Spin-orbit correlations in the nucleon

0.17 MB·

by  Cédric Lorcé

Spin-orbit correlations in the nucleon
C´edric Lorc´e
∗
IPNO, Universit´e Paris-Sud, CNRS/IN2P3, 91406 Orsay, France and
IFPA, AGO Department, Universit´e de Li`ege, Sart-Tilman, 4000 Li`ege, Belgium
(Dated: June25, 2014)
Weinvestigatethecorrelationsbetweenthequarkspinandorbitalangularmomentuminsidethe
nucleon. Similarly to the Ji relation, we show that these correlations can be expressed in terms of
specificmomentsofmeasurablepartondistributions. Thisprovidesawholenewpieceofinformation
about the partonic structureof the nucleon.
PACSnumbers: 11.15.-q,12.38.Aw,13.88.+e,13.60.Hb,14.20.Dh
4
1 I. INTRODUCTION measurable parton distributions. In section IV, we pro-
0
vide an estimate of the various contributions, and con-
2
clude the paper with section V.
One of the key questions in hadronic physics is to un-
n
ravelthespinstructureofthe nucleon,averyinteresting
u
playground for understanding many non-pertubative as-
J II. QUARK SPIN-ORBIT CORRELATION
pects of quantum chromodynamics (QCD). So far, most
4
of the efforts have focused on the proper decomposition
2
ofthenucleonspinintoquark/gluonandspin/orbitalan- Itiswellknownthatthelight-frontoperatorgivingthe
] gular momentum (OAM) contributions (see Ref. [1] for total number of quarks can be decomposed into the sum
h a detailed recent review) and their experimental extrac- of right- and left-handed quark contributions
p
tion. The spin structure is however richer than this.
-
p Since the spin and OAM have negative intrinsic par- Nˆq = d3xψγ+ψ (1)
Z
e ity, the only non-vanishing single-parton (a = q,G) lon-
h
[ ghSitauSdNinia,lhLcoarSreNlaitaionndshLalalSowaie,dwbhyerepahritiydiennvoatreiasntcheeaapre- =Z d3xψRγ+ψR+Z d3xψLγ+ψL, (2)
z z z z z z
2 propriate average, Sq,G is the quark/gluon longitudinal
v spin, Lq,G is the quarzk/gluonlongitudinalOAMandSN | Nˆ{qzR } | Nˆ{qzL }
z z
4 is the nucleon longitudinal spin. Since we are interested whereψ = 1(1±γ )ψ,a = 1 (a0+a3)forageneric
8 R,L 2 5 ± √2
7 in the intrinsic correlations only, the global orbital mo- four-vector a, and d3x = dx−d2x . The quark longi-
7 tion of the system LNz is not considered. The first two tudinal spin operator simply corres⊥ponds to half of the
. kindsofcorrelationareusuallyjustcalledspinandOAM difference betweenright-andleft-handedquarknumbers
1
contributions of parton a to the nucleon spin. The last
0
4 type is simply the parton spin-orbit correlation. Sˆq = d3x1ψγ+γ ψ (3)
1 Even though generalized parton distributions (GPDs) z Z 2 5
v: atinodnst(rTanMsvDesr)sea-rmeonmateunrtaulmly sdeenpseitnidveenttopthaertponartdoinstsrpibinu-- = 21(NˆqR −NˆqL). (4)
i
X orbitcorrelations,noquantitativerelationbetweenthem Similarly, we decompose the local gauge-invariant
r has been derived so far. The only quantitative relation light-front operator for the quark longitudinal OAM [5]
a we areawareofhas been obtainedin Ref. [2] atthe level into the sum of right- and left-handed quark contribu-
of generalizedTMDs (GTMDs) [3, 4], also known as un- tions
integrated GPDs (uGPDs), which are unfortunately not
yet related to any experimental process. Lˆq = d3x1ψγ+(x×iD↔) ψ (5)
z Z 2 z
In this Letter we provide the relation between the
quark spin-orbit correlation and measurable parton dis- =LˆqR +LˆqL, (6)
z z
tributions. Our approach is similar to the one used in
Ref. [5] in the case of quark OAM, but this time in the where D↔ = →∂− ←∂− 2igA is the symmetric covariant
parity-odd sector and with an asymmetric tensor. The derivative, and LˆqR,L = d3x1ψ γ+(x×iD↔) ψ .
