in the chiral quark-soliton model

PHYSICAL REVIEW D 67, 114010 ~2003!

Chirally-odd twist-3 distribution function ea„x… in the chiral quark-soliton model

P. SchweitzerDipartimento di Fisica Nucleare e Teorica, Universita` degli Studi di Pavia, Pavia, Italy

~Received 4 March 2003; published 9 June 2003!

The chirally-odd twist-3 nucleon distributionea(x) is studied in the large-Nc limit in the framework of thechiral quark-soliton model at a low normalization point of about 0.6 GeV. The remarkable result is that in themodelea(x) contains ad-function-type singularity atx50. The regular part ofea(x) is found to be sizable atthe low scale of the model and in qualitative agreement with bag model calculations.

DOI: 10.1103/PhysRevD.67.114010 PACS number~s!: 12.39.Ki, 12.38.Lg, 13.60.2r, 14.20.Dh

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I. INTRODUCTION

In deeply inelastic scattering~DIS! processes the nucleostructure up to twist-3 is described by six parton distributfunctions, the twist-2f 1

a(x), g1a(x), h1

a(x) and the twist-3ea(x), gT

a(x), hLa(x). Among these functions the least co

sidered one is probably the twist-3 chirally odd distributifunctionea(x) @1,2#, which contrasts the fact that it is relateto several interesting phenomena. E.g., a known@3–7# butrarely emphasized@8# fact is that ea(x) contains ad-function-type singularity atx50, as follows from theQCD equations of motion. The existence of ad(x) contribu-tion also was concluded from perturbative calculations@9#.The first Mellin moment ofea(x) is due to thed(x) contri-bution only. Another interesting phenomenon is connectethe first Mellin moment of the flavor-singlet (eu1ed)(x)which is proportional to the pion-nucleon sigma-termspN .The latter gives rise to the so-called ‘‘sigma-term puzzlThe large value extracted from pion-nucleon scattering dspN'(50–70) MeV@10,11#, implies that about 20% of thenucleon massMN is due to the strange quark—an unexpeedly large number from the point of view of the OkubZweig-Iizuka rule.

The reason whyea(x) has received only little attention sfar is related to its chiral odd nature, which means thatea(x)can enter an observable only in connection with anotchirally odd distribution or fragmentation function, antherefore is difficult to access in experiments. Only recenit was shown thatea(x) can be accessed by means of t‘‘Collins effect’’ @12#, i.e. the left-right asymmetry in thefragmentation of a transversely polarized quark into a piThis effect is described by the chirally andT-odd fragmen-tation functionH1

'a(z), which is ‘‘twist-2’’ in the sense thatits contribution to an observable is not power suppres@12,13#. First experimental indications toH1

' were reportedin @14#. Assuming factorization it was shown that the Collieffect gives rise to a specific azimuthal~with respect to theaxis defined by the exchanged hard virtual photon! distribu-tion of pions produced in DIS of longitudinally polarizeelectrons off an unpolarized proton target. The observasingle ~beam! spin asymmetry is proportional t(aea

2ea(x)H1'a(z) @15# (ea56 2

3 ,6 13 are the quark electric

charges!. The process was studied in the HERMES expement and the effect found consistent with zero within er

0556-2821/2003/67~11!/114010~13!/$20.00 67 1140

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bars@16#.1 However, in the CLAS experiment, in a differenkinematics, a sizable asymmetry was observed@17,18#. If theinterpretation applies, that the CLAS data@17,18# are due tothe Collins effect, thenea(x) is definitely not small. Usingestimates ofH1

'(z) from HERMES data@19# it was shown in@20# that ea(x) could be about half the magnitude of thunpolarized twist-2 distributionf 1

a(x) at Q2;1.5 GeV2 inthe region 0.15<x<0.4 covered in the CLAS experiment.

