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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
2
In DIS experiments, longitudinal proton spin sum rule
gqgq LGLJJ 2
1
2
1
1
0
1
0
)(
)]()()([
dxxgG
dxxsxdxusdu
is tested by measuring polarized parton distribution functions
In non-relativistic QM it is expected that =u+d=4/3-1/3=1Relativistic QM ~ 0.65, Lq ~ 0.35 /2
q(x,Q2)=
is identical to flavor-singlet axial coupling gA0 related to the
axial vector current A0 = u 5u +d5d +s5s
3
EMC (European Muon Collaboration ’87) measured g1p(x) = ½∑ei
2qi(x) with 0.01<x<0.7, <Q2>=10.7 GeV2 and its first moment. Combining thiswith the couplings gA
3=u-d, gA8=u+d-2s measured in low-
energy neutron & hyperon decays = 0.140.18, u = 0.770.06, d = -0.490.06, s = -0.150.06
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)
4
Anomalous gluon interpretation
Consider QCD corrections to order s : Efremov, Teryaev; Altarelli, Ross; Leader, Anselmino; Carlitz, Collins, Muller (88’)
Gqe s
qsp
21
2
1 21
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
from (a)
from (b)
Why is this pQCD correction so special ?
5
QCD corrections imply that
08.02
08.0
42.02
42.0
85.02
85.0
Gss
Gdd
Guu
s
s
s
34.02
3
2
3 GsduG ss
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 L⇒ q+G also increases with lnQ2 with fine tuning
This anomalous gluon interpretation became very popular after 1988
GqGq LGLJJ 2
1
2
1
6
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
]4[9
1
9
1
9
4
2
1
9
1
9
1
9
4
2
1
||2
1)(
1
0 352
11
sssvv
qpp
sdudu
sdu
pqqpedxxg
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)
J5 has anomalous dimension at 2-loop (Kodaira ’79) ⇒ q is Q2 dependent,
against parton-model intuition
7
A hot debate between anomalous gluon & sea quark interpretations before 1996 !
anomalous gluon sea quarkEfremov, Teryaev
Altarelli, Ross
Carlitz, Collins, Muller
Soffer, Perparata
Strirling
Roberts
Ball, Forte
Gluck, Reya, Vogelsang
Lampe
Mankiewicz
Gehrmann
….
Anselmino, Efremov, Leader [Phys. Rep. 261, 1 (1995)]
Jaffe, Manohar
Bodwin, Qiu
Ellis, Karlinear
Bass, Thomas
…
As a consequence of QCD, a measurement of 10g1(x) does
not measure . It measures only the superposition -3s/(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.
- Anselmino,Efremov,Leader (’95)
8
Two hot debates in the past years:
1988 ~ 1995: anomalous gluon or sea quark interpretation of smallness of or gA
0
2008 ~ now: gauge-invariant decomposition of proton spin and gluon angular momentum Jg = Sg + Lg
9
Factorization scheme dependence
It was realized by Bodwin, Qiu (’90) and by Manohar (’90) that hard gluonic contribution to 1
p is a matter of convention used for defining q
)()()()()(2
1)( 2
1 xGxCxqxCxqexg Gqip
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
fact. scheme dependent
Int. J. Mod. Phys. A11, 5109 (1996)
)(xGhard
softhard),(,, 2/2
22
fGq
fqG x
QxCQxC
10
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 1
0 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
GIq
sCIq
p qeGqedxxg 21
0
21 2
1
22
1)(
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. It is usually done in MS scheme.
)()1()()( xGxxqxq sCIs
GIs
HYC (’95); Muller, Teryaev (’97) parton model OPE
11
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 axial-
anomaly ambiguity is presumably well known to all the
practitioners in the field, especially to those QCD experts
working in the area. hep-ph/0002157
My conclusion:
Dust is settled down after 1995 !
12
How to probe gluon polarization ?
