Transcript PowerPoint 簡報 - Institute of Physics Academia Sinica

Proton Spin Decomposition :
The Second Hot Debate in Proton Spin Physics
Hai-Yang Cheng
Academia Sinica
 Anomalous gluon & sea-quark
interpretation of smallness of 
 Proton spin decomposition
October 15, 2012
Journal Club, AS
In DIS experiments, longitudinal proton spin sum rule
1
1
 J q  J g    Lq  G  Lg
2
2
is tested by measuring polarized parton distribution functions
1
  u  d  s   [u( x )  d ( x)  s( x )]dx
0
1
G   g ( x )dx
0
q(x,Q2)=
In non-relativistic QM it is expected that =u+d=4/3-1/3=1
Relativistic QM   ~ 0.65, Lq ~ 0.35 /2
 is identical to flavor-singlet axial coupling gA0 related to the
axial vector current A0 = u 5u +d5d +s5s
2
p
EMC (European Muon Collaboration ’87) measured g1 (x) = ½∑ei2qi(x)
with 0.01<x<0.7, <Q2>=10.7 GeV2 and its first moment. Combining this
with the couplings gA3=u-d, gA8=u+d-2s measured in lowenergy neutron & hyperon  decays 
 = 0.140.18,
u = 0.770.06, d = -0.490.06, s = -0.150.06
Two surprises:
 strange sea polarization is sizable & negative
 very little of the proton spin is carried by quarks
⇒
Proton Spin Crisis
(or proton helicity decomposition puzzle)
3
Anomalous gluon interpretation
Consider QCD corrections to order s : Efremov, Teryaev; Altarelli, Ross;
Leader, Anselmino; Carlitz, Collins, Muller (88’)
from (a)
1  



1p  1  s  eq2  q  s G 
2
 
2


from (b)
Anomalous gluon contribution (s/2)G arises from photon-gluon
scattering. Since G(Q2)  lnQ2 and s(Q2)  (lnQ2)-1 ⇒ s(Q2)G(Q2) is
conserved and doesn’t vanish in Q2→ limit
Why is this pQCD correction so special ?
4
QCD corrections imply that
s
G  0.85
2

d  0.42  d  s G  0.42
2

s  0.08  s  s G  0.08
2
u  0.85  u 
 
3 s
3
G  u  d  s  s G  0.34
2
2
If G is positive and large enough, one can have s  0 and =u+d 
0.60 ⇒ proton spin problem is resolved provided that G  (2/s)(0.08) 
1.9 ⇒ Lq+G also increases with lnQ2 with fine tuning
1
1
 J q  J G    Lq  G  LG
2
2
This anomalous gluon interpretation became very popular after 1988
5
Operator Product Expansion
moments of structure function= 10 xn-1F(x)dx = ∑ Cn(q)<p,s|On|p,s>
= short-distance  long-distance
No twist-2, spin-1 gauge-invariant local gluonic operator for first moment
1
1
eq2  p  | q   5q | p    3

2
14
1
1 
  u  d  s 
29
9
9 
   g1p ( x )dx 
p
1
0

14
1
1

 uv  d v  [4us  d s  ss ] 
29
9
9

OPE ⇒ Gluons do not contribute to 1p ! One needs sea quark polarization to
account for experiment (Jaffe, Manohar ’89)
 It is similar to the naïve parton model
 How to achieve s  -0.08 ? Sea polarization (for massless quarks) cannot
be induced perturbatively from hard gluons (helicity conservation ⇒ s=0 for
massless quarks)
 J5 has anomalous dimension at 2-loop (Kodaira ’79) ⇒ q is Q2 dependent,
6
against parton-model intuition
A hot debate between anomalous gluon & sea quark
interpretations before 1996 !
anomalous gluon
sea quark
Efremov, Teryaev
Jaffe, Manohar
Altarelli, Ross
Bodwin, Qiu
Carlitz, Collins, Muller
Ellis, Karlinear
Soffer, Perparata
Bass, Thomas
Strirling
…
Roberts
Ball, Forte
Gluck, Reya, Vogelsang
Lampe
Mankiewicz
As a consequence of QCD, a measurement of 10g1(x) does
not measure . It measures only the superposition
-3s/(2)G and this combination can be made small by a
cancellation between quark and gluon contributions. Thus
EMC results ceases to imply that  is small.
Gehrmann
- Anselmino,Efremov,Leader (’95)
….
Anselmino, Efremov, Leader [Phys. Rep. 261, 1
(1995)]
7
Two hot debates in the past years:
 1988 ~ 1995: anomalous gluon or sea quark interpretation
of smallness of  or gA0
 2008 ~ now: gauge-invariant decomposition of proton
spin and gluon angular momentum Jg = Sg + Lg
8
Factorization scheme dependence
 It was realized by Bodwin, Qiu (’90) and by Manohar (’90) that hard
gluonic contribution to 1p is a matter of convention used for defining q
g1p ( x) 

