Anomalous magnetic moments and quark orbital angular momentum

Matthias BurkardtDepartment of Physics, New Mexico State University, Las Cruces, New Mexico 88003-0001, USA

Gunar SchnellSubatomaire en Stralingsfysica, Universiteit Gent, 9000 Gent, Belgium

(Received 1 March 2006; published 5 July 2006)

We derive an inequality for the distribution of quarks with nonzero orbital angular momentum, and thusdemonstrate, in a model-independent way, that a nonvanishing anomalous magnetic moment requires botha nonzero size of the target as well as the presence of wave function components with quark orbitalangular momentum Lqz > 0.

DOI: 10.1103/PhysRevD.74.013002 PACS numbers: 13.40.Em, 13.60.�r, 13.88.+e, 14.20.�c

I. INTRODUCTION

Detailed measurements of the spin structure of the nu-cleon [1–11] have revealed that only a small fraction of thenucleon spin is carried by the quark spin. This resultimmediately raised the question, which degrees of freedomcarry the rest. Unfortunately, both orbital angular momen-tum and the gluon spin are difficult to access experimen-tally, and therefore little rigorous information exists aboutquark orbital angular momentum.

Meanwhile, many qualitative statements regarding orbi-tal angular momentum have been made. For example,when one expresses the matrix element for the anomalousmagnetic moment of the nucleon in terms of light conewave functions (summed over all Fock components), anonzero anomalous magnetic moment can only resultwhen there is a nonzero probability that the vector currentflips the nucleon helicity [12,13]. Since the same matrixelement conserves the spin of the quarks it is evident thatsome orbital angular momentum must be transferred to thequarks. Hence a nonzero anomalous magnetic moment canonly occur when the target wave function contains compo-nents with nonzero orbital angular momentum. While thisargument is rigorous, it leaves open quantitative questionsregarding the norm of those wave function components orperhaps the resulting net Lqz . Within models, it has alsobeen found that a pointlike object cannot produce a non-zero anomalous magnetic moment [12,14] and within thismodel one can even derive quantitative bounds.

Similarly, the observation of a nonzero Sivers [15] effectby the HERMES collaboration [16] seems to indicate wavefunction components with nonzero Lqz since the effectrequires an interference between initial nucleon statesthat have opposite helicity. Furthermore, orbital angularmomentum seems to play a central role in all models forthe Sivers function [17–21].

The main purpose of this paper is to make some of thestatements regarding the anomalous magnetic momentmore quantitative. In order to accomplish this goal, westart from the matrix element that yields the generalizedparton distribution (GPD) Eq and apply the Cauchy-

Schwarz inequality which will then provide a lower boundon the norm of wave function components with nonzeroorbital angular momentum.

II. DECOMPOSITION OF THE NUCLEON SPIN

In QCD, there is no unique way to decompose thenucleon spin into quark spin, quark angular momentum,gluon spin, and gluon orbital angular momentum. Forexample, Ji has considered a decomposition based on theM012 component of the angular momentum tensor [22]

M012 �i2qy�~r� ~D�zq�

1

2qy�zq� 2 TrEj� ~r� ~D�zAj

� Tr� ~E� ~A�z: (1)

The matrix elements of the four terms in Eq. (1) areinterpreted as the quark orbital angular momentum, quarkspin, gluon orbital angular momentum, and gluon spin,respectively. One advantage of this decomposition is gaugeinvariance, another is the fact that the matrix elements ofthe first term in Eq. (1) can be probed in deeply virtualCompton scattering. A disadvantage of this decompositionis that the orbital angular momenta in Eq. (1) containinteractions through the gauge covariant derivative.