The indication thatea(x) could be large in the valence-xregion is not surprising, if one considers results from the bmodel @2,21#, the only model whereea(x) has been studiedso far. In this paperea(x) will be studied in the chiral quark-soliton model (xQSM). A subtle question is whether modewith no gluon degrees of freedom~bag model,xQSM) candescribe twist-3 distributions functions. The answer given@1,2# is yes, becauseea(x), gT

a(x) and hLa(x) are special

cases of more general quark-gluon-quark correlation futions; special inasmuch they do not containexplicit gluonfields. However, implicitly gluons do contribute and it is important and instructive to carefully interpret the results. Ein the bag modelea(x) is due to the bag boundary@2#. Thiscan be understood considering that the bag boundary~in amost intuitive way! models confinement, and thus ‘‘mimicsgluons.

The xQSM was derived from the instanton model of thQCD vacuum. An important small parameter in this derivtion is the ‘‘instanton packing fraction’’ which characterizethe diluteness of the instanton medium@22,23#. Gluon de-grees of freedom appear only at next-to-leading order ofparameter@24#. In leading order of the instanton packinfraction thexQSM quark degrees of freedom can be idenfied with the QCD quark degrees of freedom. This allowsconsistently describe twist-2 quark and antiquark distributfunctions f 1

a(x), g1a(x) andh1

a(x) at a low scale around 600MeV @25#. The numerical results@25–29# agree to within~10–30!% with parametrizations for the ‘‘known’’ distribu-tion functions performed at low scales@30#.

The twist-3 distribution functionsgTa(x) and hL

a(x) were

1The prominent result of the HERMES experiment@16# is theobservation of sizable azimuthal asymmetries in pion producfrom DIS of unpolarized electronsoff longitudinally polarized pro-tons, which contain information onH1

'a(z) and the chirally odddistribution functionsh1

a(x) andhLa(x) @15#.

©2003 The American Physical Society10-1

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P. SCHWEITZER PHYSICAL REVIEW D67, 114010 ~2003!

studied in the instanton vacuum model in Refs.@31,32#. Theremarkable conclusion was that the pure twist-3 interacdependent partsgT

a(x) and hLa(x) in the Wandzura-

Wilczek~-like! decompositions ofgTa(x) and hL

a(x) arestrongly suppressed by powers of the instanton packing ftion. As gT

a(x) andhLa(x) do not contain explicit gluon fields

it is possible to evaluategTa(x) and hL

a(x) directly in thexQSM. This was done in Refs.@33,34#, and it was observedthat the pure twist-3 partsgT

a(x) andhLa(x) are indeed small.

Thus in these cases thexQSM respects the results foundirectly in the instanton vacuum model—i.e. the theory frowhich it was derived. This experience encourages to atackle the study ofea(x) in thexQSM. However, one has tokeep in mind that the results and their interpretation psented here should also be reexamined in the instavacuum model. This is out of the scope of this paper andfor future studies.

ThexQSM describes the nucleon as a chiral soliton ofpion field in the limit of a large number of colorsNc . Thisnote focuses on the flavor singlet distributions (eu1ed)(x)and (eu1ed)(x)—the leading flavor combinations in thlarge-Nc limit. The consistency of the approach is checkeby demonstrating that the model expressions foreu

1ed)(x) satisfy the QCD sum rules. It is shown that thmodel expressions are~quadratically and logarithmically!UV divergent, and a consistent regularization is defined.markably, it is found that thexQSM expression for (eu

1ed)(x) contains ad(x) contribution. The UV behavior andthed(x) contribution make an exact numerical evaluation(eu1ed)(x) and (eu1ed)(x) particularly involved. There-fore (eu1ed)(x) is evaluated using an approximation, th‘‘interpolation formula’’ of Ref. @25#.

This paper is organized as follows. In Sec. II the twisdistribution functionea(x) and some of its properties ardiscussed. Section III contains a brief introduction into txQSM. In Sec. IV the flavor-singlet distribution (eu

1ed)(x) is discussed and evaluated in the model usinginterpolation formula. In Sec. Vea(x) is discussed in thenon-relativistic limit. Section VI contains a summary anconclusions.

II. THE DISTRIBUTION FUNCTION ea„x…

The chirally odd twist-3 distribution functionseq(x) forquarks of flavorq and eq(x) for antiquarks of flavorq aredefined as@1,2#

eq~x!51

2MNE dl

2peilx^Nucq~0!

