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
the spin and orbital motion
of quarks and gluons
--- Chairman Mao
Actual practice: Challenge and Controversy
Elliot Leader (2011)
Jaffe-Manohar [NPB337:509 (1990)]
1
1
J total   d 3 x     d 3 xx      d 3 xE  A   d 3 xx  E iAi
2
i
Ji [PRL78:610 (1997)], Chen-Wang [CTP27:212 (1997)]
J total   d 3 x 

1
1
   d 3 xx   D   d 3 xx  E  B
2
i

Gauge
Invariance!
Chen-Lu-Sun-Wang-Goldman [PRL100:232002 (2008); 103:062001 (2009)]
1
1
i
J total   d 3 x     d 3 xx   Dpure   d 3 xE  Aphys   d 3 xx  E i Dpure Aphy
s
2
i
Wakamatsu [PRD81:114010(2010); 83:014012 (2011); 84:037501 (2011)]
J total   d 3 x 
1
1
a
   d 3 xx   D   d 3 xE  Aphys   d 3 xx  ( E i D pure Api hys  Aphys
a)
2
i
Outline (of lecture series)
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 finetuning in actual application
IV. Another complementary example:
graviton (spin-2 gauge particle)
V. Prospect
Related recent papers
1) 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
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)
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
1988-1996: Dark age, no gauge-invariance
J total
1
3
1
  d x
   d xx     d 3 xE  A   d 3 xE i x Ai
2
i

Sq

Lq

Sg

Lg
3

Jaffe-Manohar [NPB337:509 (1990)]
Concentration on quark spin,
the only gauge-invariant piece,
from ~0% to ~30%
1997: Manifestly gauge-invariant
decomposition of the nucleon spin
1
1
   d 3 xx      d 3 xE  A   d 3 xE i x  Ai
2
i
Sq

Lq

Sg

Lg
1) J total   d 3 x 

2) J total
Ptotal

1
3
1
  d x
   d xx  D   d 3 xx  E  B
2
i

Sq

L'q

J'g

3

1
3
3
1
  d x D   d xE  B   d x    d 3 xE i Ai
i
i
3

X. Ji, Phys. Rev. Lett. 78, 610 (1997)
X.S. Chen, F. Wang, Commun.Theor. Phys. 27:212 (1997)
1998: A delicate and appealing possibility:
gauge-invariant matrix element of gaugedependent operators in certain states
J total
1
3
1
  d x
   d xx     d 3 xE  A   d 3 xE i x Ai
2
i

Sq

Lq

Sg

Lg

3
Ptotal
1
  d x    d 3 xE i Ai  Pq  Pg
i
 L  [J
z
q
3
z
total

, O],
 P  [P , O ']
z
q
z
total
X.S. Chen, F. Wang, hep-ph/9802346: a path-integral proof
M. Anselmino, A. Efremov, E. Leader, Phys. Rep. 261:1 (1995).
Problem with the covariant derivative
Electron in
a magnetic
field

ˆ
ˆ
ˆ
 

1  1
PK  D    qA  PK  PK  iq  A  iqB
i
i
ˆ
ˆ
ˆ
ˆ
 1 
   
LK  r  D  LK  LK  iLK  ir [r  (  A)]
i
LK is not quantized, thus
does not help to solve/label
a quantum state
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).
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
The recent proof of Elliot Leader by
canonical quantization
Limitation to covariant quantization
in the covariant gauge!
E. Leader, PRD 83:096012 (2011)
2001-2007: Another miserable stage
Mixed use of different decompositions!
In both theory and experiments!
1) J total
2) J total
1
3
1
  d x
   d xx     d 3 xE  A   d 3 xE i x  Ai
2
i

Sq

Lq

Sg

Lg
3


1
3
1
  d x
   d xx  D   d 3 xx  E  B
2
i

Sq

L'q

J'g
3


A typical confusion: Sg~0, Lg~0, L’q~0,
then where is the nucleon spin?!
2008-2010: The field-separation method
Key Observation: Dual Role of the Gauge Field
   eig
1. Conpensate phase freedom of  : 
 A  A    
1 

L   i     igA   m   F F
4
2. Physical coupling to :
Physical decomposition of
the gauge field and its dual role