An alternative decomposition of the total angular mo-mentum starts from M�12 in light cone gauge A� � A0 �Az � 0 [23]

M�12 � 12qy��~r� i ~@�

zq� �12qy��5q�

� 2 TrF�j� ~r� i ~@�zAj � "��ij TrF�iAj: (2)

Here q� �12����q is the dynamical component of the

quark field operators. The obvious disadvantage of usingEq. (2) to interpret the total angular momentum of thenucleon, which we all know is equal to 1

2 @, as a sum ofquark and gluon orbital and spin angular momentum,respectively, is the fact that Eq. (2) is not gauge invariant.However, it is invariant under the residual group of gaugetransformations that leave A� � 0 and where the Aj satisfyantisymmetric boundary conditions at x� � �1. The

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main advantages of using Eq. (2) to decompose the nucleonspin is the fact that all terms in Eq. (2) are quadratic in thefields. Unlike Eq. (1) there is therefore no ambiguity as tothe interpretation of interaction terms. Another advantageis the fact that the matrix elements of the various terms inEq. (2) can be easily expressed in terms of light cone wavefunctions.

We should emphasize that although we have listed heresome of the most obvious advantages and disadvantages ofthese two possibilities for decomposing the angular mo-mentum, we do not consider one or the other superior ingeneral. However, for specific applications one of thesetwo decompositions may be advantageous. In particular, ifone wants to place constraints on light cone wave func-tions, which find many applications in hadron phenome-nology, then one may prefer the decomposition of theorbital angular momentum based on M�12 (2). In the restof this paper we will exclusively study the light conedecomposition of Jz, based on Eq. (2).

We thus consider in the following the orbital angularmomentum of quarks with flavor q

Lqz �Zdx�

Zd2x?

1

2qy��~r� i ~@�

zq� (3)

and we will investigate its role in nucleon spin-flip matrixelements, such as the anomalous magnetic moment.

As it stands, Eq. (3) is gauge dependent. We thereforeneed to specify the gauge. First of all, we impose the lightcone gauge condition A� � 0. However, this does not yetcompletely fix the gauge since A� remains zero undergauge transformations with a phase that only depends onx?. We thus impose as an additional condition that the ?component of the gauge field satisfies antiperiodic bound-ary conditions at light cone infinity

A?�x� � 1;x?� � �A?�x� � �1;x?�: (4)

With this additional condition, the definition (3) of Lqzbecomes unique.

Although all results that we will be deriving will be validin this gauge only, we believe they will nevertheless beuseful since a lot of hadron phenomenology is based onlight cone wave functions and their use usually implies theuse of light cone gauge. It is therefore fair to say thatalthough our results will not have a gauge-independentinterpretation, they will nevertheless provide additionalinsights about light cone wave functions of hadrons.

III. ANOMALOUS MAGNETIC MOMENT ANDGPDS

We first consider the generalized parton distributionE�x; 0;��2

?�, which appears in nonforward nucleonspin-flip matrix elements of light cone correlation func-tions. For purely transverse momentum transfer p� � p0�

one finds

hP� �; " jOq�x; 0?�jP; "i � Hq�x; 0;��2?�; (5)

hP� �; " jOq�x; 0?�jP; #i � ��x � i�y

2MEq�x; 0;��2

?�;

(6)

where

Oq�x;b?� �Z dx�

4�eip

�x�x �q�0�;b?���q�x�;b?� (7)

(in gauges other than light cone gauge A� � 0 one needs toinsert a Wilson line gauge link in Eq. (7)). In the forwardlimit, integration over x yields the anomalous magneticmoment contribution from quarks with flavor q

ZdxEq�x; 0; 0� � �q: (8)

Since the operator in Eq. (6) is chirally even, the matrixelement is diagonal in quark spin. Since the matrix elementinvolves a nucleon spin-flip, angular momentum conserva-tion thus requires a change in quark orbital angular mo-mentum. As a result, the mere fact that the anomalousmagnetic moment of the nucleon is nonzero implies thatthere must be components in the nucleon wave functionthat have a nonzero orbital angular momentum [13] (ofcourse, this does not necessarily mean that there is any netorbital angular momentum). In the following we will try tomake this statement more quantitative, i.e., we will attemptto place constraints on the probability to find nonzeroorbital angular momentum components in the nucleonwave function.