DIS via scaling violation in g1(x,Q2)
photon or jet or heavy quark production in polarized pp collider, lepton-
proton 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
13
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
14
SU(3) symmetry implies gA8= 3F-D = 0.585 while gA
3= F+D = 1.2701
Using gA8=0.585, COMPASS (’07) & HERMES (’07) obtained
gA0(3 GeV2) = 0.350.030.05 (COMPASS)
gA0(5 GeV2) = 0.3300.00110.0250.028 (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 = -0.020.03 by COMPASS
)()(2
1)()( 2222 QGQQqQq sCIGI
Sea polarization should be small due to smallness of G
15
When SU(3) symmtry is broken, gA8 may be reduced. For example,
gA8 = 0.460.05 in cloudy bag model
gA8 gA
0 -s = 1/3 (gA8 - gA
0)
For gA= 0.46, s = - 0.03sensitive to SU(3) breaking
Three lattice calculations in 2012 :
1.QCDSF s = - 0.0200.0100.004 at Q = 2.7 GeV
2.Engelhardt s = - 0.0310.017 at Q = 2 GeV
3.Babich et al s = GAs(0) = - 0.0190.017 not renormalized yet
Bass, Thomas (’10)
The smallness of G implies a small s. Hence, SU(3) symmetry should be broken in gA
8
16
Second hot debate on gauge-invariant decomposition of the proton spin
17
Conservation of energy-momentum is governed by T, while conservation of angular momentum is described by rank-3 angular momentum tensor M= Tx-Txwith symmetric T which can be achieved by Belinfante symmetrized expression
)4
1(2)(
2
1 2FgFFTriDiDTTT gq
52
1)( iDxiDxM q
03xTdP
)( )1
( 2
1
)(2
1
2
1
333
00303
BExxdDi
xxdxdJ
xTxTxdxMdJ jkkjijkjkijki
][2
1][2 2
gxgxTrFFFxFFxTrM g
18
gqq JLS
BExxdDi
xxdxdJ
)( )1
( 2
1 333
Ji spin sum rule (PRL ’97)
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
)]()()([ )( 33 AxEAxEAxEAExdBExxd iiii
AxgxdAxExdgE
)( 33
)()1
( 2
1 3
ggqq
ii
LSLS
AxEAEi
xxdJ
Jaffe-Manohar decomposition (’90)
19
)()1
( 2
1 3
ggqq
ii
LSLS
AxEAEi
xxdJ
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
purephys AAA
Aphys carries the physical d.o.f. while A
pure carries the pure gauge d.o.f. To achieve this goal, they demand
Aphys transforms covariantly as F
Apure transforms as A and gives zero F, i.e. F
pure=0
20
Apure is used to construct covariant derivative & it doesn’t contribute to F
21
In QED, one can impose the conditions
0 physA
0 pureA
to ensure Aphys has no longitudinal component
to ensure Apure has only longitudinal component
Under gauge transformation:
Aphys A
phys, Apure A
pure +
In QCD, replace the two conditions in QED in terms of covariant derivatives
Under gauge transformation:
Aphys U(x)A
phys U(x)+, Apure U(A
pure+i/g)U(x)+
0
0],[
purepurepurepurepure
physpurephysphysadjpure
AAigAAD
AAigAAD
Aphys=A, Apure =A‖
Gauge fixing A = 0 Apure = 0. Hence we can set Apure=0
22
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
|'|4
),'(''),( , 3
2 xx
txFxdtxAAAA ii
physphys
ggqq
iphys
adjpure
iphyspure
LSLS
ADxEAEDi
xxdJ
1
2
13
Chen et al. decomposition (’08)
Each term is gauge invariant, Sg and Lg have operator definition !
pureq Di
xL 1
satisfies the commutation relation LiLL
Momentum p-gApure is neither the canonical momentum p nor the dynamical momentum p-gA; it is gauge-invariant and satisfies canonical commutation relation 0pp
23
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
24
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
)]()([)( xxAxA
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
25
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
33
2 |'|4
)'(),'('
1
0 ,
xx
xxtxBxdBA
ABA
phys
physphys
Hence, Aphys is as measurable (physical) as E and B are.