Decomposition: A  Aphys
 Apure


Aphys
and Apure
are to be expressed in terms of A




A
transforms
as
does
A
,
and
gives
zero
F
 pure
Desired goal:  

A
transform
covaria
ntly
as
does
F

 phys


A
 pure solely carries the pure-gauge degrees of freedom
Namely:  

 Aphys solely carries the physical degrees of freedom
Advantage (usage)
of the decomposition

 Apure
is used instead of A to construct covariant derivative
or the gauge link  Wilson line  to achieve gauge invariance

 Aphys
is used instead of F  as the canonical variable
Physical quantity = f(Aphys, Dpure,…)
Application: Consistent separation
of nucleon momentum and spin
L   d 3x r  P
van Enk, Nienhuis, J. Mod. Opt. 41:963 (1994)
Chen, Sun, Lü, Wang, Goldman, PRL 103:062001 (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
Its first moment gives the gauge-invariant local operator:
j
M g ij  F  i ij  Aphys
,
which is the + component of the gauge-invariant gluon spin
S g  E  Aphys
Physical separation of the Abelian Field:
Prescription
Boundary condition:
F | x   0, Aˆ  | x   0
Physical separation of the Abelian Field:
Solution
F | x   0,
Aˆ  | x   0
Physical separation of the Abelian Field:
Uniqueness


 1

ˆ
ˆ
  A  0  A  (   )  F
Initial condition required!
Aˆ z  0  Aˆ ( x, y, z, t )   Fz ( x, y, z ', t )dz '
z


Aˆ  0  Aˆ ( x, y,  ,  )   F  ( x, y,  ,  ')d  '


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
Aˆ z  0  Aˆ ( x, y, z, t )   Fz ( x, y, z ', t )dz '
z

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
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
The new controversy and Leader’s Criteria
Jaffe-Manohar [NPB337:509 (1990)]
1
1
J total   d 3 x     d 3 xx      d 3 xE  A   d 3 xx  E i Ai
2
i
Chen-Lu-Sun-Wang-Goldman [PRL100:232002 (2008); 103:062001 (2009)]
1
1
i
J total   d 3 x     d 3 xx   Dpure   d 3 xE  Aphys   d 3 xx  E i Dpure Aphys
2
i
Wakamatsu [PRD81:114010(2010); 83:014012 (2011); 84:037501 (2011)]
1
1
i
J total   d 3 x     d 3 xx   D   d 3 xE  Aphys   d 3 xx  ( E i D pure Aphys
 Apha ys  a )
2
i
Interacting theory:
Structure of Poincare generators
Lagrangian: L  La  Lb  Lint
"Good" generators
"bad" generators
 P  Pa  Pb
 H = H a  H b  H int


 J  J a  J b
 K  K a  Kb  Kint
Spatial translation and rotation are kinematic
Time translation and Lorentz boost are dynamic
Interacting theory:
Poincare (sub)algebra
[ J i , J (ja ,b ) ]  i ijk J (ka ,b )
 i j
k
Kinematic transformation [ J , P( a ,b ) ]  i ijk P( a ,b )
[ P i , P j ]  0
( a ,b )

[ K i , J aj,b ]  i ijk K ak,b
Dynamic transformation  i j
[ K , Pa ,b ]  iH a ,b ij
[ K i , J j ]  i ijk K k
Only total J and P are covariant: 
i
j
[
K
,
P
]  iH  ij

Generators for the gauge-invariant
physical fields - translation
Generators for the gauge-invariant
physical fields - Rotation
The quark-gluon system
Generator for the gaugeinvariant quark field
Generator for the gaugeinvariant gluon field
Some detail in the proof
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
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




3
J ' g   d x r  ( E  B)
0
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
Canonical expression of spin and OAM
Complete tensor gauge conditions
Vanishing of angular momentum for a
stationary tensor gauge field
No spurious timedependence
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 (offforward) 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
l=1
m=1
e
B
LY11
ikr
i
E    B  i A
k
E Flux
EB
J Flux
E  A  Ei x Ai

(rad. gauge)
Dipole rad.
ikr
x EB

dP
dJ z
dJ z
d
 1  cos  
d
 1  cos  
d
2
2
 2sin 2 
Hadron physics is the best subject
to educate people
--- Chairman Mao