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

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

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

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)

Related recent papers 1)Art of spin decomposition Xiang-Song Chen, Wei-Min Sun, Fan Wang, T. Goldman, Phys. Rev. D 83, (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, (2011). 4) Spin and orbital angular momentum of the tensor gauge field. Xiang-Song Chen, Ben-Chao Zhu, Niall Ó Murchadha, arXiv:

 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)

 : Dark age, no gauge-invariance  : Two approaches towards gauge- invariance: Operator/Matrix Element  : Another miserable stage  : 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

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

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

X.S. Chen, F. Wang, hep-ph/ : 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

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

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: (1999); P. Hoodbhoy, X. Ji, PRD 60, (1999).

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

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

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

Key Observation: Dual Role of the Gauge Field : The field-separation method

Physical decomposition of the gauge field and its dual role

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

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

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 modified quark PDF With a second moment:

The conventional gluon PDF Relates to the Poynting vector:

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

Physical separation of the Abelian Field: Prescription

Physical separation of the Abelian Field: Solution

Physical separation of the Abelian Field: Uniqueness

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

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

Separation of non-Abelian field

Perturbative solution

The explicit expressions

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

 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

The new controversy and Leader’s Criteria

Interacting theory: Structure of Poincare generators

Interacting theory: Poincare (sub)algebra

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  Fine-tuning for the gluon spin and OAM III. The issue of convenience and fine-tuning in actual application

Hint from a forgotten practice: Why photon is ignored for atomic spin? Do these solution 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

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

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

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

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

The tensor gauge field

Canonical expression of spin and OAM

Complete tensor gauge conditions

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

The same property of momentum

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

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.

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

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