Letter is organized as follows: In section II, we define z 2 R,L z R,L
The difference betweenRthese right- and left-handed
thequarkspin-orbitcorrelationoperatorandexpressthe
quark contributions will be referred to as the quark lon-
correspondingexpectationvalueintermsofformfactors.
gitudinal spin-orbit correlation operator which reads
In section III we relate these form factors to moments of
Cˆq = d3x1ψγ+γ (x×iD↔) ψ (7)
z Z 2 5 z
∗Electronicaddress: [email protected];[email protected] =LˆqzR −LˆqzL. (8)
2
The quark spin and OAM operators attracted a lot of whereM isthenucleonmass,sands aretheinitialand
′
attention because they enter the Ji decompositionof the finalrest-framepolarizationvectorssatisfyings2 =s2 =
′
total angular momentum operator in QCD [5] 1,P = p′+p istheaveragefour-momentum,andt=∆2is
2
thesquareofthefour-momentumtransfer∆=p −p. For
Jˆz =Sˆzq+Lˆqz+JˆzG. (9) convenience,weusedthe notationsa{µbν} =aµb′ν+aνbµ
and a[µbν] =aµbν −aνbµ.
Though, as we have seen, a complete characterization
Since we are interested in the matrix element of
of the nucleon longitudinal spin structure requires us to
Eq. (13) which involves only one explicit power of x, we
go beyond this and to consider the quark number and
needtoexpandEq.(16)onlyuptolinearorderin∆[1,6].
spin-orbit correlation operators as well. Contrary to the
Consideringinitialandfinalnucleonstateswiththesame
quark number, the quark spin-orbit correlation defined
rest-frame polarization s = s = (s ,s ) and using the
by Eq. (7) has, to the best of our knowledge, never been light-front spinors (see e.′g. Append⊥ix Az of Ref. [7]), we
studied so far. The purpose of this Letter is to fill this
arrive at the following expression
gap and to show that such a quantity is actually related
to measurable quantities. hp,s|Tˆµν|p,si= P µSν − P{µiǫν}+∆P A˜
′ q5 h { } 2P+ i q
We basically follow the same strategy as Ji in Ref. [5], P[µSν]− P[µiǫν]+∆P (C˜ −2F˜ )
exceptforthefactthatwedirectlyconsiderthemoregen- h 2P+ i q q
eral asymmetric gauge-invariant energy-momentum ten- +iǫµν∆P F˜q+O(∆2) (18)
sor instead of the symmetric gauge-invariant (or Belin- with ǫ = +1 and the covariant spin vector Sµ =
0123
fpaonitnet)toonSe.ecWtioenpoIVst.pTonheetqhueadrkiscOuAssMionopofertahtiosrpcaarntictuhleanr i[ssfzyPin+g,−Ps·zPS−=+0PPa⊥+n·d(MS2s⊥=+−PM⊥2sz−),sM2(sP⊥2+−PM⊥2s)z.] sFaotr-
z
conveniently be expressed as follows
convenience, we removed the argument of the FFs when
evaluated at t=0, i.e. X˜ =X˜ (0).
q q
Lˆq = d3x(x1Tˆ+2−x2Tˆ+1), (10) Substituting the expansion (18) into the matrix ele-
z Z q q
mentofEq.(13)andworkinginthesymmetriclight-front
where Tˆµν is the quark energy-momentumtensor opera- frame, i.e. with P =0 , we find
⊥ ⊥
tor [1] Czq ≡ hPh,Pe,ze|zCˆ|zqP|,Pe,zeizi = 12(A˜q +C˜q). (19)
Tˆµν = 1ψγµiD↔νψ (11) Thus, to determine the quark spin-orbit correlation, one
q 2 hastomeasuretheA˜ (t)andC˜ (t)FFs,whichareanalo-
=TˆqµRν +TˆqµLν (12) gous to the axial-vectqorFF GqAq(t). The B˜q(t) and D˜q(t)
FFs, which are analogous to the induced pseudoscalar
with TˆqµRν,L = 12ψR,LγµiD↔νψR,L. Similarly, we rewrite FF GqP(t), are not needed since they contribute only to
the quark spin-orbit operator as higherx-momentsofTˆµν,asonecanseefromtheexpan-
q5
sion u(p,s)Pµ∆νγ5 u(p,s)=O(∆2).
