Spin-orbit correlations in the nucleon

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Spin–orbit

correlations

in

the

nucleon

Cédric Lorcé

a

,

b

,

∗

a_{IPNO,UniversitéParis-Sud,CNRS/IN2P3,91406Orsay,France}

b_{IFPA,AGODepartment,UniversitédeLiège,Sart-Tilman,4000Liège,Belgium}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received20May2014

Receivedinrevisedform19June2014 Accepted24June2014

Availableonline30June2014 Editor:J.-P.Blaizot

We investigate the correlations between the quark spin and orbital angular momentum inside the nucleon.SimilarlytotheJirelation,weshowthatthesecorrelationscanbeexpressedintermsofspecific momentsofmeasurablepartondistributions.Thisprovidesawholenewpieceofinformationaboutthe partonicstructureofthenucleon.

©2014PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/3.0/).FundedbySCOAP3_.

1. Introduction

Oneofthekeyquestionsinhadronicphysics istounravelthe spin structure of the nucleon, a very interesting playground for understandingmanynon-pertubativeaspectsofquantum chromo-dynamics (QCD).So far, mostof the effortshave focused on the proper decomposition of the nucleon spin into quark/gluon and spin/orbitalangularmomentum (OAM)contributions (seeRef. [1]

foradetailedrecentreview)andtheirexperimentalextraction.The spinstructureishoweverricherthanthis.

SincethespinandOAMhavenegativeintrinsicparity,theonly non-vanishingsingle-parton(a

=

q

,

G)longitudinalcorrelations al-lowedby parityinvarianceare



Sa

zSNz



,



LazSzN



and



LazSaz



, where





denotestheappropriateaverage, Sqz,G isthequark/gluon longi-tudinal spin, Lqz,G isthequark/gluon longitudinalOAM and SNz is the nucleon longitudinalspin. Since we are interested in the in-trinsic correlations only, the globalorbital motion of the system

LN_z isnotconsidered.Thefirsttwokindsofcorrelationareusually justcalledspinandOAMcontributionsofpartona tothenucleon spin.Thelasttypeissimplythepartonspin–orbitcorrelation.

Eventhoughgeneralizedpartondistributions(GPDs)and trans-verse-momentumdependentpartondistributions(TMDs)are natu-rallysensitivetothepartonspin–orbitcorrelations,noquantitative relationbetweenthemhasbeenderivedsofar.Theonly quantita-tiverelationwe areawareofhasbeenobtainedinRef.[2]atthe level of generalized TMDs (GTMDs) [3,4], also known as uninte-grated GPDs (uGPDs), whichare unfortunately not yet relatedto anyexperimentalprocess.

*

Correspondence to: IPNO, Université Paris-Sud, CNRS/IN2P3, 91406 Orsay, France.

E-mailaddresses:lorce@ipno.in2p3.fr,C.Lorce@ulg.ac.be.

InthisLetterweprovidetherelationbetweenthequark spin– orbit correlation and measurable parton distributions. Our ap-proachis similarto theone usedinRef. [5]inthecaseofquark OAM, butthis time in the parity-oddsector and withan asym-metrictensor.TheLetter isorganized asfollows:InSection 2,we define the quark spin–orbit correlation operator andexpress the corresponding expectationvalue interms offormfactors.In Sec-tion 3 we relate these form factors to moments of measurable parton distributions. InSection 4,we provide an estimate of the variouscontributions,andconcludethepaperwithSection5. 2. Quarkspin–orbitcorrelation

It is well known that the light-front operator giving the total numberofquarks canbe decomposedinto thesum ofright- and left-handedquarkcontributions

ˆ

Nq

=



d3x

ψγ

+

ψ

(1)

=



d3x

ψ

R

γ

+

ψ

R







ˆ NqR

+



d3x

ψ

L

γ

+

ψ

L







ˆ NqL

,

(2)

where

ψR

,L

=

1₂

(

1

±

γ

5

)ψ

,a±

=

√1₂

(

a0

+

a3

)

for a generic four-vectora,andd3x

=

dx−d2x_⊥.Thequarklongitudinalspinoperator simply corresponds to half of the difference between right- and left-handedquarknumbers

ˆ

Sq_z

=



d3x1 2

ψγ

+

_γ

5

ψ

(3)

=

1 2

 ˆ

N qR

− ˆ

_NqL



_.