In order to facilitate the separation between intrinsicorbital angular momentum and the orbital angular momen-tum due to the motion of the entire nucleon, we first switchto a representation of states that are eigenstates of thetransverse center of momentum

jp�;R?; �i �NZd2p?jp�;p?; �ieib?R? ; (9)

where N is a normalization constant. In this basis, we candefine impact parameter dependent parton distributions as

H q�x;b?� � hp�;R?; " jOq�x;b?�jp�;R?; "i; (10)

�@@bx� i

@@by

�Eq�x;b?�

2M�hp�;R?;" jOq�x;b?�jp�;R?;#i:

(11)

The impact parameter dependent parton distribution func-tions (PDFs) are related to GPDs via a simple Fouriertransform [24–27]

H �x;b?� �Z d2�?�2��2

eib?�?H�x; 0;��2?�; (12)

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E �x;b?� �Z d2�?�2��2

eib?�?E�x; 0;��2?�; (13)

and the normalization is such thatRd2b?q�x;b?� � q�x�.

Since the GPDs provide simultaneous information aboutthe longitudinal momentum and the transverse position ofpartons, it is instructive to introduce creation operators inthis ‘‘hybrid space.’’ For example, for x > 0 we definedestruction operators for quarks with momentum fractionx at transverse position b?

b�xp�;b?� �Zdx�eip�x

�xq��x�;b?� (14)

and similarly for antiparticles (x < 0). For now we willsuppress the helicity of the quarks and consider onlyquantities that have been summed over the helicities ofthe quarks. In terms of these, the impact parameter depen-dent PDFs take on a particularly simple form (x > 0)

H �x;b?��hp�;R?;" jby�xp�;b?�b�xp�;b?�jp�;R?;"i;

(15)

1

2M

�@@bx� i

@@by

�E�x;b?� � hp�;R?; " jby�xp�;b?�

� b�xp�;b?�jp�;R?; #i;

(16)

which emphasizes their physical interpretation as densities.Upon introducing

Bq��x� �Zd2b?�bx � iby�by�xp�;b?�b�xp�;b?�

(17)

and integration by parts we find

hp�;R?; " jBq��x�jp

�;R?; #i �1

2MEq�x; 0; 0�: (18)

We can use this result to provide a formal proof that �Lqz �1 in the matrix element defining Eq�x; 0; 0�. Indeed, oneeasily verifies the commutation relation

Lqz ; Bq��x�� � Bq��x�; (19)

which proves that Bq� has nonvanishing matrix elementsonly between states that differ by one unit of orbital

angular momentum Lqz for flavor q. This observation isconsistent with results based on overlap integrals of lightcone wave functions [13].

IV. ANGULAR MOMENTUM DECOMPOSITIONOF PDFS

In order to derive some quantitative constraints on thelight cone wave functions of hadrons, we first introduce anangular momentum decomposition for parton distributions,i.e.,

by�xp�;b?� �Xm

bym�xp�;b?�; (20)

where

bym�xp�;b?� � e�im�b

Z 2�

0

d�0b2�

e�im�0bby�xp�;b0?�;

(21)

where bx � jb?j cos� and by � jb?j sin�, and b0x �jb?j cos�0b and b0y � jb?j sin�0b, respectively.

Of course, the above decomposition into orbital angularmomentum components is scale dependent due to the scaledependence of light cone wave functions [28]. However,we imagine performing such a decomposition at a fixedscale at which we perform the analysis of thisdecomposition.