Cˆq = d3x(x1Tˆ+2−x2Tˆ+1), (13) ′ 4M
z Z q5 q5
where Tˆµν can be considered as the parity-odd partner III. LINK WITH PARTON DISTRIBUTIONS
q5
of the quark energy-momentum tensor operator
Likeinthecaseoftheenergy-momentumtensor,there
Tˆqµ5ν = 12ψγµγ5iD↔νψ (14) is no fundamental probe that couples to Tˆqµ5ν in particle
physics. However,it is possible to relate the correspond-
=Tˆµν −Tˆµν. (15)
qR qL ingvariousFFstospecificmomentsofmeasurableparton
distributions. From the component Tˆ++, we find
Justlike in the caseofthe genericasymmetric energy- q5
momentum tensor [1, 6], we find that the non-forward
dxxH˜ (x,ξ,t)=A˜ (t), (20)
matrix elements of Tˆµν can be parametrized in terms of Z q q
q5
five form factors (FFs)
dxxE˜ (x,ξ,t)=B˜ (t), (21)
Z q q
hp′,s′|Tˆqµ5ν|p,si=u(p′,s′)Γµq5νu(p,s) (16) where H˜ (x,ξ,t) and E˜ (x,ξ,t) are the GPDs
q q
with parametrizing the non-local twist-2 axial-vector light-
front quark correlator [8]
Γµν = P{µγν}γ5 A˜ (t)+ P{µ∆ν}γ5 B˜ (t)
q5 2 q 4M q 1 dz
+ P[µγ2ν]γ5 C˜q(t)+ P[µ4∆Mν]γ5 D˜q(t) 2Z 2π− eixP+z−hp′,s′|ψ(−z2−)γ+γ5Wψ(z2−)|p,si
+Miσµνγ5F˜q(t), (17) = 2P1+ u(p′,s′)Γ+qAu(p,s) (22)
3
with W = Pexp[ig z−−z/−2/2dy−A+(y−)] a straight light- the fraction of longitudinal momentum, as discussed by
front Wilson line anRd Burkardt in Ref. [11].
From the components Tˆj+ with j = 1,2 we also find
q5
Γ+ =γ+γ H˜ (x,ξ,t)+ ∆+γ5 E˜ (x,ξ,t), (23) the following relations
qA 5 q 2M q
the skewness variable being given by ξ =−∆+/2P+. dxxG˜q(x,ξ,t)=−1[B˜ (t)+D˜ (t)], (37)
Z 1 2 q q
The relations for the other FFs can be obtained using
the following QCD identity dxxG˜q(x,ξ,t)=−1[A˜ (t)+C˜ (t)]+(1−ξ2)F˜ (t),
Z 2 2 q q q
ψγ[µγ iD↔ν]ψ =2mψiσµνγ ψ−ǫµναβ∂ (ψγ ψ), (24) (38)
5 5 α β
where m is the quark mass. Taking the matrix elements Z dxxG˜q3(x,ξ,t)=−2ξ F˜q(t), (39)
ofbothsidesandusingsomeGordonandǫ-identities,we
find dxxG˜q(x,ξ,t)=−1F˜ (t), (40)
Z 4 2 q
C˜ (t)= m Hq(t)−Fq(t), (25)
q 2M 1 1 where G˜q(x,ξ,t) are the GPDs parametrizing the non-
D˜q(t)= 2mM H2q(t)−F2q(t), (26) localtwisit-3axial-vectorlight-frontquarkcorrelator[12].