1
2
e

i q ( x )  Cq ( x )  q ( x )  CG ( x )  G ( x )
2
fact. scheme dependent

G
 hard
(x)
 Q2 
 CG x, Q   Cq  x, 2   q / G ( x,  2f )  hard  soft
  
f 

2
Consider polarized photon-gluon cross section
1.
Its hard part contributes to CG and soft part to qs. This
decomposition depends on the choice of factorization scheme
2.
It has an axial QCD anomaly that breaks down chiral symmetry
Int. J. Mod. Phys. A11, 5109 (1996)
9
Two extreme schemes of interest (HYC, ’95)
 gauge-invariant (GI) scheme (or MS scheme)
-- Axial anomaly is at soft part, i.e. qG, which is non-vanishing due to
chiral symmetry breaking and 10 CG(x)=0 (but G  0 !)
-- Sea polarization is partially induced by gluons via axial anomaly
 chiral-invariant (CI) scheme (or “jet”, “parton-model”, “kT cut-off’,
“Adler-Bardeen” scheme)
Axial anomaly is at hard part, i.e. CG, while hard gluons do not
contribute to qs due to chiral symmetry

q ( x )  q ( x )   s (1  x )  G ( x )

GI
s
CI
s
HYC (’95); Muller, Teryaev (’97)
1

0
g1p ( x )dx 
1
s
 1
2
2
e

q


G

e




q
CI
q qGI
2
2

 2
parton model
OPE
 Hard gluonic contribution to  g1p is matter of factorization
convention used for defining q.
 It is necessary to specify the factorization scheme for data analysis.
10
It is usually done in MS scheme.
My conclusion:
In retrospect, the dispute among the anomalous gluon and
sea-quark explanations…before 1996 is considerably
unfortunate and annoying since the fact that g1p(x) is
independent of the definition of the quark spin density and
hence the choice of the factorization scheme due to the axialanomaly ambiguity is presumably well known to all the
practitioners in the field, especially to those QCD experts
working in the area.
hep-ph/0002157
Dust is settled down after 1995 !
11
How to probe gluon polarization ?
 DIS via scaling violation in g1(x,Q2)
 photon or jet or heavy quark production in polarized pp collider, leptonproton collider or lepton-proton fixed target
RHIC (at BNL):
via direct high-pT prompt ,  production,
jet production
HERMES (at DESY):
via open charm production
COMPASS (at CERN): via open charm production
Direct measurement of G:
Photon-Gluon-Fusion process
12
Adolph et al. arXiv:1202.4064
G/G is very small and cannot explain the smallness of  via
anomalous gluon effect, but G  0.2 - 0.3 makes a significant
contribution to proton spin
13
SU(3) symmetry implies gA8= 3F-D = 0.585 while gA3= F+D = 1.2701
Using gA8=0.585, COMPASS (’07) & HERMES (’07) obtained
gA0(3 GeV2) = 0.350.030.05 (COMPASS)
gA0(5 GeV2) = 0.3300.00110.0250.028 (HERMES)
 u = 0.85, d = -0.42, s = -0.08 at Q2  4 GeV2
Semi-inclusive DIS data of COMPASS & HERMES show no
evidence of large negative s: s = -0.020.03 by COMPASS
qGI (Q 2 )  qCI (Q 2 )  
1
 s (Q 2 )G (Q 2 )
2
Sea polarization should be
small due to smallness of G
14
When SU(3) symmtry is broken, gA8 may be reduced. For example,
gA8 = 0.460.05 in cloudy bag model
Bass, Thomas (’10)
gA8   gA0   -s = 1/3 (gA8 - gA0) 
For gA8 = 0.46, s = - 0.03 sensitive to SU(3) breaking
Three lattice calculations in 2012 :
1.QCDSF
s = - 0.0200.0100.004 at Q = 2.7 GeV
2.Engelhardt
s = - 0.0310.017 at Q = 2 GeV
3.Babich et al
s = GAs(0) = - 0.0190.017 not renormalized yet
The smallness of G implies a small s. Hence, SU(3)
symmetry should be broken in gA8
15
Second hot debate on gauge-invariant
decomposition of the proton spin
16
Conservation of energy-momentum is governed by T, while
conservation of angular momentum is described by rank-3 angular
momentum tensor M = Tx -Tx with symmetric T which can be
achieved by Belinfante symmetrized expression
T   Tq