The physical interpretation of the creation and destruc-tion operators (21) is that they create quarks withm units ofangular momentum (in the z-direction), i.e.,

Lqz ; bym�xp�;b?�� � mbym�xp�;b?�: (22)

In terms of these angular momentum projected creationoperators, the operator Bq� appearing in the impact parame-ter space representation of the matrix element for theanomalous magnetic moment takes on the form

Bq� �Xm

Zd2b?�bx � iby�b

ym�1�xp

�;b?�bm�xp�;b?�:

(23)

Inserting the angular mode expansion into Eq. (18) andapplying the Cauchy-Schwarz inequality to scalar productsbetween states yields

Eq�x; 0; 0�2M

�Xm

Zd2b?�bx � iby�hp�; 0?; " jb

ym�1�xp

�;b?�bm�xp�;b?�jp�; 0?; #i

�Xm

Zd2b?jb?j

����������������������������������������������������������������������������������������������������������hp�; 0?; " jb

ym�1�xp

�;b?�bm�1�xp�;b?�jp�; 0?; "iq

�

�������������������������������������������������������������������������������������������������hp�; 0?; # jb

ym�xp�;b?�bm�xp�;b?�jp�; 0?; #i

q: (24)

In order to simplify the notation, we now introduce the distribution of partons with orbital angular momentumm in a targetwith spin "

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q"m�x��Zd2b?hp�;0?;" jb

ym�xp�;b?�bm�xp�;b?�j

�p�;0?;"i (25)

as well as the b2?-weighted distribution of partons with

orbital angular momentum m

b2"m�x� �

Zd2b?hp�; 0?; " jb

ym�xp�;b?�bm�xp�;b?�j

� p�; 0?; "ib2?: (26)

Obviously we have

q#m�x� � q"�m�x�; (27)

b2;#m �x� � b2;"

�m�x�: (28)

To each term in the sum in Eq. (24) we now apply theCauchy-Schwarz inequality for integrals

Zd2b?

�������������������������f�b?�g�b?�

q�

�����������������������������Zd2b?f�b?�

s �����������������������������Zd2b?g�b?�

s:

(29)

For m 0 we apply this inequality with

f�b?� �Zd2b?hp�;0?; " jb

ym�1�xp

�;b?�bm�1�xp�;b?�j

�p�;0?; "i � q"m�1�x�;

g�b?� �Zd2b?b2

?hp�;0?; # jb

ym�xp�;b?�bm�xp�;b?�j

�p�;0?; #i � b2;#m �x�; (30)

while for m< 0 we identify

f�b?��Zd2b?b2

?hp�;0?;" jb

ym�1�xp

�;b?�bm�1�xp�;b?�j

�p�;0?;"i�b2"m�1�x�;

g�b?��Zd2b?hp�;0?;# jb

ym�xp�;b?�bm�xp�;b?�j

�p�;0?;#i�q#m�x�; (31)

yielding

Eq�x; 0; 0�2M

�Xm 0

�����������������������������q"m�1�x�b

2;#m �x�

q�Xm<0

����������������������������b2;"m�1�x�q

#m�x�

q

�Xm 0

�������������������������������q"m�1�x�b

2"�m�x�

q�Xm<0

�������������������������������b2"m�1�x�q

"�m�x�

q

�Xm 0

�������������������������������q"m�1�x�b

2;"�m�x�

q�Xm>0

����������������������������q"m�x�b

2;"1�m�x�

q

� 2Xm 0

�������������������������������q"m�1�x�b

2;"�m�x�

q(32)

� 2

��������������������������������������������������������Xm 0

q"m�1�x���X

n 0

b2;"�n�x�

�s; (33)

where in the last step we used (fm; gm 0)

Xm

������������fmgm

p�

����������������������������������Xm

fm

��Xn

gn

�s: (34)

If we now introduce the distribution of quarks with positiveangular momenta in a nucleon with spin in the�z direction

qLz 1�x� �Xm 1

q"m�x� (35)

as well as the contribution to b2?�x� from quarks with zero

or negative Lqz

b2Lz�0�x� �

Xm�0

b2;"m �x� (36)

our result can be cast into the form

�Eq�x; 0; 0�

4M

�2� qLz 1�x�b2

Lz�0�x�; (37)

which is the main result of this paper. A slightly strongerinequality can be obtained by keeping track of the quarkhelicity. For this purpose we note that both quark helicitiescontribute equally in Eq. (18). Upon repeating the aboveanalysis starting from quarks with helicity " , one thusarrives at