₍₄₎ http://dx.doi.org/10.1016/j.physletb.2014.06.068

Similarly, we decompose the local gauge-invariant light-front operatorforthequarklongitudinalOAM[5]intothesum of right-andleft-handedquarkcontributions

ˆ

Lqz

=



d3x1 2

ψγ

+

₍

_x

_×

_i↔_D

₎

z

ψ

(5)

= ˆ

LqR z

+ ˆ

LqzL

,

(6)

where ↔D

=

→

∂

−

←

∂

−

2ig A is the symmetric covariant derivative, and L

ˆ

qR,L z

=

d3_x1 2

ψ

R,L

γ

+

(

x

×

i ↔

D

)zψR

,L. The difference between theseright- andleft-handed quark contributions will be referred toasthequarklongitudinalspin–orbitcorrelationoperatorwhich reads

ˆ

Cqz

=



d3x1 2

ψγ

+

_γ

5

(

x

×

i ↔ D

)

z

ψ

(7)

= ˆ

LqR z

− ˆ

LqzL

.

(8)

Thequark spinandOAMoperators attracteda lotofattention becausethey entertheJi decompositionofthe totalangular mo-mentumoperatorinQCD[5]

ˆ

Jz

= ˆ

Sqz

+ ˆ

Lqz

+ ˆ

JGz

.

(9)

Though, aswe haveseen, a completecharacterization ofthe nu-cleonlongitudinalspinstructurerequiresustogobeyondthisand toconsiderthequarknumberandspin–orbitcorrelationoperators aswell.Contrarytothequarknumber,thequarkspin–orbit corre-lationdefinedbyEq.(7)has,tothebestofourknowledge,never beenstudied so far. The purpose ofthis Letter is to fillthisgap andtoshowthatsuchaquantityisactuallyrelatedtomeasurable quantities.

Webasically followthe samestrategy as Ji inRef. [5],except forthefactthatwedirectlyconsiderthemoregeneralasymmetric gauge-invariantenergy–momentumtensorinsteadofthe symmet-ricgauge-invariant(orBelinfante)one.Wepostponethediscussion ofthisparticularpointtoSection 4.Thequark OAMoperatorcan thenconvenientlybeexpressedasfollows

ˆ

Lqz

=



d3x



x1T

ˆ

_q+2

−

x2T

ˆ

_q+1



,

(10)

whereT μν is

ˆ

thequarkenergy–momentumtensoroperator[1]

ˆ

Tqμν

=

1 2

ψγ

μ_i↔_Dν

_ψ

₍₁₁₎

= ˆ

TqμνR

+ ˆ

T μν qL (12) with T

ˆ

qμνR,L

=

1 2

ψ

R,L

γ

μi ↔

Dν

ψR

,L. Similarly, we rewrite the quark spin–orbitoperatoras

ˆ

Cqz

=



d3x



x1T

ˆ

_q5+2

−

x2T

ˆ

_q5+1



,

(13)

where T

ˆ

_q5μν can be considered as the parity-odd partner of the quarkenergy–momentumtensoroperator

ˆ

T_q5μν

=

1 2

ψγ

μ

_γ

5i ↔ Dν

ψ

(14)

= ˆ

TqμνR

− ˆ

T μν qL

.

(15)

Just like in the case of the generic asymmetric energy– momentum tensor [1,6], we find that the non-forward matrix elements of T

ˆ

_q5μν can be parametrized in termsof five form fac-tors(FFs)



p

,

s

| ˆ

T_q5μν

|

p

,

s

 =

u



p

,

s



Γ

_q5μνu

(

p

,

s

)

(16) with

Γ

_q5μν

=

P {μ

_γ

ν}

_γ

5 2 A

˜

q

(

t

)

+

P{μ



ν}

γ

5 4M B

˜

q

(

t

)

+

P[μ

γ

ν]

γ

5 2 C

˜

q

(

t

)

+

P[μ



ν]

γ

5 4M D

˜

q

(

t

)

+

Mi

σ

μν

γ

5F

˜

q

(

t

),

(17)

where M isthe nucleon mass, s and s are the initial andfinal rest-frame polarization vectors satisfying s2

₌

_s2

₌

_{1, P}

₌

p+p 2 is the average four-momentum, and t

= 

2 is the square of the four-momentumtransfer



=

p

−

p.Forconvenience,weusedthe notationsa{μbν}

=

aμbν

+

aν bμ anda[μbν]

=

aμbν

−

aν bμ.

