Gluon Spin and OAM with Different Definitions

INT Workshop Feb 6-17, 2012Orbital 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

t3 3 3 3

3 3

otal

tota3

l

Jaffe-Manohar [NPB337:509 (1990)]

Ji [PRL78:610 (1997)], Chen-Wang [CTP27:

1 1

2

1

212 (1997)]

Chen

1

2

-Lu-S

i id x d x dx E A x E Ai

x D x E B

J x d x

d x d x d xi

J

3 3pure phys pure phy

3 3total s

t

un-Wang-Goldman [PRL100:232002 (2008); 103:062001 (2009)]

Wakamatsu [PRD81:114010(2010); 83:014012 (2011); 84:037501

1 1D

2

(2011)]

i ix D E A x E Ai

d x x dJ

J

d x d x

3 3phys pure potal hys phys

3 31 1( D )

2i i a ax D E A x Ed x d x d x d

ix A A

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, 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

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)

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

History of theoretical efforts: Brief Review

1988-1996: Dark age, no gauge-invariance

3 3 3

q q

t l

g g

ta3

o

1 1

2

S L S L

i id x d x d x dJ x A E xxE Ai

Jaffe-Manohar [NPB337:509 (1990)]

Concentration on quark spin, the only gauge-invariant piece,

from ~0% to ~30%

3 3

q q g g

total

tota

3

3 3 3l

31 1

2

S L S

1)

2)

L

1 1

2

i iJ

J

d x d x d x d x

d x d x

x E A E x Ai

x D di

xx E B

q q g S L' J'

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

3 3 33total

1 1 i id x D d xE d x d xEP Ai

Bi

3 3 3

q q

t l

g g

ta3

o

1 1

2

S L S L

i id x d x d x dJ x A E xxE Ai

X.S. Chen, F. Wang, hep-ph/9802346: a path-integral proofM. 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

3 3total

1q g

i idP P Px d xE Ai

total total[ , ], [ , ']z z z zq qL J O P P O

)]([ˆˆˆ1ˆ

ˆˆ11ˆ

ArriLiLLDi

rL

BiqAiqPPAqi

Di

P

KKKK

KKK

Problem with the covariant derivative

LK 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 calculationP. Hoodbhoy, X. Ji, W. Lu, PRD 59:074010 (1999); P. Hoodbhoy, X. Ji, PRD 60, 114042 (1999).

Revealing the unreliability of the utilized conventional path-integral approachX.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 83:096012 (2011)

The recent proof of Elliot Leader by canonical quantization

Mixed use of different decompositions!In both theory and experiments!

2001-2007: Another miserable stage

3 3

q q g g

total

tota

3

3 3 3l

31 1

2

S L S

1)

2)

L

1 1

2

i iJ

J

d x d x d x d x

d x d x

x E A E x Ai

x D di

xx E B

q q g S L' J'

A typical confusion: Sg~0, Lg~0, L’q~0, then where is the nucleon spin?!

Key Observation: Dual Role of the Gauge Field

1. Conpensate phase freedom of

1

4

:ige

A A

i m FigA F

L

2. Physical coupling to :

2008-2010: The field-separation method

Physical decomposition of the gauge field and its dual role

phys pureDecomposition : A A A

pure

phys

transforms as does , and gives zero

transform covariaDesir

ntly as doesed a

go l:

A A F

A F

pure

phys

solely carries the pure-gauge degrees of freedom

solely carries the physical degrees of frNamely

edom:

e

A

A

phys pure and are to be expressed in terms of A A A

Advantage (usage) of the decomposition

pure

phys

or the gauge link Wils

is used instead of to construct covariant derivative

to achieve gauge invariance

is used instead of as t

on lin

he canonical variable

e

A A

A F

Physical quantity = f(Aphys, Dpure,…)

Application: Consistent separation of nucleon momentum and spin

3 L d x 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

phys

phys

Its first moment gives the gauge-invariant local operator:

which is the + component of the gauge-invariant gluon spi

,

n

ij i jg ij

g

M F A

S E A

Physical separation of the Abelian Field: Prescription

Boundary condition:

ˆ| 0, | 0x xF A

Physical separation of the Abelian Field: Solution

| 0,

ˆ | 0

x

x

F

A

1

Initial condition required

ˆ ˆ0 ( )

!

A A F

Physical separation of the Abelian Field: Uniqueness

ˆ ˆ0 ( , , , ) ( , , ', ) 'z

z zA A x y z t F x y z t dz

ˆ ˆ0 ( , , , ) ( , , , ') 'A A 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

ˆ ˆ0 ( , , , ) ( , , ', ) 'z

z zA A x y z t F x y z t dz

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

p

3 3 3 3

3 3ure

total

to3

tal

Jaffe-Manohar [NPB337:509 (1990)]

Chen-Lu-Sun-Wang-Goldman [PRL100:232002 (2008); 103:062001 (2009)]

1 1

2

1 1

2

i iJ

J

d x x E A xd x d x E Ai

x Di

d x

d x d x d

phys pure phys

p

3

3 3hys pure phys phto

3ys

3tal

Wakamatsu [PRD81:114010(2010); 83:014012 (2011); 84:037501 (

D

1 1( D

2011)]

)2

i i

i i a a

E A x E A

x D E

x d x

d x d x d x d xA Ai

J x E A

Interacting theory: Structure of Poincare generators

int

i t

n

n

i t

"bad" genera

Lagrangian:

Spatial translation and rotation are

"Good" generators

kine

tors

a b

a b

a b

a

a b

b

P P P

J

H = H H H

K K KJ KJ

L L L L

Time translation and Lorentz boost ar d

matic

e ynamic

Interacting theory: Poincare (sub)algebra

( , ) ( , )

( , ) ( , )

( ,

,

)

,

, ,

[ , ]

Kinematic transformation [ , ]

[ , ] 0

[ , ]Dynamic transformat

Only total

ion [ , ]

and are

i j ka

i j ka b ijk

b ijk a bi j k

a b ijk

a bi

a b

ja b a b ij

i ja b

K J i K

K P

J

J J i J

J P i P

P

i

P

H

P

[ , ] covariant:

[ , ]

i j kijk

i jij

K J i K

K P iH

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

0

)( ' 3

BErxdJ g

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 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

i iE A E x A

Dip

ole rad

.

11

ikreB LY

ikri

E B i Ak

(ra

d.

gau

ge)

l=1

m=1

E B 21 cosdP

d

E

Flu

x

J Flu

x

x E B

21 coszdJd

22sinzdJd

Hadron physics is the best subject to educate people

--- Chairman Mao