 Tg

1
1
  (  iD    iD )  2Tr( F  F  g  F 2 )
2
4
1
M q     ( x iD   x  iD )        5
2
1
M g  2Tr[ x F  F  x  F  F ]  TrF 2 [ x g   x  g  ]
2
P    d 3 xT 0 
1
1
J i    ijk  d 3 xM 0 jk    ijk  d 3 x (T 0 j x k  T 0 k x j )
2
2


 
1
3
 1
3
 
3 
J   d x
   d x  ( x  D )   d x x  ( E  B )
2
i
17



 
3
 1
3
  1
3 
J   d x     d x  ( x  D)   d x x  ( E  B)
2
i
  
Ji spin sum rule (PRL ’97)
 Sq  Lq  J g
Gauge-invariant decomposition, but Jg cannot be further
decomposed into spin and OAM parts. However, gluon spin G
has been measured in many experiments. In QED, S & L are
measurable.
Using the identity
 i
   

 
  
3
i 
i
i
 d x x  ( E  B)   d x [ E  A  E ( x  ) A  (  E ) x  A   ( E x  A)]
3
 
  E    g  
   

3

 d x(  E ) x  A   d x g x  A
3
 i


 
1
3   1
 
i 
J   d x 
   ( x  )  E  A  E ( x  ) A 
i
 2

  

 Sq  Lq  S g  Lg
Jaffe-Manohar decomposition (’90)
18
 i


 
1
3   1
 
i 
J   d x 
   ( x  )  E  A  E ( x  ) A 
i
 2

  

 Sq  Lq  S g  Lg
Each term is not separately gauge invariant except for Sq.
Xiang-Song Chen, Xiao-Fu Lu, Wei-Min Sun, Fan Wang, T. Goldman
(PRL, 2008, 2009) proposed to solve the gauge-invariance problem
by decomposing gauge field into
A  A phys  A pure
Aphys carries the physical d.o.f. while Apure carries the pure gauge
d.o.f. To achieve this goal, they demand
Aphys transforms covariantly as F
Apure transforms as A and gives zero F, i.e. Fpure=0
19
Apure is used to construct covariant derivative & it doesn’t
contribute to F
20
In QED, one can impose the conditions
 
  Aphys  0
 
  Apure  0
to ensure Aphys has no longitudinal component
to ensure Apure has only longitudinal component
Under gauge transformation:
Aphys Aphys,
 Aphys=A, Apure =A‖
Apure Apure + 
Gauge fixing  A = 0   Apure = 0. Hence we can set Apure=0
In QCD, replace the two conditions in QED in terms of covariant
derivatives
 


 adj 
D pure  Aphys    Aphys  ig[ Apure , Aphys ]  0

 



D pure  Apure    Apure  igApure  Apure  0
Under gauge transformation:
Aphys U(x)Aphys U(x)+,
Apure U(Apure+i/g)U(x)+
21
The separation of A into physical and pure gauge parts is possible
at the cost of introducing nonlocality. Aphys & Apure are nonlocally
related to the total A. For example, in QED