3@0,ln#cq~ln!uN&, eq~x!5eq~2x!, ~1!

where@0,ln# denotes the gauge link. The scale dependeis not indicated for brevity. The light-like vectorsna in Eq.~1! and pa are defined such thatnapa51 and the nucleonmomentum is given byP N

a 5na1paMN2 /2. The matrix ele-

ment in Eq. ~1! is averaged over nucleon spin, i.^Nu . . . uN&[ 1

2 (S3^N,S3u . . . uN,S3&.

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The renormalization scale evolution ofea(x) was studiedin Refs. @3–5#, see also Refs.@6,7# for reviews. It evolvesaccording to an evolution pattern typical for twist-3 quanties. It is not sufficient to know thenth Mellin momentMn@ea#(Q0

2)5*dxxn21ea(x,Q02) at an initial scaleQ0

2, inorder to computeMn@ea#(Q2) for Q2.Q0

2. Instead, theknowledge of all momentsMk@ea#(Q0

2) with k<n is re-quired. In the limit of a large number of colorsNc the evo-lution of ea(x) simplifies to a DGLAP-type evolution—as idoes for the other two proton twist-3 distributionshL

a(x) and~the flavor non-singlet! gT

a(x).The QCD equations of motion allow to decomposeeq(x)

in a gauge-invariant way as@3–5#, see also@6,7# and @8#,

eq~x!5d~x!1

2MN^Nucq~0!cq~0!uN&1etw3

q ~x!

1mq

MN

f 1q~x!

xdn.1 . ~2!

The d(x) contribution has no partonic interpretation. Somauthors cancel it out by multiplyingeq(x) by x, while othersprefer to consider from the beginning an alternative defition of eq(x) with an explicit factor ofx on the right handside of Eq.~1!. The existence of ad(x) contribution also wasconcluded in Ref.@9#, whereeq(x) was constructed explicitly for a one-loop dressed massive quark. The contributetw3

q (x) in Eq. ~2! is a quark-gluon-quark correlation function, i.e. the actual ‘‘pure’’ twist-3~‘‘interaction dependent’’!contribution toea(x), and has a partonic interpretation asinterference between scattering from a coherent quark-glpair and from a single quark@1,2#. Its first two momentsvanish. The third~‘‘mass’’! term in Eq.~2! vanishes in thechiral limit. The ‘‘Kronecker symbol’’dn.1 accompanying ithas the following meaning. The first moment of the materm vanishes. ForxÞ0 the expression in Eq.~2! is, how-ever, correct and can be used literally to take higher momen.1.

The first and the second moment ofeq(x) satisfy the sumrules @1,2#

E21

1

dxeq~x!51

2MN^Nucq~0!cq~0!uN&, ~3!

E21

1

dxxeq~x!5mq

MNNq . ~4!

In Eq. ~4! Nq denotes the number of the respective valenquarks~for protonsNu52 andNd51). The sum rules~3!,~4! follow immediately from the decomposition in Eq.~2!and the above mentioned properties ofetw3

q (x) and the massterm. In particular, the sum rule~3! is saturated by thed(x)contribution in Eq.~2!.

The flavor singlet distribution function (eu1ed)(x) is re-lated to the scalar isoscalar nucleon form factors(t) definedas

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CHIRALLY-ODD TWIST-3 DISTRIBUTION FUNCTION . . . PHYSICAL REVIEW D67, 114010 ~2003!

s~ t !uN~P8!uN~P!5m^N~P8!u@cu~0!cu~0!

1cd~0!cd~0!#uN~P!&, t5~P82P!2,

~5!

whereuN denotes the nucleon spinor~normalized asuNuN52MN) andm5 1

2 (mu1md). In Eq. ~5! a term proportionalto (mu2md)(cucu2cdcd) is neglected. At the pointt50the form factors(t) is referred to as the pion-nucleon sigmterm spN and related to (eu1ed)(x) by means of the sumrule ~3! as

spN[s~0!5mE21

1

dx~eu1ed!~x!. ~6!

The form factors(t) describes the elastic scattering off thnucleon due to the exchange of a spin-zero particle and ismeasured yet. However, low-energy theorems allow toduce its value at the Cheng-Dashen pointt52mp

2 from pion-nucleon scattering data. One finds

s~2mp2 !5H ~6468!MeV Ref. @10#

~7967!MeV Ref. @11#.~7!