F˜ (t)= m Hq(t)− 1Gq (t), (27) As a check, we inserted in Eqs. (37)-(40) the expres-
q 2M 3 2 E sions for dxxG˜q(x,ξ,t) derived in Ref. [12] within the
i
where the electric Sachs FF is givenby Gq (t)=Fq(t)+ WandzuraR-Wilczekapproximation(whichisexactforthe
E 1
t Fq(t). The FFs on the right-hand side parametrize lowest two x-moments in the chiral limit [12]) and con-
4M2 2
the vector and tensor local correlators as follows sistently recovered Eqs. (25)-(27) in the massless quark
limit. The analogue of the Penttinen-Polyakov-Shuvaev-
hp′,s′|ψγµψ|p,si=u(p′,s′)ΓµqVu(p,s), (28) Strikman relation [12, 13]
hp,s|ψiσµνγ ψ|p,si=u(p,s)Γµνu(p,s) (29)
′ ′ 5 ′ ′ qT
Lq =− dxxG (x,0,0) (41)
z Z 2
with
Γµ =γµFq(t)+ iσµν∆ν Fq(t), (30) in the parity-odd sector is therefore
qV 1 2M 2
Γµν = P[µγν]γ5 Hq(t)+ P[µ∆ν]γ5 Hq(t) Cq =− dxx[G˜q(x,0,0)+2G˜q(x,0,0)]. (42)
qT 2M 1 4M2 2 z Z 2 4
+iσµνγ Hq(t). (31)
5 3
For completeness, we note that the quark spin-orbit
OurtensorFFsarerelatedtotheonesusedinRefs.[9,10] correlationcan also be expressed in terms of GTMDs
in the following way
AqT10(t)=−12[H1q(t)+ 4Mt2 H2q(t)]+H3q(t), (32) Czq =Z dxd2k⊥ Mk2⊥2 Gq11(x,0,k⊥2,0,0). (43)
Bq (t)= 1[Hq(t)+Hq(t)], (33)
T10 2 1 2 As discussed in Refs. [14–17], the shape of the Wilson
A˜q (t)=−1Hq(t), (34) line entering the definition of the GTMDs determines
T10 4 2
the type of OAM. Since in this Letter we are interested
andcanalsobeexpressedintermsofmomentsoftwist-2 in the Ji OAM, the GTMD Gq (x,ξ,k2,k ·∆ ,∆2)
11
chiral-odd GPDs. in Eq. (43) has to be defined with a dir⊥ect s⊥traigh⊥t W⊥il-
The quark spin-orbit correlation is therefore given by son line. On the contrary, the spin-orbit correlation in-
the simple expression troduced in Ref. [2] dealt with the Jaffe-Manohar OAM
which is obtained by defining the GTMDs with a staple-
Cq = 1 dxxH˜ (x,0,0)− 1[Fq(0)− m Hq(0)] (35) like light-front Wilson line.
z 2Z q 2 1 2M 1
which is the analogue in the parity-odd sector of the Ji
IV. DISCUSSION
relation [5] for the quark OAM
In the previous section, we have obtained three dif-
Lq = 1 dxx[H (x,0,0)+E (x,0,0)]−1Gq(0). (36)
z 2Z q q 2 A ferent expressions for the quark spin-orbit correlation
in terms of parton distributions. From an experimental
It may seem counter-intuitive that the spin-orbit cor- pointofview,Eq.(35)isclearlythe mostusefulone. By
relation Cq is related to the spin-spin correlation H˜ . equating the right-hand side of Eq. (35) with the right-
z q
But,justlikeintheJirelation,the“orbital”information handsidesofEqs.(42)and(43), weobtaintwonewsum
is actually provided by the extra x-factor representing rules among parton distributions.
4
Inordertodetermine the quarkspin-orbitcorrelation,
TABLE I: Comparison between the lowest two axial moments
we need to know three quantities. The first quantity is
the Dirac FF evaluated at t = 0 which simply corre- ∆(n)q ≡ R−11dxxnH˜q(x,0,0) for q = u,d as predicted by
the naive quark model (NQM), the light-front constituent quark
sponds to the valence number, namely F1u(0) = 2 and model (LFCQM) and the light-front chiral quark-soliton model
Fd(0) = 1 in a proton, and therefore does not require (LFχQSM)atthescaleµ2∼0.26GeV2 [24],withthecorrespond-
1
any experimental input. The second quantity is the ten- ing values obtained from the LSS fit to experimental data [18] at
sor FF Hq(0) which is not known so far. However, since the scale µ2 = 1 GeV2 and Lattice calculations [23] at the scale
it appears1 multiplied by the mass ratio m/4M ∼ 10 3 µ2=4GeV2 andpionmassmπ =293MeV.