SinceweareinterestedinthematrixelementofEq.(13)which involvesonlyoneexplicitpowerofx,weneedtoexpand Eq.(16)

onlyuptolinearorderin



[1,6].Consideringinitialandfinal nu-cleonstateswiththesamerest-framepolarizations

=

s

= (

s_⊥

,

sz) andusingthelight-frontspinors (seee.g. Appendix AofRef.[7]), wearriveatthefollowingexpression



p

,

s

| ˆ

T_q5μν

|

p

,

s

 =

P{μSν}

−

P {μ_i

_

ν}+P 2P+



˜

Aq

P[μSν]

−

P [μ_i

_

ν]+P 2P+



( ˜

Cq

−

2F

˜

q

)

+

i



μνP_Fq

˜

₊

_O



_

2



₍₁₈₎

with



0123

= +

1 and the covariant spin vector Sμ

= [

szP+

,

−

szP−

+

P_P⊥+

· (

Ms⊥

+

P_⊥sz),Ms_⊥

+

P_⊥sz

]

satisfying P

·

S

=

0 and S2

_{= −}

_M2

₋

_s2

z

(

P2

−

M2

)

.Forconvenience, we removed the argumentoftheFFswhenevaluatedatt

=

0,i.e.X

˜

q

= ˜

Xq(0

)

.

Substituting the expansion (18) into the matrix element of Eq.(13)andworkinginthesymmetriclight-frontframe,i.e. with

P_⊥

=

0_⊥,wefind Cqz

≡



P

,

ez

| ˆ

Cqz

|

P

,

ez





P

,

ez

|

P

,

ez



=

1 2

( ˜

Aq

+ ˜

Cq

).

(19)

Thus, to determine the quark spin–orbit correlation, one has to measurethe A

˜

q(t

)

andC

˜

q(t

)

FFs,whichareanalogoustothe axial-vector FFGq_A

(

t

)

.The B

˜

q(t

)

and D

˜

q(t

)

FFs,whichare analogousto theinducedpseudoscalarFFGq_P

(

t

)

,arenotneededsincethey con-tributeonlytohigherx-momentsofT

ˆ

_q5μν,asonecanseefromthe expansionu

(

p

,

s

)

Pμνγ5

4M u

(

p

,

s

)

=

O(

2

₎

_. 3. Linkwithpartondistributions

Like in thecaseof theenergy–momentum tensor,there is no fundamental probethat couples to T

ˆ

_q5μν inparticle physics. How-ever,itispossibleto relatethecorresponding variousFFs to spe-cific momentsof measurablepartondistributions.From the com-ponentT

ˆ

_q5++,wefind



dx xHq

˜

(

x

, ξ,

t

)

= ˜

Aq

(

t

),

(20)



dx xEq

˜

(

x

, ξ,

t

)

= ˜

Bq

(

t

),

(21)

where H

˜

q(x

,

ξ,

t

)

and E

˜

q(x

,

ξ,

t

)

are the GPDs parametrizing the non-localtwist-2axial-vectorlight-frontquarkcorrelator[8]

1 2



_dz− 2

π

e ixP+z−

_

_p

_,

_s

_|ψ



−

z− 2

γ

+

γ

5

W

ψ



z− 2

|

p

,

s



=

1 2P+u



p

,

s



Γ

_{q A}+u

(

p

,

s

)

(22)

with

W = P

exp

[

ig

_z−−z_/−₂/2d y−A+

(

y−

)

]

astraightlight-front

Wil-sonlineand

Γ

_{q A}+

=

γ

+

γ

5Hq

˜

(

x

, ξ,

t

)

+



+

γ

5

2M Eq

˜

(

x

, ξ,

t

),

(23)

theskewnessvariablebeinggivenby

ξ

= −

+

/

2P+.