The differences(2mp2 )2s(0) was obtained from a disper

sion relation analysis@35# and chiral perturbation theory caculations @36# with the consistent result of 14 MeV. Thiyields

spN'~50–70! MeV. ~8!

With m'(5 –8) MeV one obtains a large number for thfirst moment of (eu1ed)(x),

E21

1

dx~eu1ed!~x!'10. ~9!

One should keep in mind, however, that the large numbeEq. ~9! is not due to a large ‘‘valence’’ structure in (eu

1ed)(x), but solely due to thed(x) contribution.

III. THE CHIRAL QUARK SOLITON MODEL „xQSM…

The xQSM is based on the effective chiral relativistquantum field theory given by the partition functio@22,37,38#

Zeff5E DcDcDU expS i E d4xc~ i ]”2MUg52m!c D ,

~10!

U5exp~ i tapa!, Ug55 12 ~U1U†!1 1

2 ~U2U†!g5 .~11!

In Eq. ~10! M is the ~generally momentum dependent! dy-namical quark mass, which is due to spontaneous breakdof chiral symmetry.U denotes theSU(2) chiral pion field.The current quark massm in Eq. ~10! explicitly breaks thechiral symmetry and can be set to zero in many applicatioFor certain quantities, however, it is convenient or even nessary to consider finitem. The effective theory~10! contains

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the Wess-Zumino term and the four-derivative GassLeutwyler terms with correct coefficients. It has been derivfrom the instanton model of the QCD vacuum@23,37#, and isvalid at low energies below a scale set by the inverse ofaverage instanton size,

rav21'600 MeV. ~12!

In practical calculations it is convenient to take the mometum dependent quark mass constant, i.e.M (p)→M (0)5350 MeV. In this caserav

21 is to be understood as thcutoff, at which quark momenta have to be cut off withsome appropriate regularization scheme. It is importanremark that (Mrav)

2 is proportional to the parametricallsmall instanton packing fraction

~Mrav!2}S rav

RavD 4

!1, ~13!

with Rav denoting the average distance between instantoThe smallness of this quantity has been used in the dertion of the effective theory~10! from the instanton vacuummodel @23,37#.

The xQSM is an application of the effective theory~10!to the description of baryons@37#. The large-Nc limit allowsto solve the path integral over pion field configurations in E~10! in the saddle-point approximation. In the leading ordof the large-Nc limit the pion field is static, and one cadetermine the spectrum of the one-particle Hamiltonianthe effective theory~10!:

Hun&5Enun&, H52 ig0gk]k1g0MUg51g0m. ~14!

The spectrum consists of an upper and a lower Dirac ctinuum, distorted by the pion field as compared to continof the free Dirac Hamiltonian

H0un0&5En0un0&, H052 ig0gk]k1g0M1g0m, ~15!

and of a discrete bound state level of energyElev , if the pionfield is strong enough. By occupying the discrete level athe states of the lower continuum each byNc quarks in ananti-symmetric color state, one obtains a state with unbaryon number. The soliton energyEsol is a functional of thepion field,

Esol@U#5NcS Elev1 (En,0

~En2En0! D U

reg

. ~16!

Esol@U# is logarithmically divergent and has to be regulaized appropriately, as indicated in Eq.~16!. Minimization ofEsol@U# determines the self-consistent soliton fieldUc . Thisprocedure is performed for symmetry reasons in the so-cahedgehog ansatz

pa~x!5eraP~r !, U~x!5cosP~r !1 i taer

asinP~r !, ~17!

with r 5uxu and era5xa/r , in which the variational problem

reduces to the determination of the self-consistent sol

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profile Pc(r ). The nucleon massMN is given byEsol@Uc#.The momentum and the spin and isospin quantum numof the baryon are described by considering zero modes osoliton. Corrections in the 1/Nc expansion can be includeby considering time dependent pion field fluctuations arouthe solitonic solution. A good and for many purposes sucient approximation to the self-consistent profilePc(r ) isgiven by the analytical ‘‘arctan-profile’’