26
Different decompositions
ggqq LSLS
)( 1
2
1BExD
ix
)( 1
2
1 ii AxEAEi
x
1
2
1 iphys
adjpure
iphyspure ADxEAED
ix
1
2
1 iphys
adjpure
iphyspure ADxEAED
ix
)(
1
2
1
phys
iphys
adjpure
iphys
Ax
ADxEAEDi
x
Ji (’97)
Jaffe-Manohar(’90):
Barshinsky-JaffeLC gauge (’98):
Chen et al. (’08,’09):
Wakamatsu(’10,’11):
Also decompositions by Cho, Ge, Zhang; Leader; Guo, Schmidt; Lorce,…
27
Wakamatsu decomposition
To decompose A=Aphys + A
pure, Wakamatsu imposed two conditions
alone: (i) Fpure=0, (ii) A
phys transforms as F and Apure as A. Gauge
fixing will be done in a later stage
)()()(
)()(
333
33
Wakag
Wakaq
physiphys
adjpure
i
iphys
adjpure
ipure
Cheng
Chenq
LL
AxxdADxxEdAgpxxd
ADxxEdAgpxxdLL
canonical OAM potential OAM
His decomposition differs from that of Chen et al. in quark & gluon OAMs.
2828
Debate between canonical & dynamical variables
Quark OAM extracted from GPD analysis is dynamical OAM; useful in classical picture
BExdDi
xdP
33 1
canonical variables dynamical variables
quark momentum: pureAgpp
or quark momentum: Agp
quark OAM: )(or pureAgpxpx
quark OAM: )( Agpx
Jaffe-Manohar; Chen et al.Bashinsky-Jaffe; Cho et al.Leader,…
Ji; Wakamatsu
In QM, they correspond to generators of translation & rotation;suitable for quantization
iphys
adjpure
ipure
ii
ADxEdDi
xd
AxEdi
xdP
33
33
1
or 1
kijkjikijkjiji PiPJJiJJPP ],[ ,],[ ,0],[
29
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
Wakamatsu argued that there exist only two physically inequivalent decompositions (I) & (II)
(Wakamatsu)
30
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
GIE at Coulomb gauge: Chen et al at LC gauge : Bashinsky-Jaffe
Coulomb GIE
LC GIE
)()1
( 2
1 3
ggqqii LSLSAxEAE
ixxdJ
The decompositions of Chen et al. and Bashinsky-Jaffe correspond to different GIE and hence they are not necessarily gauge equivalent
31
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.
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(EAphys)3 in the gauge-
invariant 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
Hoodbhoy, Ji, Lu (’99)
32
Gluon momentum fraction
BExdDi
xdP
33 1
fg
fgs
nn
nn
3
4
9
8
3
4
9
8
4
iphys
adjpure
ipure ADxEdD
ixdP
33 1
fg
fgs
nn
nn
3
4
9
2
3
4
9
2
4
Conventional approach (Ji) Chen et al. (PRL, 2009)
totaltotalfg
gG PP
nn
nP
5
1
6
totaltotal
fg
gG PP
nn
nP
2
1
32
2
Chen et al. thus claimed the standard textbook statement that gluons carry half of the nucleon momentum is wrong !
33
T++= T++q
+ T++g + i+ + 2Tr(+ A)2
)4
1(2)(
2
1 2FgFFTriDiDTTT gq
Momentum sum rule follows from <P|T++|P>/2(P+)2=1
In A+=0 gauge, D+ = + -igA+ +
F+= + A - A+ + g[A+,A] +A
Hence, 1 = dx x [q(x)+g(x)] = <x>q + <x>g
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.
T++q (Chen)-T++
q (Ji) = g+ A+phys = 0 in LC gauge
34
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.