The relationsfor theother FFs can be obtainedusingthe fol-lowingQCDidentity

ψγ

[μ

γ

5i ↔

Dν]

ψ

=

2m

ψ

i

σ

μν

γ

5

ψ

−



μναβ

∂

α

(ψ

γ

β

ψ ),

(24)

wherem is the quark mass.Taking the matrix elements of both sidesandusingsomeGordonand



-identities,wefind

˜

Cq

(

t

)

=

m 2MH q 1

(

t

)

−

F q 1

(

t

),

(25)

˜

Dq

(

t

)

=

m 2MH q 2

(

t

)

−

F q 2

(

t

),

(26)

˜

Fq

(

t

)

=

m 2MH q 3

(

t

)

−

1 2G q E

(

t

),

(27)

wheretheelectricSachsFFisgivenby Gq_E

(

t

)

=

Fq₁

(

t

)

+

_4Mt2F q 2

(

t

)

. TheFFs ontheright-handside parametrizethevector andtensor localcorrelatorsasfollows



p

,

s

|ψ

γ

μ

ψ

|

p

,

s

 =

u



p

,

s



Γ

_qVμu

(

p

,

s

),

(28)



p

,

s

|ψ

i

σ

μν

γ

5

ψ

|

p

,

s

 =

u



p

,

s



Γ

_qTμνu

(

p

,

s

)

(29) with

Γ

_qVμ

=

γ

μF₁q

(

t

)

+

i

σ

μν

_

ν 2M F q 2

(

t

),

(30)

Γ

_qTμν

=

P [μ

_γ

ν]

_γ

5 2M H q 1

(

t

)

+

P[μ



ν]

γ

5 4M2 H q 2

(

t

)

+

i

σ

μν

γ

5Hq3

(

t

).

(31)

OurtensorFFs are relatedtotheonesused inRefs. [9,10]inthe followingway Aq_{T 10}

(

t

)

= −

1 2

Hq₁

(

t

)

+

t 4M2H q 2

(

t

)



+

Hq₃

(

t

),

(32) Bq_{T 10}

(

t

)

=

1 2



Hq₁

(

t

)

+

Hq₂

(

t

)



,

(33)

˜

Aq_{T 10}

(

t

)

= −

1 4H q 2

(

t

),

(34)

andcanalsobe expressedintermsofmomentsoftwist-2 chiral-oddGPDs.

Thequarkspin–orbitcorrelationisthereforegivenby the sim-pleexpression Cq_z

=

1 2



dx xH

˜

q

(

x

,

0

,

0

)

−

1 2

F₁q

(

0

)

−

m 2MH q 1

(

0

)



(35)

whichistheanalogueintheparity-oddsectoroftheJirelation[5]

forthequarkOAM Lqz

=

1 2



dx x



Hq

(

x

,

0

,

0

)

+

Eq

(

x

,

0

,

0

)



−

1 2G q A

(

0

).

(36) It may seem counter-intuitive that the spin–orbit correlation Cqz isrelated to thespin–spin correlation H

˜

q.But, justlike inthe Ji relation,the“orbital”informationisactuallyprovidedbytheextra

x-factor representing the fraction of longitudinal momentum, as discussedbyBurkardtinRef.[11].

From thecomponents T

ˆ

_q5j+ with j

=

1

,

2 we alsofind the fol-lowingrelations



dx xG

˜

q₁

(

x

, ξ,

t

)

= −

1 2

 ˜

Bq

(

t

)

+ ˜

Dq

(

t

)



,

(37)



dx xG

˜

q₂

(

x

, ξ,

t

)

= −

1 2

 ˜

Aq

(

t

)

+ ˜

Cq

(

t

)



+



1

− ξ

2

 ˜

Fq

(

t

),

(38)



dx xG

˜

q₃

(

x

, ξ,

t

)

= −

ξ

2F

˜

q

(

t

),

(39)



dx xG

˜

q₄

(

x

, ξ,

t

)

= −

1 2F

˜

q

(

t

),

(40)

whereG

˜

q_i

(

x

,

ξ,

t

)

aretheGPDsparametrizingthenon-localtwist-3 axial-vector light-front quark correlator [12]. As a check, we in-sertedinEqs.(37)–(40)theexpressionsfor

dxxG

˜

q_i

(

x

,

ξ,

t

)

derived in Ref. [12] within the Wandzura–Wilczekapproximation (which is exact for the lowest two x-moments in the chiral limit [12]) and consistently recovered Eqs. (25)–(27) in the massless quark limit. The analogue of the Penttinen–Polyakov–Shuvaev–Strikman relation[12,13]

Lqz

= −



dx xG2

(

x

,

0

,

0

)

(41)

intheparity-oddsectoristherefore Cqz

= −



dx x

 ˜

Gq₂

(

x

,

0

,

0

)

+

2G

˜

q₄

(

x

,

0

,

0

)



.