P~r ,mp!522 arctanS Rsol2

r 2~11mpr !e2mpr D ,

Rsol5M 21, ~18!

g

gk.ioguq.ef

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ysall

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whereRsol is the soliton size. In the chiral limitm→0 in Eq.~10! the self-consistent profilePc(r )}1/r 2 for r→`, andthen mp50 in Eq. ~18!. If mÞ0 in Eq. ~10! the self-consistent profile exhibits a Yukawa tailPc(r )}e2mpr /r forr→` and mp is the physical pion mass connected to tcurrent quark massm in Eq. ~10! by the Gell-Mann–Oakes–Renner relation@see below Eq.~58!#. The analytical profile~18! simulates this and the Gell-Mann–Oakes–Renner rtion holds approximately.

The xQSM allows to evaluate in a parameter-free wnucleon matrix elements of the QCD quark bilinear operatas ~schematically!

^Nuc~z1!Gc~z2!uN&5cG2MNNc(nocc

E d3XFn~z12X!GFn~z22X!eiEn(z102z2

0)1 . . . ~19!

52cG2MNNc(nnon

E d3XFn~z12X!GFn~z22X!eiEn(z102z2

0)1 . . . . ~20!

-

ee.

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ad

In Eqs. ~19!, ~20! G is some Dirac- and flavor-matrix,cG aconstant depending onG and the spin and flavor quantumnumbers of the nucleon stateuN&5uS3 ,T3&, and Fn(x)5^xun& are the coordinate space wave functions of the sinquark statesun& in Eq. ~14!. The sum in Eq.~19! goes overoccupied levelsn ~i.e. n with En<Elev), and vacuum sub-traction is implied forEn,Elev analogue to Eq.~16!. Thesum in Eq.~20! goes over non-occupied levelsn ~i.e. n withEn.Elev), and vacuum subtraction is implied for allEn

.Elev .2 The dots in Eqs.~19!, ~20! denote terms subleadinin the 1/Nc expansion, which will not be needed in this worDepending on the Dirac and flavor structures the express~19!, ~20! can possibly be UV divergent and need to be relarized. If in QCD the quantity on the left hand side of E~19! is normalization scale dependent, the model results rto a scale roughly set byrav

21 in Eq. ~12!.In the way sketched in Eqs.~19!, ~20! static nucleon prop-

erties~form factors, axial properties, etc., see@39# for a re-view!, twist-2 @25–29# and twist-3@33,34# quark and anti-quark distribution functions, and off-forward distributiofunctions @40# have been studied in thexQSM. As far asthose quantities are known, the model results agree to wi~10–30!% with experiment or phenomenology. It is impotant to note the theoretical consistency of the approachparticular the quark and antiquark distribution functionsthe model satisfy all general QCD requirements~sum rules,positivity, inequalities, etc.!.

2The possibility of computing model expressions in the two waEq. ~19! or ~20!, has a deep connection to the analyticity and locity properties of the model@25#. In practice it provides a powerfucheck of numerical results.

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IV. ea„x… IN THE xQSM

A. Expressions and consistency

~a! Model expressions.The model expressions for the flavor combinations (eu6ed)(x) ‘‘follow’’ from the expres-sions for the unpolarized twist-2 distributions (f 1

u6 f 1d)(x)

derived in Ref.@25# by ‘‘replacing’’ the relevant Dirac struc-ture naga in the definition of f 1

a(x) by 1/MN . This can bechecked by an explicit calculation which closely follows thderivation given in@25# and can therefore be skipped herThis ‘‘analogy’’ betweenea(x) and f 1

a(x) is due to the factthat both are ‘‘spin average’’ distributions, and the relevaDirac and flavor structures exhibit the same properties unthe hedgehog symmetry transformations. As a consequethe flavor combinations (eu6ed)(x) have the same large-Nc

behavior as (f 1u6 f 1

d)(x) @25#, namely

~eu1ed!~x!5Nc2d~Ncx!

