1. Sichuan U @ Chengdu → Huazhong U of Sci. & Tech.@ Wuhan
Hadron Structure in Terms of Physical Fields Towards a “Good” Structure of the Nucleon
Xiang-Song Chen
陈相松
Sichuan Chengdu → Huazhong U of Sci. & Wuhan →
X.S. Chen, X.F. Lü, W.M. Sun, F. Wang, and T. Goldman
Phys. Rev. Lett. 100, (2008)
X.S. Chen, W.M. Sun, X.F. Lü, F. Wang, and T. Goldman
Phys. Rev. Lett. 103, in press (2009)

2. Outline
Some disharmonies in hadron structure studies (due to gauge invariance)
An efficient and economic reconciliation: Physical decomposition of the gauge field
Gauge invariant gluon spin and polarized gluon distribution function Δg(x)
Gauge invariant and scale-invariant quark spin (quark axial vector current)
Gauge invariant Chern-Simons current
Towards a coherent picture of the nucleon

3. Measurements of nucleon spin structure
 Promising?
 Incoherent investigations!!!

4. Mixed use of different spin decompositions

5. Mismatch between momentum and angular momentum

6. The common obstacle: Gauge invariance
If only without the gauge-invariance restrictions…

7. Key Observation: Dual Role of the Gauge Field
A clever way out …
Key Observation: Dual Role of the Gauge Field

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

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

10. Example: Consistent separation of nucleon momentum and spin

11. Simple example: Abelian gauge field
Criteria: 1) unique solution 2) desired transformation

12. Explicit construction: Express Apure and Aphys in terms of A

13. Yang-Mills Theory: non-Abelian field

14. Solution by power expansion

15. Some clarifications to avoid confusion
Independence from gauge condition f(A)=0 and canonical quantization
Locality: Aphys (x)=f[A(x)]
Lorentz covariance: ∂iAiphys=0
Comparison with gauge-fixing approach

16. Deep analogy between Yang-Mills theory and Gravitation
Comparison
Yang-Mills
Gravitation
Field equation
Gluon charge & gravity energy
Ordinary conservation law
Covariant derivative
Covariant conservation law

17. Decompose the field and its dual role
Gauge Theory
Gravitation
A=Apure +Aphys
Γ= Γpure+ Γphys
Apure generates no
field strength
Γpure generates no
curvature
Dpure=∂ + Apure
Dpure=∂ + Γpure
Aphys is
gauge-covariant
Γphys is
coordinate-covariant
Physical quantity
= f(Aphys,Dpure,…)
= f(Γphys,Dpure,…)

18. Gravity: Einstein’s Theory

19. Physical quantity and conservation law

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

21. Quark spin (quark axial vector current) and the axial anomaly

22. Gauge invariant and scale-invariant quark spin (quark axial vector current)

23. Gauge invariant Chern-Simons Current and its implications
Contribution to OPE and g1 sum rule?

24. Break down of PCAC argument for pion decay suppression

25. Towards a coherent nucleon structure
Proper definition of (polarized) quark/gluon PDFs related to consistent quark/gluon momentum and angular momentum
Theoretical calculation of the PDFs and
their moments
Examination of factorization formulae and
experimental measurement of the PDFs

26. Consistent separation of nucleon momentum and spin

27. 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:

28. The novel “proper” quark PDF
With a second moment:

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

30. Relates to the gauge-invariant canonical gluon momentum
The novel gluon PDF
Relates to the gauge-invariant canonical gluon momentum

31. To measure the novel 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

32. Example of theoretical calculation: Quark/gluon momentum in the nucleon
Gluons carry about half of the nucleon momentum in the asymptotic limit

33. Partition of gauge-invariant proper momentum
“Do gluons carry half of the nucleon momentum?” X.S. Chen, W.M. Sun, X.F. Lü, F. Wang, and T. Goldman Phys. Rev. Lett., scheduled to appear the next week

34. Summary Dual role of the gauge field
A basis for consistent and systematic investigation of nucleon internal structure
Gluon spin is indeed meaningful and measurable
The nucleon can be quark-dominated

35. Prospect Computation of asymptotic partition of nucleon spin
Reanalysis of the measurements of unpolarized quark and gluon PDFs
Reanalysis and further measurements of polarized gluon distributions
Orbital contributions, Generalized Parton Distributions, and Deeply Virtual Compton Scattering

36. Thanks