(42)

Forcompleteness,wenotethatthequarkspin–orbitcorrelation canalsobeexpressedintermsofGTMDs

Cqz

=



dx d2k_⊥k 2 ⊥ M2G q 11



x

,

0

,

k2_⊥

,

0

,

0



.

(43)

As discussed in Refs. [14–17], the shape of the Wilson line en-tering the definitionof the GTMDsdetermines the type of OAM. Since in this Letter we are interested in the Ji OAM, the GTMD

Gq₁₁

(

x

,

ξ,

k2_⊥

,

k_⊥

· 

_⊥

,



2_⊥

)

in Eq. (43) has to be defined with a direct straightWilson line. Onthecontrary, thespin–orbit corre-lation introduced in Ref. [2] dealt with the Jaffe–Manohar OAM whichisobtainedby definingtheGTMDswithastaple-like light-frontWilsonline.

4. Discussion

In the previous section, we have obtainedthree different ex-pressions for the quark spin–orbit correlation in terms of par-ton distributions.From an experimentalpoint ofview,Eq.(35)is clearly the most useful one. By equating the right-hand side of Eq.(35)withtheright-handsidesofEqs.(42)and

(43)

,weobtain twonewsumrulesamongpartondistributions.

Inordertodeterminethequarkspin–orbitcorrelation,weneed to knowthree quantities.Thefirstquantity istheDiracFF evalu-ated at t

=

0 which simply corresponds to the valence number, namely Fu

1

(

0

)

=

2 and F1d

(

0

)

=

1 ina proton,andthereforedoes notrequireanyexperimentalinput.Thesecondquantityisthe ten-sorFF Hq₁

(

0

)

whichisnotknownsofar.However,sinceitappears multipliedbythemassratiom

/

4M

∼

10−3 foru andd quarksin Eq.(35),wedonotexpectittocontributesignificantlytoCqz.The last quantityisthesecondmomentofthequark helicity distribu-tion 1



−1 dx xH

˜

q

(

x

,

0

,

0

)

=

1



0 dx x





q

(

x

)

− 

q

(

x

)



.

(44)

Contrary tothelowestmoment

₋1₁dxH

˜

q(x

,

0

,

0

)

=

₀1dx

[

q

(

x

)

+



q

(

x

)

]

, this second moment cannot be extracted from deep-inelastic scattering (DIS) polarized data without additional as-sumptions about the polarized sea quark densities. The separate

Table 1

Comparisonbetweenthelowesttwoaxialmoments(n)_q≡ 1

−1dxxnH˜q(x,0,0)for

q=u,d aspredictedbythenaivequarkmodel(NQM),thelight-frontconstituent quarkmodel(LFCQM)andthelight-frontchiralquark-solitonmodel(LFχQSM)at thescaleμ2_∼0

.26GeV2_{[24],withthecorrespondingvaluesobtainedfromtheLSS} fittoexperimentaldata[18]atthescaleμ2_=1GeV2_{andLatticecalculations[23]} atthescaleμ2_=4GeV2_andpionmassm

π=293MeV. Model (0)_{u }(0)_{d }(1)_{u }(1)_d NQM 4/3 −1/3 4/9 −1/9 LFCQM 0.995 −0.249 0.345 −0.086 LFχQSM 1.148 −0.287 0.392 −0.098 Exp. 0.82 −0.45 ≈0.19 ≈ −0.06 Latt. 0.82(7) −0.41(7) ≈0.20 ≈ −0.05

quark and antiquark contributions can however be obtained e.g.

in a combined fit to inclusive and semi-inclusive DIS. From the Leader–Sidorov–Stamenov(LSS)analysisofRef.[18],weobtain

1



−1 dx xHu

˜

(

x

,

0

,

0

)

≈

0

.

19

,

(45) 1



−1 dx xH

˜

d

(

x

,

0

,

0

)

≈ −

0

.

06

,

(46)

atthescale

μ

2

₌

_1GeV2_{,leading to C}u

z

≈ −

0

.

9 and Cdz

≈ −

0

.

