hint from a forgotten practice: Why photon is ignored for atomic spin? Do these solution make sense?!

45. The atom as a whole

46. Close look at the photon contribution The static terms!

47. Justification of neglecting photon field

48. A critical gap to be closed

49. The same story with Hamiltonian

50. The fortune of using Coulomb gauge

51. Momentum of a moving atom A stationary electromagnetic field carries no momentum

52. Gauge-invariant revision – Angular Momentum

53. Gauge-invariant revision -Momentum and Hamiltonian

54. The covariant scheme spurious photon angular momentum

55. Gluon angular momentum in the nucleon: Tree-level One-gluon exchange has the same property as one-photon exchange

56. Beyond the static approximation

57. Fine-tuning for the gluon spin and OAM Possible convergence in evolution

58. Another complementary example: graviton (spin-2 gauge particle)

59. The tensor gauge field

60. Canonical expression of spin and OAM

62. Complete tensor gauge conditions

63. Vanishing of angular momentum for a stationary tensor gauge field No spurious time- dependence

64. The same property of momentum

65. Prospect of measuring the new quantities  The same experiments as to “measure” the conventional PDFs  New factorization formulae and extraction of the new PDFs  Quark and gluon orbital angular momentum can in principle be measured through generalized (off- forward) PDFs

66. Reminder on the goal of studying nucleon structure The ultimate goal ： A complete description of the nucleon Completeness ： sufficiency in predicting all reaction involving nucleon Intermediate goal: to learn from the nucleon internal dynamics by looking at the origins of mass, momentum, spin, magnetic moment, etc.

67. Possibly a real final solution Dipole rad. (rad. gauge) l=1 m=1 E Flux J Flux

68. Hadron physics is the best subject to educate people --- Chairman Mao