   

'
F
(
x
' , t)


Aphys  A  2   A,
Aphys
( x, t )   d 3 x' i i  

 4 | x  x ' |
Chen et al. decomposition (’08)


 
 adj i 
1
3   1

i
J   d x 
   x  D pure   E  Aphys  E x  D pure Aphys 
i
 2

  

 Sq  Lq  S g  Lg
Each term is gauge invariant, Sg and Lg have operator definition !
Momentum p-gApure is neither the canonical momentum p nor the
dynamical momentum p-gA; it is gauge-invariant and satisfies
 
canonical commutation relation p  p  0

 1
Lq  x  D pure
i
  
satisfies the commutation relation L  L  i L
22
Criticism (mainly due to Ji)
 clashes with locality, Aphys is non-local
 lack of Lorentz covariance: The decomposition A=Aphys+Apure is
not Lorentz covariant. In other Lorentz frame, Aphys and Apure may
mix together
 Aphys is not unique even after a gauge fixing
 How to quantize the theory with both Aphys & Apure as quantum
mechanical degrees of freedom ? It seems quantization makes
sense only if Apure=0.
 gauge invariance not in the textbook sense
 limited physical significance; cannot be measured experimentally.
For example, Sg is not the gluon spin G measured in high energy
DIS experiments or in pp collisions
 Gluons carry only 1/5 of the proton momentum in Q2 limit
23
Lorentz covariance
Q: Is the separation A=Aphys+Apure Lorentz covariant ? In the other
Lorentz frame, will Aphys and Apure remain physical & pure gauge ?
A: A cannot transform as a 4-vector
A ( x)   [ A ( x)    ( x)]
Chen et al.; Leader,
Lorce,…
One can choose  in such a way that Aphys remains
physical in any Lorentz frame. However, the Lorentz
transformation law is complicated.
For example,  Aphys=0 in one frame, then   restores
’ A’phys=0 in a transformed frame
24
Uniqueness of Aphys ?
Ji:
Aphys is not unique even after a gauge fixing
In QED,  Aphys=0 doesn’t fix Aphys. Consider Aphys Aphys+
with 2=0. Hence, there are infinite numbers of Aphys
Chen et al.: Since we demand Apure transforms as A, Aphys is
invariant under guage transformation. With Aphys  0 at spatial
infinity, it can be solely expressed in terms of E & B fields in QED
 

  Aphys  B,

 
  Aphys  0
 
 

 1 
B
(
x
'
,
t
)

(
x
 x' )
Aphys    2 B   d 3 x '
 

4 | x  x ' |3
Hence, Aphys is as measurable (physical) as E and B are.
25
Different decompositions



Sq
Lq
Sg
1
 
Ji (’97)
2

 1

Jaffe-Manohar 
2
(’90):

 1
Barshinsky-Jaffe 

2
LC gauge (’98):

 1
Chen et al.
 
2
(’08,’09):

Wakamatsu
(’10,’11):
1
 
2

 1
 x  D
i

 1
 x  
i

 1
 x  Dpure
i

 1
 x  Dpure
i

 1
 x  D
i


Lg
  
x  ( E  B)
 
EA
  i
E ( x  ) A
 
E  Aphys
  adj i
E x  Dpure Aphys
 
E  Aphys
  adj i
E x  Dpure Aphys
 
E  Aphys
  adj i
E x  Dpure Aphys
 
  ( x  Aphys )
i
i
i
i
Also decompositions by Cho, Ge, Zhang; Leader; Guo, Schmidt; Lorce,…26
Wakamatsu decomposition
To decompose A=Aphys + Apure, Wakamatsu imposed two conditions
alone: (i) Fpure=0, (ii) Aphys transforms as F and Apure as A. Gauge
fixing will be done in a later stage
His decomposition differs from that of Chen et al. in quark & gluon
OAMs.
Chen
q
L
L
Chen
g

 adj i
 
3
i 
  d x x  ( p  gApure )   d xE ( x  D pure ) Aphys

 adj i

 
3

3
i 
3
  d x x  ( p  gA)   d xE ( x  D pure ) Aphys   d x ( x  Aphys )
3

Waka
 LWaka

L
q
g
canonical OAM
potential OAM
27
Debate between canonical & dynamical variables
canonical variables


