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Gluon Spin and OAM with Different Definitions

INT Workshop Feb 6-17, 2012 Orbital Angular Momentum in QCD. Gluon Spin and OAM with Different Definitions. Xiang-Song Chen Huazhong University of Science & Technology 陈相松 • 华中科技大学 • 武汉. A universally correct statement for the nucleon spin. Nucleon spin comes from

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Gluon Spin and OAM with Different Definitions

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

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

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

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

  5. Related recent papers • 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. I. Chief theoretical framework and key issues (uniqueness, applicability) • 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)

  7. History of theoretical efforts: Brief Review • 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

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

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

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

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

  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. Questioning the path-integral proof of gauge-invariant matrix element for gauge-dependent operators---continued 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). The common practices can be wrong: Averaging over the gauge group; Interchange of the integration order

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

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

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

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

  18. Advantage (usage) of the decomposition Physical quantity = f(Aphys, Dpure,…)

  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. II. Leader’s criteria of separating momentum and angular momentum • 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

  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. III. The issue of convenience and fine-tuning in actual application • 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

  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

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