53. Note that these estimates are however not particularly reliable since it follows from the new HERMES [19] and COMPASS [20]

data on multiplicities that the fragmentation functions given in Ref. [21] and used in the LSS analysis are presumably not cor-rect [22]. Further experimental data and dedicated analyses are thereforerequired.Nevertheless,thesevaluesseemconsistentwith recentLatticecalculationsbytheLHPCCollaboration[23],see

Ta-ble 1.

Thesecond moment ofthequark helicity distribution beinga valence-like quantity withsuppressed low-x region,we may ex-pectphenomenological quark model predictionsto be more reli-ableforthis second momentthan forthelowest one. In

Table 1

we providethe first two moments ofthe u and d-quark helicity distributions obtained within the naive quark model (NQM), the light-front constituent quark model (LFCQM) and the light-front chiralquark-solitonmodel(LF

χ

QSM)atthescale

μ

2

_∼

₀

_.

_26GeV2_, seeRef. [24] formoredetails. Fromtheseestimates,we expect a negativequark spin–orbitCqz forbothu andd quarks(Cuz

≈ −

0

.

8 andCd

z

≈ −

0

.

55),meaningthatthequarkspinandJiOAMare ex-pectedtobe,inaverage,anti-correlated.Thishastobecontrasted withthe modelresultsobtainedinRef. [2]wherethequark spin andJaffe–ManoharOAMare,inaverage,correlated.

Finally, we would like to comment about the difference be-tweensymmetric andasymmetric tensors.The canonicalenergy– momentum tensor obtainedfrom Noethertheorem is conserved, but is usually neither gauge-invariant nor symmetric. One has however the freedom to define alternative energy–momentum tensors by adding so-called superpotential terms [1,5,25] to the canonicalenergy–momentumtensor.Thesetermsareoftheform

∂

λGλμν withGμλν

= −

Gλμν ,sothatthenewtensorsremain

con-servedandthetotal four-momentum(i.e. integratedoverallspace) remains unchanged. Physically, adding superpotential terms cor-responds to relocalizing/redefining what we mean by density of energyandmomentum[26].

The Belinfante procedure simply makes use of this freedom to define a symmetric andgauge-invariant tensor. It is however importantto realizethat the symmetry property ofthe energy– momentumtensorisnotmandatoryinparticlephysics,butis

ba-sicallymotivatedbyGeneralRelativitywheregravitationiscoupled toa symmetricenergy–momentumtensor.From theconservation of totalangular momentum

∂

μ

(

xν T μρ

−

xρ T μν

+

Sμνρ

)

=

0,one

seesthattheantisymmetricpartoftheenergy–momentumtensor isintimatelyrelatedtothespindensity

Tνρ

−

Tρν

= −∂

μSμνρ

.

(47)

GeneralRelativityisapurelyclassical theorywherethenotionof intrinsicspindoesnotexist,andsoitishardlysurprisingthatonly asymmetricenergy–momentumtensorisneededinthatcontext.

So,assoonasonedealswithaspindensity,thenatural gauge-invariant energy–momentum tensorwill be asymmetric.The fact thatonecanstilldefineaconsistentsymmetric(orBelinfante) ten-sorisaconsequenceofthefollowingQCDidentity

ψγ

[μi↔Dν]

ψ

= −



μναβ

∂

α

(ψγ

β

γ

5

ψ ),

(48)

which is the parity-even analogue of Eq. (24). It tells us that the difference between the symmetric and asymmetric tensors (namely the antisymmetric part) is nothing but a superpotential term. This means that one can effectively absorb the contribu-tionofthespindensityinarelocalization/redefinitionofenergy– momentumdensity.

In Ref. [5], Ji considered the symmetric (Belinfante) energy– momentumtensor,discardingfromtheverybeginningthenotion ofintrinsicspindensity.Accordingly,Jiwasabletorelateonlythe

total angularmomentumtomomentsofGPDs

Jq_z,G

=

1 2



dx x



Hq,G

(

x

,

0

,

0

)

+

Eq,G

(

x

,

0

,

0

)



.