1
P   d 3 x     d 3 xE i Ai or
i


3
1
3
i adj
i
d
x

D


d
xE
D
pure
pure Aphys


i
quark momentum:
quark OAM:



p or p  gApure

 
 
x  p or x  ( p  gApure )
In QM, they correspond to
generators of translation & rotation;
suitable for quantization
dynamical variables


 
3
1
3
P   d x  D   d x E  B
i
quark momentum:
quark OAM:


p  gA

 
x  ( p  gA )
Quark OAM extracted from GPD
analysis is dynamical OAM;
useful in classical picture
[ Pi , P j ]  0, [ J i , J j ]  i ijk J k , [ J i , P j ]  i ijk Pk
Jaffe-Manohar; Chen et al.
Bashinsky-Jaffe; Cho et al.
Leader,…
Ji; Wakamatsu
28
Wakamatsu argued that there exist only two physically inequivalent
decompositions (I) & (II)
(Wakamatsu)
In the gauge Apure=0, A=Aphys, Bashinsky-Jaffe and Chen et al.
decompositions are reduced to Jaffe-Manohar one as Dpure ,
E Aphys E A.
Wakamatsu claimed that these 3 decompositions are all gauge
equivalent, provided that gauge fixing procedure is done
consistently with the general conditions for Aphys and Apure
29
Gauge invariant extension (GIE)
Ji et al. claimed that gauge-invariant decomposition of proton spin is
just a GIE of gauge-variant quantities which generalizes the fixed
gauge result extrapolated to any other gauge. Consider gaue-variant
Jaffe-Manohar decomposition

 i    


 
3   1
  1
i 
J   d x    ( x  )  E  A  E ( x  ) A   Sq  Lq  S g  Lg
i
 2

GIE at Coulomb gauge: Chen et al
at LC gauge
: Bashinsky-Jaffe
Coulomb GIE
LC GIE
The decompositions of Chen et al.
and Bashinsky-Jaffe correspond to
different GIE and hence they are not
necessarily gauge equivalent
30
 In Jaffe-Manohar decomposition, gluon spin Sg=d3x(E A)3
is gauge dependent. Its values in light-cone, covariant and Coulomb
gauge fixings are different.
Hoodbhoy, Ji, Lu (’99)
 According to Ji, Chen et al. decomposition is a GIE of JM in
Coulomb gauge, while Bashinsky-Jaffe is a GIE of JM in LC gauge.
Hence, Sg is different in these two schemes. Indeed, Chen et al.
found Sg(Chen) = 5/9 Sg(BJ)
 Wakamatsu argued that since Sg=d3x(EAphys)3 in the gaugeinvariant decomposition is gauge invariant, it should be same in
Chen et al and Bashinsky-Jaffe, provided that gauge fixing procedure
is done consistently with the general conditions for Aphys and Apure
31
Gluon momentum fraction
Conventional approach (Ji)


 
3
1
3
P   d x  D   d x E  B
i
 8
 n
s  9 g
 
4  8
ng

 9

PG 
4
nf
3
4
 nf
3







1
Ptotal  Ptotal
2ng  3n f
2
2n g
Chen et al. (PRL, 2009)


 adj i
3
1
3
i
P   d x D pure   d xE D pure Aphys
i
 2
 n
s  9 g
 
4  2
ng

 9

PG 
4
nf
3
4
 nf
3







1
Ptotal  Ptotal
ng  6n f
5
ng
Chen et al. thus claimed the standard textbook statement
that gluons carry half of the nucleon momentum is wrong !
32
T   Tq