−
for u and d quarks in Eq. (35), we do not expect it to Model ∆(0)u ∆(0)d ∆(1)u ∆(1)d
contribute significantly to Cq. The last quantity is the
z
second moment of the quark helicity distribution NQM 4/3 −1/3 4/9 −1/9
1 dxxH˜ (x,0,0)= 1dxx[∆q(x)−∆q(x)]. (44) LFCQM 0.995 −0.249 0.345 −0.086
Z q Z
−1 0 LFχQSM 1.148 −0.287 0.392 −0.098
Contrary to the lowest moment 1 dxH˜ (x,0,0) =
1 q Exp. 0.82 −0.45 ≈0.19 ≈−0.06
1 R−
dx[∆q(x) + ∆q(x)], this second moment cannot be
0
Rextracted from deep-inelastic scattering (DIS) polarized Latt. 0.82(7) −0.41(7) ≈0.20 ≈−0.05
datawithoutadditionalassumptionsaboutthepolarized
sea quark densities. The separate quark and antiquark
contributions can however be obtained e.g. in a com-
bined fit to inclusive and semi-inclusive DIS. From the Finally,wewouldliketocommentaboutthedifference
Leader-Sidorov-Stamenov(LSS) analysis of Ref. [18], we betweensymmetricandasymmetrictensors. Thecanoni-
obtain calenergy-momentumtensorobtainedfromNoetherthe-
oremis conserved,but is usually neither gauge-invariant
1
dxxH˜ (x,0,0)≈0.19, (45) norsymmetric. Onehashoweverthefreedomtodefineal-
Z 1 u ternative energy-momentum tensors by adding so-called
−
1 superpotential terms [1, 5, 25] to the canonical energy-
Z dxxH˜d(x,0,0)≈−0.06, (46) momentum tensor. These terms are of the form ∂λGλµν
−1 with Gµλν = −Gλµν, so that the new tensors remain
at the scale µ2 = 1 GeV2, leading to Cu ≈ −0.9 and conserved and the total four-momentum (i.e. integrated
z
Cd ≈−0.53. Note that these estimates are howevernot overallspace)remainsunchanged. Physically,addingsu-
z
particularly reliable since it follows from the new HER- perpotentialtermscorrespondstorelocalizing/redefining
MES[19]andCOMPASS[20]dataonmultiplicities that whatwe meanby density ofenergyandmomentum [26].
the fragmentation functions given in Ref. [21] and used TheBelinfanteproceduresimplymakesuseofthisfree-
intheLSSanalysisarepresumablynotcorrect[22]. Fur- domtodefineasymmetricandgauge-invarianttensor. It
therexperimentaldataanddedicatedanalysesarethere- is howeverimportantto realizethat the symmetry prop-
fore required. Nevertheless, these values seem consistent erty of the energy-momentum tensor is not mandatory
with recent Lattice calculations by the LHPC collabora- inparticlephysics,butisbasicallymotivatedbyGeneral
tion [23], see table I. Relativity where gravitation is coupled to a symmetric
The second moment of the quark helicity distribution energy-momentum tensor. From the conservation of to-
being a valence-like quantity with suppressed low-x tal angular momentum ∂µ(xνTµρ−xρTµν +Sµνρ)= 0,
region, we may expect phenomenological quark model one sees that the antisymmetric part of the energy-
predictions to be more reliable for this second moment momentum tensor is intimately related to the spin den-
than for the lowest one. In table I we provide the first sity
two moments of the u and d-quark helicity distributions Tνρ−Tρν =−∂ Sµνρ. (47)
µ
obtained within the naive quark model (NQM), the
light-front constituent quark model (LFCQM) and the General Relativity is a purely classical theory where the
light-front chiral quark-soliton model (LFχQSM) at the notionofintrinsic spindoes not exist,andso it is hardly
scale µ2 ∼ 0.26 GeV2, see Ref. [24] for more details. surprisingthatonlyasymmetricenergy-momentumten-
From these estimates, we expect a negative quark sor is needed in that context.
spin-orbit Cq for both u and d quarks (Cu ≈ −0.8 and So,assoonasonedealswithaspindensity,thenatural
z z
Cd ≈−0.55), meaning that the quark spin and Ji OAM gauge-invariant energy-momentum tensor will be asym-
z
are expected to be, in average, anti-correlated. This metric. The fact that one can still define a consistent
has to be contrasted with the model results obtained in symmetric (or Belinfante) tensor is a consequence of the
Ref. [2] where the quark spin and Jaffe-Manohar OAM following QCD identity
are, in average,correlated.
ψγ[µiD↔ν]ψ =−ǫµναβ∂ (ψγ γ ψ), (48)
α β 5
5
which is the parity-even analogue of Eq. (24). It tells us Indeed, this mass term does not have the form of a su-
thatthedifferencebetweenthesymmetricandasymmet- perpotential, which in turn implies that d3xTˆ+ν 6=
q5
rbiucttaenssuoprser(pnoatmenetliyalttheerman.tTishymismmeetarnicspthaartt)oinsencoatnhienfg- d3x12Tˆq{5+ν}. We arguedin section II thatRCzq can nat-
Rurally be interpreted as (twice) the quark spin-orbit cor-
fectively absorbthe contribution of the spin density in a
relationhLqSqi. By analogywith the parity-evensector,
relocalization/redefinitionof energy-momentum density. it is temptizngzto interpret Cq as (twice) the correlation
In Ref. [5], Ji considered the symmetric (Belinfante) z
between quark spin and quark total angular momentum
energy-momentum tensor, discarding from the very be-
hJqSqi. This is actually true if one gets rid of the mass
ginning the notion of intrinsic spin density. Accordingly, z z
term, i.e. work in the chiral limit m = 0. In this case,
Ji was able to relate only the total angular momentum
justlikeintheparity-evensectorwiththeJirelation,we
to moments of GPDs
can understand Eq. (35) as hLqSqi = hJqSqi−hSqSqi.