Eq�x; 0; 0�4M

����������������������������������������q"Lz 1;"�x�b

2;"Lz�0;#�x�

q�

���������������������������������������q"Lz 1;#�x�b

2;"Lz�0;"�x�

q; (38)

where for example q"Lz 1;#�x� denotes the distribution ofquarks with helicity # and positive orbital angular momen-tum in a nucleon with helicity " . An even stronger versionof Eq. (38) can be achieved following Ref. [29] by startingwith ���1� �5�=2 instead of only �� in Eq. (7). In thatcase only one of the two terms on the right-hand side(r.h.s.) of (38) remains, and E

4M gets replaced by E8M [30].

Additional simplified inequalities can be derived if oneneglects quarks with orbital angular momenta jLqz j 2.

V. DISCUSSION

First of all, our result illustrates that a nonvanishinganomalous magnetic moment (and hence nonvanishingE�x; 0; 0�) implies both

(i) wave function components with positive quark or-bital angular momentum (for Sz � �

12 )

(ii) a nonvanishing ? sizeThe fact that the size of hadrons needs to be nonzero andthat nonzero angular momentum components need to bepresent if a state has an anomalous magnetic moment hasbeen observed before in light cone wave function modelsof hadrons [13,14]. What is new in our paper is the fact that

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our result provides a model-independent, quantitativelower bound for the ? size distribution, which unlike inRef. [29] is connected to the quark orbital angular momen-tum. Furthermore, we have shown that there must bequarks with orbital angular momentum in the same direc-tion as the nucleon spin—regardless of the sign of theanomalous magnetic moment. This is at first surprising,since one might have expected that if the anomalous mag-netic moment is due to an orbital angular momentum then,when E�x; 0; 0�< 0 the quark should orbit opposite to thenucleon spin. However, one needs to keep in mind that wemake no statement about the net orbital angular momen-tum. We only derived a lower bound on contributions to thenet Lqz from modes with Lqz > 0.

The necessity of quark orbital angular momentum for ananomalous magnetic moment is particularly surprisingfrom the nonrelativistic quark model (NRQM) point ofview. In the NRQM the anomalous magnetic moment isentirely due to the spins of the quarks and no orbitalangular momentum is needed. However, in this point theNRQM is not completely consistent: For example, when amassive Dirac particle is in some bound state then, due tothe localization, the particle must necessarily possess anonzero momentum and thus a nonvanishing lower Diraccomponent. The lower component for a massive s-wavequark has orbital angular momentum Lqz � 1. Since eventhe d-quark appears in the NRQM wave functions withboth spin up and down, this relativistic effect gives rise towave function components with both positive and negativeorbital angular momentum. In our inequality (37), thelower bound on orbital angular momentum is proportionalto the inverse radius squared of the quark distribution.Strictly speaking the nonrelativistic limit requires Rmq !

1 and hence RM ! 1. In this limit the lower bound onqLz 1 goes to zero. Therefore, in a system where the non-relativistic approximation would be allowed, a nonzeroanomalous magnetic moment would not necessarily re-quire nonzero orbital angular momentum.