 Tg

1
1
  (  iD    iD )  2Tr( F  F  g  F 2 )
2
4
Momentum sum rule follows from <P|T++|P>/2(P+)2=1
In A+=0 gauge,
T++= T
++
q
D+ = + -igA+  +
F+ = + A - A+ + g[A+,A]  +A
+ T++g  + i+ + 2Tr(+ A)2
Hence, 1 =  dx x [q(x)+g(x)] = <x>q + <x>g
T++q (Chen)-T++q (Ji) = g+ A+phys = 0 in LC gauge
Wakamatsu (’02) has shown explicitly that the anomalous dimension
matrix in Chen et al. decomposition is the same as in the conventional
approach. This makes him wondering if the result Sg(Chen) = 5/9
Sg(BJ) obtained by Chen et al. is also wrong.
33
Observables must be gauge-invariant. Doesn’t it mean that GIE of
gauge-variant quantity becomes measurable experimentally if one
is lucky enough ?
Ji: problems with GIE:
 GIE operators are in general nonlocal and hence doesn’t have
clear physical meaning in general gauges, although they do in the
fixed gauge; cannot be calculated in lattice QCD
 Do not transform simply under Lorentz transformation
 infinite number of non-local operators
Wakamatsu: GIE is not a correct way of handling gauge symmetry.
Color SU(3) symmetry is an intrinsic property of QCD Lagrangian, no
need of GIE.
34
Gluon helicity
Gluon spin ( E  A) & OAM are not gauge invariant, but helicity is.
Gluon polarization in IMF G=dxg(x) is a measurable quantity.
Experimentally, the polarized gluon distribution is given by
i
 ixP 

 ~
Manohar;
g ( x ) 
d

e

P
,
S
|
F
(
0
)

(
0
,

) F (  ) | P, S 
 
2xP
Collins, Soper
 
0
where (0,   )  P exp[ig 
invariance

A  ds]
is a gauge link to ensure local gauge
In light-cone gauge A+=0, g(x) has a simple interpretation: it
measures the distribution of gluon polarization
~
F  F  F  R F  R  F  L F  L  g( x)  g  ( x)  g  ( x)
first moment:
G   g ( x )dx 
1
 P, S | F   (0)   A (0) | P, S 

2P
Bashinsky & Jaffe claimed that G is gauge invariant, but it is not
obvious why it is so. Wakamatsu showed that A above can be
replaced by Aphys without making any approximation.
In gauge-invariant decomposition of the proton spin
 
 
M g

2
Tr
[
F
A

F
Aphys ]
spin
phys
This leads to
G 
1
1
12

P
,
S
|
M
|
P
,
S


 P, S | F   (0)   Aphys (0) | P, S 
g  spin


2P
2P
The above expression is not in contradiction to the usual statement:
Gluon helicity cannot be expressed in terms of gauge-invariant
twist-2 local operator as Aphys is not a local operator
However, Ji et al. claimed that G is meaningful only in LC
gauge and in infinite momentum frame
36
Criticism from Leader:
None of these (Chen et al. Ji, Wakamatsu,…) is acceptable as not
enough attention is paid to the difference between classical &
quantum field theory.
 Gauge invariance of operators is not an important criterion.
Physical m.e. of measurable operators must be gauge invariant
 Previous treatment is classical and use has been made of
classical EOM.
 It is OK to use non-local field operator, but not OK if they are
dynamical variables. In Coulomb gauge A0 is not an independent
dynamical variable.
37
Quark orbital angular momentum
At Q2→, Ji, Tang & Hoodbhoy found (’96)
J q (Q 2 ) 
1
1 3n f
1
(Q 2 )  Lq (Q 2 ) 
 (0.53)
2
2 16  3n f 2
1 16
1
J G (Q 2 )  G (Q 2 )  LG (Q 2 ) 
 (0.47)
2 16  3n f 2
for nf=6
1
 J q  J G  0.26  0.24
2
Analogous to the nucleon’s momentum partition: half of the proton’s
momentum is carried by gluons
Experimentally, how to measure Jq ?
38
Jq is related to the GPDs by the Ji sum rule
1
J q  lim  dxx[ H q ( x, t ,  )  Eq ( x, t ,  )]
t 0 2
Ji (’97)
1
Lq  J q  q
2
Study of hard exclusive processes leads to a new class of PDFs:
~ ~
four independent GPDs (at twist-2): H , E (unpol), H
, E (pol)
H q ( x,0,0)  q( x ),
1