z z z z z z
The quarkspin-spincorrelationhSqSqiis ofcoursequite
Jq,G = 1 dxx[H (x,0,0)+E (x,0,0)]. (49) z z
z 2Z q,G q,G trivialanddoesnotdependonthespinorientation. That
is the reason why the unpolarized quark FF Fq(0) ap-
1
In order to get the quark OAM, one has to subtract as pears in the expression for Cq.
z
a second step the quark spin contribution Sq = 1Gq(0)
z 2 A
fromthetotalquarkangularmomentumcontributionJq.
z
So Eq. (36) can be understoodas Lq =Jq−Sq, or more
z z z
precisely as hLqSNi = hJqSNi−hSqSNi. Starting with V. CONCLUSIONS
z z z z z z
the more general asymmetric energy-momentum tensor,
one can directly define the OAM operator as in Eq. (10)
We provided a local gauge-invariant definition of the
and get the final result (36) without invoking any addi-
quarkspin-orbitcorrelation,whichis a new independent
tional step [1, 6, 27].
piece of information about the nucleon longitudinal spin
Similarly,one candefine another quarkcorrelationus-
ingonlythe symmetricpartofTˆµν incloseanalogywith structure. We derived several expressions for the expec-
q5 tation value of this correlation in terms of measurable
Eq. (13)
parton distributions, leading to new sum rules. Using
estimates from fits to available experimental data, Lat-
Cˆqz =Z d3x(x112Tˆq{5+2}−x221Tˆq{5+1}) (50) ticecalculationsandphenomenologicalquarkmodels,we
concluded that the quark spin-orbit correlation is very
= 1Cˆq + d3x1ψiD↔+(x×γ) γ ψ. likely negative, meaning that the quark spin and Ji or-
2 z Z 4 z 5
bitalangularmomentumare,inaverage,anti-correlated.
However,amoreprecisedeterminationofthisquarkspin-
Contrary to Cˆq, the operator Cˆq is time-independent
z z orbit correlation requires further experimental data and
since the corresponding current Mˆqµ5νρ = xν21Tˆq{5µρ} − dedicatedanalysesin orderto disentanglequarkand an-
xρ21Tˆq{5µν} is conserved ∂µMˆqµ5νρ = 0, as a consequence tfoiqrumarfkacctoonrtsraibruetaiolsnositnopthrienhcieplilceitnyeeddisetdr,ibbuuttiotnh.eTceonrsroer-
of the symmetry of Tˆq{5µν}. Note however that even if sponding contribution can safely be neglected for light
the operator Cˆq is in general time-dependent, its expec- quarks.
z
tation value is time-independent because the initial and
final nucleon states have the same energy [1].
It is pretty simple now to get the expressions for the
matrix element of Cˆqz. We need to keep only the sym- Acknowledgements
metric part of Eq. (16), i.e. set the FFs C˜ , D˜ and F˜
q q q
to zero in the derivation of Section II. We then find
I am thankful to H. Moutarde, B. Pasquini and
Cqz ≡ hPh,Pe,ze|zCˆ|qzP|,Pe,zeizi = 21A˜q(0), (51) MI a.mPoallysaokvoevryforgruasteeffuulldtoiscEu.ssLioeandserre,laAt.edV.toSitdhoisrosvtuadnyd.
1 D. B. Stamenov for providing me with their extraction
= 1 dxxH˜ (x,0,0). (52)
2Z q ofthe quarkhelicity distributions,andtoM.Engelhardt
1
− forinformingmeaboutrecentLatticeresults. This work
Contrary to Cq, the physical interpretation of Cq is not wassupportedbytheBelgianFundF.R.S.-FNRSvia the
z z
particularly clear owing to the mass term in Eq. (24). contract of Charg´e de recherches.
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arXiv:1309.4235 [hep-ph]. (2011).
6
[3] S. Meissner, A. Metz and M. Schlegel, JHEP 0908, 056 [17] M. Burkardt,Phys. Rev.D 88, 014014 (2013).
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