Poincare sub-algebra and gauge invariance in nucleon structure Xiang-Song Chen Huazhong University of Science & Technology 陈相松 华中科技大学 武汉 10 July KITPC-Beijing

I.Controversy in nucleon (spin) structure II.Elliot Leader’s criteria of separating momentum and angular momentum III.Reconciling Poincare sub-algebra with gauge invariance IV.A practical thinking about nucleon structure V.A critical thinking about gauge invariance Outline

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

Controversy in nucleon spin structure Leader [PRD 83: (2011)]

Interacting theory: Structure of Poincare generators Without gauge symmetry, the issue is trivial:

Interacting theory: Poincare (sub)algebra

Further criteria by Elliot Leader Corollary

Examination of various decompositions by Leader’s criteria Leader [PRD 83: (2011)]

Generators for the gauge-invariant physical fields - translation

Generators for the gauge-invariant physical fields - Rotation

The quark-gluon system

Generator for the gauge- invariant quark field

Generator for the gauge- invariant gluon field

Some detail in the proof

 Hint from a forgotten practice: Why photon is ignored for atomic spin?  The fortune of choosing Coulomb gauge  Quantitative differences  Another example: momentum of a moving atom and nucleon A practical thinking about nucleon structure

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

The atom as a whole

Close look at the photon contribution The static terms!

Justification of neglecting photon field

A critical gap to be closed

The same story with Hamiltonian

The fortune of using Coulomb gauge

Gauge-invariant revision – Angular Momentum

Gauge-invariant revision -Momentum and Hamiltonian

The covariant scheme spurious photon angular momentum

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

Beyond the static approximation

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

Quark and gluon momentum in the nucleon

Weinberg’s approach: derivation of QED with physical photons S. Weinberg, Phys. Rev. 138 (1965) B988

The non-covariant propagator of Physical photon

A delicate point: the contact term and its effect

Cancelation of the contact term

Is gauge-invariance a “Compromise”, or even “illusion”?  First step in Physics ： Complete Description  Classic Physics: r and p （ controllable ）  Quantum Mechanics ： Wave Function （ Not completely controllable ）  Gauge Theory ： Gauge potentials (Completely uncontrollable ） Need for the physical variable: Real emergence of a photon

A possible real difference Dipole rad. (rad. gauge) l=1 m=1 E Flux J Flux

If we never need physical gluons …… Then QCD is a true gauge theory, and the only try gauge theory so far known And all quark and gluon quantities are a matter of definition

Do we sometimes need physical gluons? Probably, in early universe Then color gauge invariance may also be an illusion!

What about SU(2) X U(1) and Higgs?

Derivation of non-Abelian gauge theory with physical gluons by requiring Lorentz invariance??? Derivation of QED with physical photons by requiring Lorentz invariance

I.Nucleon spin and momentum can be separated gauge invariantly, with quark/gluon part acting as the rotation and translation generators for the physical quark/gluon field. II.If adopting the naive free-form expression, Coulomb gauge gives the simplest pictures for atomic and nucleon structure. III.Gauge symmetry might be an illusion. QED can be derived from physical photons by requiring Lorentz invariance of S matrix, but the situation for non- Abelian theory is more tricky and not yet proven. demonstrated. Summary

Thank you! 谢谢 !