~eu2ed!~x!5Ncd~Ncx!, ~21!

where the functionsd(y) are stable in the limitNc→` forfixed argumentsy5Ncx, and different for the different flavorcombinations. Though derived in thexQSM, the relations inEq. ~21! are of general character, considering that thexQSMis a particular realization of the large-Nc picture of thenucleon@41#. The relations~21! are already the end of thstory of ‘‘analogies’’ betweenea(x) and f 1

a(x) in a relativis-tic model. In the non-relativistic limit, however,ea(x) andf 1

a(x) become equal, see Sec. V below.In this work only the leading order in the large-Nc limit

will be considered. At this order the model expressions re

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~eu1ed!~x!5NcMN (n occ

^nug0d~xMN2 p32En!un&

~22!

52NcMN (n non


~23!

and (eu2ed)(x)50 as anticipated in Eq.~21!.~b! Sum rule for the first moment.The first moment of the

model expression in Eq.~22! reads

E21

1

dx~eu1ed!~x!5Nc (n occ

^nug0un&[spN

m. ~24!

When integrating overx in Eq. ~24! one can substitutex→y5xMN and extend they-integration range@2MN ,MN#to the wholey axis in the large-Nc limit. The final step in Eq.~24! follows by recognizing in the intermediate step in E~24! the model expression for the scalar isoscalar form facs(t),

s~ t !5mNcE d3xJ0~A2tuxu! (n occ

Fn* ~x!g0Fn~x!,

~25!

at t50. ~The Bessel functionJ0(z)5sinz/z→1 for z→0.!The pion-nucleon sigma-termspN5s(0) was studied in thexQSM in @42# and the form factors(t) in @43#.

The model expression forspN can also be derived in aalternative way using the Feynman-Hellmann theorem~thismethod was used in@42#!,

spN5m]MN~m!

]m. ~26!

Rewriting the expression for the nucleon massMN[MN(m) in Eq. ~16! as

MN~m!5Nc(nocc

En5Nc(nocc

^nuHun&

[MN~0!1mNc(nocc

^nug0un&, ~27!

where vacuum subtraction is implied, and insertingMN(m)~27! into Eq. ~26! one recovers the model expressionsspN in Eqs.~24!, ~25!. This proof is formally correct but oneshould be careful about regularization. A comment on twill be made at the end of Sec. IV C.

~c! Sum rule for the second moment.The second momenof (eu1ed)(x) in Eq. ~22! is

E21

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dxx~eu1ed!~x!5Nc

MN(n occ

^nug0~ p31En!un&

5Nc

MN(n occ

En^nug0un&, ~28!

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r

t

where^nug0p3un& drops out due to hedgehog symmetry.QCD the sum rule~4! follows from using equations of motion. In the model the analogon is to useEnun&5Hun&. Oneobtains

E21

1

dxx~eu1ed!~x!5Nc

MN~m1bM !, ~29!

b[ (n occ

^nuU1U†

2un&, ~30!

where the relation En^nug0un&5 12 ^nu$H,g0%un&5m

1M ^nu 12 (U1U†)un& was used. Equation~29! then follows

from (nocc nun&5Sp@Q(Elev102H)2Q(2H0)#5Bwhere the vacuum subtraction is considered explicitly aB51 denotes the baryon number@25#. ~Sp is the functionaltrace which can be saturated by respectively Sp@ . . . #5(nall^nu . . . un& or (n0all^n0u . . . un0&.!

For b50 the QCD sum rule~4! would hold ‘‘literally’’ inthexQSM. However, in the model the equations of motioare modified compared to QCD and one cannot expecb50 in Eqs. ~29!, ~30!. Instead, the modified equations omotion in thexQSM suggest to interpretbM ~in the chirallimit ! as the effective mass of model quarks bound insoliton field. ~One cannot expectb51 either, which wouldimply an effective massM. It is jargon to refer toM as mass,strictly speakingM is a dimensionful coupling of the fermionfields to the chiral background fieldU.!

