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 U @ Chengdu → Huazhong U of Sci. & Tech.@ Wuhan 四川大学@成都 → 华中科技大学@武汉 X.S. Chen, X.F. Lü, W.M. Sun, F. Wang, and T. Goldman Phys. Rev. Lett. 100, 232002 (2008) X.S. Chen, W.M. Sun, X.F. Lü, F. Wang, and T. Goldman Phys. Rev. Lett. 103, in press (2009)

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

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

Mixed use of different spin decompositions

Mismatch between momentum and angular momentum

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

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

Physical decomposition of the gauge field and its dual role

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

Example: Consistent separation of nucleon momentum and spin

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

Explicit construction: Express Apure and Aphys in terms of A

Yang-Mills Theory: non-Abelian field

Solution by power expansion

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

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

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,…)

Gravity: Einstein’s Theory

Physical quantity and conservation law

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

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

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

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

Break down of PCAC argument for pion decay suppression

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

Consistent separation of nucleon momentum and spin

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:

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

The conventional gluon PDF Relates to the Poynting vector:

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

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

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

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

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

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

Thanks