1
1

1
dxH ( x,  , t )  F1 (t ),
~
dxH ( x,  , t )  G A (t ),
~
H q ( x,0,0)  q( x ),
1

1
1

1
dxE( x,  , t )  F2 (t )
DVCS in large s and small t
region can probe GPDs
~
dxE ( x,  , t )  GP (t )
39
Lq 
1 1
1 1
012
x
[
H
(
x
,
0
,
0
)

E
(
x
,
0
,
0
)]
dx


q
(
x
)
dx


p

|
M
q
q
q
OAM | p 



1

1
2
2
1 3
1 3
1
 
 
 
M q012


(
x

D
)



(
x


)

,
or

(
x

D pure )3
OAM
i
i
i
Quark OAM extracted from GPD analysis is dynamical OAM not
canonical OAM
Lg 
M
1 1
1 1
012
x
[
H
(
x
,
0
,
0
)

E
(
x
,
0
,
0
)]
dx


g
(
x
)
dx


p

|
M
g
g
g
OAM | p 



1

1
2
2
012
g OAM
 
 
3 i
 2Tr[ E ( x  Dpure ) Aphys ]  2Tr[  ( x  Aphys )3 ]
i
 canonicalOAM  potentialOAM
40
Recent development: relate OAM to Wigner or phase-space
operator as OAM is a correlation between position & momentum





W ( x , k )   ( x   / 2)( x   / 2, x   / 2) ( x   / 2)eik d
Ji (’03)
Lorce, Pasquini [1106.0139]
Hatta [1111.3547]
Lorce, Pasquini, Xiong, Yuan [1111.4827]
Ji, Xiong, Yuan [1202.2843]
Burkardt [1205.2916]
Lorce [1205.6483]
Lorce, Pasquini [1206.3143]
Hatta, Yoshida [1207.5332]
Ji, Xiong, Yuan [1207.5221]
Lorce, Pasquini [1208.3065]
Lorce [1210.2581]
41
Just like the debate between anomalous gluon & sea quark
interpretation of the proton spin, it appears all the different
decompositions are correct. Controversies mainly concern
the physical interpretation. As to which one is more
convenient and more physical is most likely a matter of taste.
42
Conclusions
What do we learn in past 25 years about the proton helicity decomposition ?
1
1
 J q  J G    Lq  G  LG
2
2
  & Lq are factorization scheme dependent, but not Jq
DIS data ⇒ GI  0.34, sGI  -0.03
RHIC, COMPASS & SIDIS data imply small G & qs
 dxg1p(x) is independent of the definition of quark spin density and the
choice of the factorization scheme due to axial-anomaly ambiguity
 Several different gauge-invariant decompositions of proton spin have
been proposed. Controversies mainly concern the physical interpretation.
As to which decomposition is more convenient and more physical is a
matter of taste.
43
Extra slides
44
Lattice QCD
Can lattice QCD shed some light on the protn spin content ?
 p, s | J 5 | p, s   p, s | J 5 | p, s con   p, s | J 5 | p, s dis
  qvGI  qsGI s 
Sea polarization from disconnected insertion
⇒
us= ds= s = -0.12±0.01
45
HERMES: hep-ex/0606061
JLab: nucl-ex/0709.0450
Ju=½ u+Lu
Jd=½ d+Ld
p-DVCS sensitive to Ju
n-DVCS sensitive to Jd
46
Lattice calculations of GPDs
arXiv:0705.4295 (LHPC,MILC): Hagler, Schroers,…
arXiv:0710.1534 (QCDSF,UKQCD): Brommel, Gockeler, Schroers,…
LHPC
QCDSF
Lu+d~0 & Jd~0
) cancellation
between Lu & Ld;
½¢d & Ld
½u+d
Lu+d
Ld
Ju
From Ju=0.230,
Jd= -0.004,
Lu+d=0.025,
)
Lu=-0.190,
Ld= 0.215
Jd
Lu
How about Ls ?
47
Alexandrou et al. (ETM, European twisted mass) 1104.1600
Syritsyn et al. 1111.0718
48