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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

Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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Page 1: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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

Page 2: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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

Page 3: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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)

Page 4: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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 ?

Page 5: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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

Page 6: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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

Page 7: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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)

Page 8: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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

Page 9: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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

Page 10: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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

Page 11: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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 !

Page 12: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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

Page 13: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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

Page 14: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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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

Page 15: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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

Page 16: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

16

Second hot debate on gauge-invariant decomposition of the proton spin

Page 17: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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

Page 18: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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)

Page 19: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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

Page 20: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

20

Apure is used to construct covariant derivative & it doesn’t contribute to F

Page 21: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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

Page 22: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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

Page 23: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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

Page 24: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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

Page 25: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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.

Page 26: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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,…

Page 27: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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.

Page 28: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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],[

Page 29: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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)

Page 30: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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

Page 31: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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)

Page 32: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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 !

Page 33: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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

Page 34: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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.

Page 35: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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

Page 36: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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

Page 37: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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.

Page 38: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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 ?

Page 39: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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

qq

DVCS in large s and small t region can probe GPDs

qJL qq 2

1

Page 40: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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

Page 41: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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)

Page 42: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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.

Page 43: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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 ?

Page 44: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

44

Extra slides

Page 45: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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

Page 46: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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

Page 47: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

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

Page 48: Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness

48

Alexandrou et al. (ETM, European twisted mass) 1104.1600

Syritsyn et al. 1111.0718