What is also surprising is that, once E�x; 0; 0� is nonzero,there need to be quarks with positive orbital angular mo-mentum—regardless of the sign of E�x; 0; 0�. This is be-cause for E�x; 0; 0� to be nonzero there ought to be anonzero matrix element between a nucleon with spin upand a nucleon with spin down in Eq. (18). For example, ifthe active quark in the initial state had Lqz � 0, then itneeds to have Lqz � �1 in the final state, i.e., Lqz in thesame direction as the nucleon spin. Likewise if Lqz � 0 inthe final state, Lqz must have been �1 in the initial state inEq. (18). Similarly one can convince oneself in the generalcase (initial and final state have nonzero Lqz ) that a nonzeroE�x; 0; 0� always requires wave function components withLqz > 1 in a nucleon that has Sz �" . This is surprising sincein Ji’s relation the sign ofE seems to suggest the sign of Lqz .However, one needs to keep in mind that Ji’s relation dealswith the net Lqz and secondly the angular momentum

decomposition to which the Ji relation applies (1) doesnot have to be the same as the angular momentum decom-position in the light cone framework (2).

In order to get some quantitative feeling for the lowerbound, we consider a model for u-quarks in the valenceregion where (� � 2)

q�x� � 8�1� x�3; (39)

E�x; 0; 0� � 6��1� x�5 � 12�1� x�5; (40)

M2b2�x� � M2R28�1� x�5; (41)

where we pick R � 0:5 fm for the transverse size, i.e.,MR � 2:5 and we let b2�x� / �1� x�2q�x�, consistentwith a finite size for large x. Since no data is available onE�x; 0; 0� and b2�x�, we simply make an educated guessregarding these functions in order to be able to make anorder of magnitude estimate for the probability to findnonzero orbital angular momentum. The ansatz forE�x; 0; 0� is motivated by the constraint that E�x; 0; 0�must vanish by two powers of (1� x) faster than q�x� forlarge x. The ansatz for b2�x� was motivated by the con-straint that the nucleon has a finite transverse size for largex. The rest of the above ansatz was guided by the ad hocrequirement to make the model as simple as possible andtherefore the idea is that this model should merely serve asa guide for what order of magnitude one should expect(‘‘back of an envelope estimate’’) with parameters that arewithin the range of what is expected in the valence region.With the above model parameters we find for the distribu-tion of quarks with positive orbital angular momentum

950 �1� x�

5 � qLz 1�x�; (42)

i.e., only some % compared to the distribution summedover all orbital angular momenta.

While this is much smaller than current estimates basedon Ji’s relation (see, for example, Ref. [31] for an up to dateestimate), one should keep in mind that the orbital angularmomentum obtained from the Ji relation differs from theorbital angular momentum from the light cone decompo-sition (2).

Although our lower bound is not very spectacular, it stillprovides the first model-independent lower bound on thedistribution of quarks with positive angular momentumdistribution. One may also wonder how such a low boundis consistent with transverse flavor dipole moments jdqy j �0:2 fm for a moving nucleon that is quite significant andalmost the same order of magnitude as the ? size of thenucleon. The important point here is that the ? distortion,which is described by our starting equation E, contains apiece involving the overlap between the Lqz � 0 compo-nent of the wave function and the Lqz � 1 component, i.e.,it is linear in the amplitude for finding a quark with nonzeroorbital angular momentum, whereas the probability forLqz > 0 is quadratic in that amplitude. The situation is

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thus somewhat analogous to the quadrupole moment of thedeuteron, which is linear in the d-wave component andquite large, while the actual d-wave probability is tiny.

VI. SUMMARY

We have derived an inequality, which provides a model-independent lower bound on the norm of wave functioncomponents with nonzero quark orbital angular momen-tum. Although the numerical bound thus obtained is notvery strong—requiring only a few % probability for non-zero orbital angular momentum—our result represents thefirst quantitative estimate for wave function componentswith nonzero Lqz . Moreover, the resulting constraints on thelight cone wave functions of the nucleon, which only enterthe distributions quadratically, are much stronger. Thebound that we derive involves the transverse size (rms-radius) times the mass of the nucleon. This quantifies theknown result that a pointlike particle cannot have a non-zero anomalous magnetic moment. Unlike Lqz itself, whichdepends quadratically on wave function components withnonzero Lqz , the GPD E�x; 0; 0� requires only that Lqz is

nonzero on one side of the matrix element. This illustrateswhy we were only able to derive a bound on the probabilityfor nonzero Lqz that depends quadratically on E�x; 0; 0�,which is why our bound is so weak. In contradistinction Lqzobtained through the Ji relation would involve E�x; 0; 0�linearly. Nevertheless, we hope that the constraints derivedin this work will be of use in developing better models forlight cone wave functions describing these interestingobservables.