B. Calculation of „eu¿ed…„x…

~d! Interpolation formula.The approximation referred toas interpolation formula@25# consists in exactly evaluatingthe contribution from the discrete level to (eu1ed)(x) in Eq.~22!, and in estimating the continuum contribution as folows. One rewrites the continuum contribution in termsthe Feynman propagator in the static background solfield U and expands it in powers of gradients of theU field,keeping the leading term~s! only.3 The interpolation formulayields exact results in three limiting cases:~i! low momenta,u“Uu!M , ~ii ! large momenta,u“Uu@M , ~iii ! any momentabut small pion field,u logUu!1. One can expect that it yielduseful estimates also in the general case. Indeed, it hasobserved that estimates based on the interpolation formagree with results from exact~and numerically much moreinvolved! calculations to within 10%@25–27#.

~e! Discrete level contribution.The HamiltonianH ~14!

commutes with the parity operatorp and the grand-spin op

3It should be noted that this is not a strict expansion in gradieof the U field. The dimensionless parameter characterizing thispansion is 1/(MRsol). Since the soliton solution is given foMRsol51, see Eq.~18!, such a strict expansion is not defined@25#.In the following higher orders in the gradient expansion willconsidered merely in order to study theUV behavior of the modelexpressions.

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eratorK , defined as the sum of the total quark angular mmentum and isospin operator. The discrete level occurs inKp501 sector of the HamiltonianH ~14!. In the notation ofRef. @25# the discrete level contribution reads

~eu1ed!~x! lev5NcMN^ levug0d~xMN2Elev2 p3!u lev&

5NcMNEuxMN2Elevu

` dk

2k@h~k!22 j ~k!2#,

~31!

whereh(k) and j (k) are the radial parts of respectively thupper and the lower component of the discrete level wfunction in momentum space,F lev(p)5^pu lev&, see@25# fordetails.

~f! Dirac continuum contribution.The two equivalent ex-pressions, Eqs.~22!, ~23!, allow to compute the contributionof the continuum states to (eu1ed)(x) in two different ways,

~eu1ed!~x!cont5NcMN (En,0


~32!

52NcMN (En.0

^nug0d~xMN2 p32En!un&.

~33!

The expressions~32! and ~33! can be rewritten by means othe Feynman propagator in the static background pion fias ~see@25#!

~eu1ed!~x!cont5 (n50

`

~eu1ed!~x!cont(n) ,

~eu1ed!~x!cont(n) 5Im NcMNE d4p

~2p!4

3d~p01p32xMN!trF ^pu~p”1MU2g5!

31

p021“

22M21 i0S 2 iM g i] iU

2g5

31

p021“

22M21 i0D n

up&GUreg

. ~34!

The vacuum subtraction is implied in Eq.~34! and meansthat the same expression but withU→1 has to be subtractedi.e. it is relevant only for the casen50. The subscript regreminds that the expression might be UV divergent andto be regularized appropriately. Closing thep0-integrationcontour to the upper half of the complexp0 plane yields Eq.~32!. Closing it to the lower half-plane yields Eq.~33!.

~g! Gradient expansion: Zeroth order.Performing thetraces overg matrices in the zeroth order contribution(eu1ed)(x)cont in Eq. ~34! yields

11401

-he

e

ld

s

~eu1ed!~x!cont(0) 5BsolA~x! ~35!

with

Bsol51

2E d3xtrFS U1U†

221D , ~36!

A~x!5NcMN8M ImE d4p

~2p!4

d~p01p32xMN!

p22M21 i0U

reg

.

~37!

The coefficientBsol is real and contains the information othe soliton structure. The ‘‘21’’ under the flavor trace in Eq.~36! is due to vacuum subtraction. The functionA(x) in Eq.~37! is well defined within some appropriate regularizatischeme to be figured out in the following.A(x) is an evenfunction of x, provided the regularization is consistent withe substitutionsp0→2p0 andp3→2p3 in Eq. ~37!. Keep-ing xÞ0 and integrating overp3 and p0 ~with the above-mentioned prescription to close the contour! one obtains

A~x!non/occ52Q~6x!

uxu2NcM

p E d2p'

~2p!2Ureg

, ~38!