(49)

InordertogetthequarkOAM,onehastosubtractasasecondstep thequarkspincontribution Sqz

=

12G

q

A

(

0

)

fromthetotalquark an-gular momentum contribution Jqz.So Eq.(36)can be understood as Lqz

=

Jqz

−

Sqz,or moreprecisely as



LqzSNz



= 

J

q zSNz



− 

S

q zSNz



. Starting with the more general asymmetric energy–momentum tensor, one can directly define the OAM operator as in Eq. (10)

andgetthefinal result(36)withoutinvokinganyadditionalstep

[1,6,27].

Similarly, one can define anotherquark correlation usingonly thesymmetricpartofT

ˆ

_q5μν incloseanalogywithEq.(13)

ˆC

q z

=



d3x



x11 2T

ˆ

{+2} q5

−

x2 1 2T

ˆ

{+1} q5

=

1 2C

ˆ

q z

+



d3x1 4

ψ

i ↔ D+

(

x

×

γ

)

z

γ

5

ψ.

(50) Contrary to C

ˆ

qz, the operator

ˆC

q

z is time-independent since the corresponding currentM

ˆ

μνρ_q5

=

xν1₂T

ˆ

_q5{μρ}

−

xρ1₂T

ˆ

_q5{μν} is conserved

∂

μM

ˆ

q5μνρ

=

0, as a consequence of the symmetry of T

ˆ

{μν} q5 . Note howeverthateveniftheoperatorC

ˆ

qz isingeneraltime-dependent, its expectationvalue is time-independent becausethe initial and finalnucleonstateshavethesameenergy[1].

Itisprettysimplenowtogettheexpressionsforthematrix el-ementof

ˆC

qz.WeneedtokeeponlythesymmetricpartofEq.(16),

i.e. settheFFsC

˜

q,D

˜

qandF

˜

q tozerointhederivationofSection2. Wethenfind

C

q z

≡



P

,

ez

| ˆC

qz

|

P

,

ez





P

,

ez

|

P

,

ez



=

1 2A

˜

q

(

0

),

(51)

=

1 2 1



−1 dx xH

˜

q

(

x

,

0

,

0

).

(52)

Contrary to Cqz, the physical interpretation of

C

qz is not particu-larly clearowing tothe massterminEq.(24).Indeed,thismass termdoesnothavetheformofasuperpotential,whichinturn im-pliesthat

d3xT

ˆ

_q5+ν

=

d3x1₂T

ˆ

_q5{+ν}.We argued inSection 2 that

Cqz can naturally be interpreted as (twice) the quark spin–orbit correlation



LqzSqz



. By analogy with the parity-even sector, it is temptingtointerpret

C

qz as(twice)thecorrelationbetweenquark spin and quark total angular momentum



JzqSqz



. This is actually trueifonegets ridofthe massterm,i.e. work inthechirallimit

m

=

0.Inthiscase, justlikein theparity-evensector withtheJi relation,wecan understandEq.(35)as



LqzSqz



= 

JqzSqz



− 

SqzSqz



. The quark spin–spin correlation



SqzSqz



is of course quite trivial anddoesnot depend on the spin orientation. Thatis the reason why the unpolarized quark FF Fq₁

(

0

)

appears in the expression for Cqz.

5. Conclusions

We provided a local gauge-invariant definition of the quark spin–orbit correlation, which is a new independent piece of in-formation aboutthe nucleon longitudinal spin structure. We de-rived several expressions for the expectation value of this cor-relation in terms of measurable parton distributions, leading to newsumrules.Using estimatesfromfitstoavailable experimen-tal data, Lattice calculations and phenomenological quark mod-els, we concluded that the quark spin–orbit correlation is very likely negative, meaning that thequark spin andJi orbital angu-lar momentum are, in average, anti-correlated. However, a more precisedeterminationofthisquark spin–orbitcorrelationrequires furtherexperimental dataanddedicatedanalyses inorderto dis-entangle quark and antiquark contributions to the helicity dis-tribution. Tensor form factors are also in principle needed, but the corresponding contribution can safely be neglected for light quarks.

Acknowledgements

Iamthankful toH. Moutarde,B. PasquiniandM. Polyakovfor usefuldiscussions relatedto thisstudy.Iam alsoverygratefulto E. Leader, A.V. Sidorov andD.B. Stamenov for providingme with their extraction ofthe quark helicitydistributions, andtoM. En-gelhardt forinformingme aboutrecentLatticeresults.Thiswork was supportedby theBelgianFundF.R.S.-FNRS(grant1.B.147.13F)

via thecontractofChargéderecherches.

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