ACKNOWLEDGMENTS

We would like to thank M. Diehl, M. Oka, and W.Vogelsang for stimulating discussions. M. B. was partiallysupported by the DOE under Grant No. DE-FG03-95ER40965. M. B. would also like to thank T.-A. Shibatafor his hospitality and support during part of this project.G. S. was partially supported by the Japan Society for thePromotion of Science, the Alexander-von-HumboldtStiftung, and the Fonds voor WetenschappelijkOnderzoek-Vlaanderen.

[1] J. Ashman et al. (EMC Collaboration), Phys. Lett. B 206,364 (1988).

[2] D. Adams et al. (SMC Collaboration), Phys. Rev. D 56,5330 (1997).

[3] D. Adams et al. (SMC Collaboration), Phys. Lett. B 396,338 (1997).

[4] B. Adeva et al. (SMC Collaboration), Phys. Rev. D 58,112001 (1998).

[5] B. Adeva et al. (SMC Collaboration), Phys. Rev. D 60,072004 (1999).

[6] E. S. Ageev et al. (COMPASS Collaboration), Phys. Lett.B 612, 154 (2005).

[7] K. Abe et al. (E143 Collaboration), Phys. Rev. D 58,112003 (1998).

[8] P. L. Anthony et al. (E155 Collaboration), Phys. Lett. B463, 339 (1999).

[9] P. L. Anthony et al. (E155 Collaboration), Phys. Lett. B493, 19 (2000).

[10] A. Airapetian et al. (HERMES Collaboration), Phys. Lett.B 442, 484 (1998).

[11] X. Zheng et al. (Jefferson Lab Hall A), Phys. Rev. C 70,065207 (2004).

[12] S. J. Brodsky and S. D. Drell, Phys. Rev. D 22, 2236(1980).

[13] S. J. Brodsky, D. S. Hwang, B.-Q. Ma, and I. Schmidt,Nucl. Phys. B593, 311 (2001).

[14] S. J. Brodsky and F. Schlumpf, Phys. Lett. B 329, 111(1994).

[15] D. W. Sivers, Phys. Rev. D 43, 261 (1991).[16] A. Airapetian et al. (HERMES Collaboration), Phys. Rev.

Lett. 94, 012002 (2005).[17] C. Boros, Z. T. Liang, and T. C. Meng, Phys. Rev. Lett. 70,

1751 (1993).[18] S. J. Brodsky, D. S. Hwang, and I. Schmidt, Phys. Lett. B

530, 99 (2002).[19] M. Burkardt, Nucl. Phys. A735, 185 (2004).[20] F. Yuan, Phys. Lett. B 575, 45 (2003).[21] A. Bacchetta, A. Schaefer, and J.-J. Yang, Phys. Lett. B

578, 109 (2004).[22] X. Ji, Phys. Rev. Lett. 78, 610 (1997).[23] R. L. Jaffe and A. Manohar, Nucl. Phys. B337, 509 (1990).[24] M. Burkardt, Phys. Rev. D 62, 071503 (2000).[25] M. Burkardt, Int. J. Mod. Phys. A 18, 173 (2003).[26] M. Diehl, Eur. Phys. J. C 25, 223 (2002).[27] A. V. Belitsky, X. Ji, and F. Yuan, Phys. Rev. D 69, 074014

(2004).[28] M. Burkardt, X. Ji, and F. Yuan, Phys. Lett. B 545, 345

(2002).[29] M. Burkardt, Phys. Lett. B 582, 151 (2004).[30] M. Diehl (private communications).[31] M. Diehl, hep-ph/0510221.

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