Gluon helicity
Experimentally, the polarized gluon distribution is given by
SPFFSPedxP
ixg ixP ,|)(
~),0()0(|,
2)(
In light-cone gauge A+=0, g(x) has a simple interpretation: it measures the distribution of gluon polarization
)()()( ~
xgxgxgFFFFFF LLRR
Manohar;Collins, Soper
SPAFSP
PdxxgG ,|)0()0(|,
2
1)(
first moment:
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.
Gluon spin ( E A) & OAM are not gauge invariant, but helicity is. Gluon polarization in IMF G=dxg(x) is a measurable quantity.
where is a gauge link to ensure local gauge invariance
0
]exp[),0(
dsAigP
36
In gauge-invariant decomposition of the proton spin
][2 physphyssping AFAFTrM
This leads to
SPAFSP
PSPMSP
PG physsping ,|)0()0(|,
2
1 ,||,
2
1
12
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
37
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.
38
Quark orbital angular momentum
At Q2→, Ji, Tang & Hoodbhoy found (’96)
24.026.02
1
)47.0(2
1
316
16
2
1)()()(
)53.0(2
1
316
3
2
1)()(
2
1)(
222
222
Gq
fGG
f
fqq
JJ
nQLQGQJ
n
nQLQQJ
Analogous to the nucleon’s momentum partition: half of the proton’s momentum is carried by gluons
for nf=6
Experimentally, how to measure Jq ?
39
Jq is related to the GPDs by the Ji sum rule
0
1lim [ ( , , ) ( , , )]
2q q qtJ dxx H x t E x t
Ji (’97)
Study of hard exclusive processes leads to a new class of PDFs: four independent GPDs (at twist-2): (pol)
~ ,
~ ,(unpol) , EHEH
1
1
1
1
1
1 2
1
1 1
)(),,(~
),(),,(~
)(),,( ),(),,(
),()0,0,(~
),()0,0,(
tGtxEdxtGtxHdx
tFtxdxEtFtxdxH
xqxHxqxH
PA
DVCS in large s and small t region can probe GPDs
qJL qq 2
1
40
1
1
1
1
012 ||)(2
1)]0,0,()0,0,([
2
1pMpdxxqdxxExHxL OAMqqqq
)1
(or ,)1
( )1
( 333012 pureOAMq Di
xi
xDi
xM
Quark OAM extracted from GPD analysis is dynamical OAM not canonical OAM
1
1
1
1
012 ||)(2
1)]0,0,()0,0,([
2
1pMpdxxgdxxExHxL OAMgggg
OAM potential OAM canonical
])([2])([2 33012
physiphyspure
iOAMg AxTrADxETrM
41
Recent development: relate OAM to Wigner or phase-space operator as OAM is a correlation between position & momentum
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]
dexxxxkxW ik)2/()2/,2/()2/(),(
Ji (’03)
42
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.
43
Conclusions
& 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.
GqGq LGLJJ 2
1
2
1
What do we learn in past 25 years about the proton helicity decomposition ?
44
Extra slides
45
Lattice QCD
Can lattice QCD shed some light on the protn spin content ?
sqq
spJspspJspspJspGIs
GIv
discon
,||,,||,,||, 555
Sea polarization from disconnected insertion
⇒ us= ds= s = -0.12±0.01
46
Ju=½ u+Lu
Jd=½ d+Ld
HERMES: hep-ex/0606061
JLab: nucl-ex/0709.0450
p-DVCS sensitive to Ju
n-DVCS sensitive to Jd
47
Lattice calculations of GPDs
arXiv:0705.4295 (LHPC,MILC): Hagler, Schroers,…arXiv:0710.1534 (QCDSF,UKQCD): Brommel, Gockeler, Schroers,…
Lu+d~0 & Jd~0 ) cancellation between Lu & Ld; ½¢d & Ld
From Ju=0.230, Jd= -0.004, Lu+d=0.025, )Lu=-0.190,Ld= 0.215
How about Ls ?
LHPC QCDSF
½u+d
Lu+d
Ju
Jd
Lu
Ld
48
Alexandrou et al. (ETM, European twisted mass) 1104.1600
Syritsyn et al. 1111.0718