with the ‘‘1 ’’ sign referring to Eq.~33! and the ‘‘2 ’’ signreferring to Eq.~32!. Thus A(x) is quadratically divergent,and depends on whether one computes it by means of~32! or ~33!. The non-equivalence of the two ways to compute a quantity in the model, Eqs.~32! and~33!, at the levelof unregularized model expressions is a known phenome@25,29#. The equivalence of Eqs.~32! and ~33!—and moregenerally of Eqs.~19! and ~20!—is a basic property of themodel ~see footnote 2!. Therefore it is necessary to restothe equivalence of Eqs.~32! and ~33! in the expression~38!by means of a suitably chosen regularization.regularization—which does this—is a Pauli-Villars subtration of the type

A~x!5A~x,M !2M

M1A~x,M1!, ~39!

where M1.M is the Pauli-Villars mass. The Pauli-Villarsubtraction is the privileged method to regularize divergdistribution functions in thexQSM. This regularization pre-serves all general properties of distribution functions~QCDsum rules, positivity, etc.! @25#, and where necessary it restores the equivalence of Eqs.~32! and~33! in the final regu-larized model expressions@25,29#. In the regularization~39!—which is sufficient atthis stage—both formulas~32!and ~33! yield the same result forA(x), namely

A~x!50 for xÞ0. ~40!

Instead of studying the functionA(x) at the pointx50 itis more convenient to consider moments ofA(x) defined asMn@A#5*dxxn21A(x). SinceA(x) is even inx, one needsto consider only odd moments 2k11 (k50,1,2, . . . ),

0-6

st

ld

on

t

e-

n-

-

of

m-

-

a

eo-

eo-


M@A# (2k11)58NcM

MN2k

Im(j 50

k S 2k

2 j D3E d4p

~2p!4

~p0!2 j~p3!2k22 j

p022p22M21 i0

Ureg

. ~41!

Performing a Wick rotation in Eq.~41!, which is well definedfor all moments under the regularization~39!, one obtains

M@A# (2k11)528NcM

MN2k (

j 50

k S 2k

2 j D ~21! j

3E d4pE

~2p!4

~pE0!2 j~pE

3!2k22 j

pE21M2 U

reg

. ~42!

Using 4D spherical coordinates

pEm5q~cosc,sinc sinu cosf,sinc sinu sinf,sinc cosu!

and substitutingk[q2 one has

M @A# (2k11)528NcMak

2~2p!4MN2kE0

` dkkk11

k1M2 Ureg

,

ak[(j 50

k S 2k

2 j D ~21! jE0

2p

dfE0

p

duE0

p

dc sinu sin2c

3~cosc!2 j~sinc cosu!2k22 j52p2dk0 . ~43!

I.e. only k50 contributes which means that only the firmoment ofA(x) is non-zero,

M @A# (1)52NcM

2p2 E0

` dkk

k1M2Ureg

,

M @A# (n)50 for n>2. ~44!

The regularization prescription~39! removes the leadingquadratic divergence in the integral in Eq.~44!, but leaves alogarithmic one unregularized. However, e.g., a twofoPauli-Villars subtraction

A~x!5A~x,M !2a1A~x,M1!2a2A~x,M2! ~45!

with

a15M

M1

M222M2

M222M1

2, a252

M

M2

M122M2

M222M1

2, M2.M1.M

~46!

is sufficient to make the first moment ofM @A# (1) finite. Inthe limit M2→` one recovers the previous regularizati~39!. It is important to note that the modification, Eqs.~45!,~46!, of the previous regularization~39! still preserves theproperty~40!. Thus, (eu1ed)(x)cont

(0) (x) satisfies

~eu1ed!~x!cont(0) ~x!50 for xÞ0,

11401

E dxxn21~eu1ed!~x!cont(0) ~x!5Cdn1 , ~47!

i.e. (eu1ed)(x)cont(0) (x) is proportional to ad function at x

50. What remains to be done is to compute the coefficienCof the d function in Eq.~47!.

The two Pauli-Villars subtractions in Eq.~45! introducetwo parametersM1 and M2 in Eq. ~46!, which have to befixed. E.g. one could first fixM1 by regularizing the loga-rithmically UV divergent model expression for the pion dcay constantf p ,

f p2 54NcE d4pE

~2p!4

M2

~pE21M2!2U

reg

, ~48!

such that it gives the experimental valuef p593 MeV. Thenone could fixM2, e.g., by means of the quark vacuum codensate which is given in the effective theory~10! by thequadratically divergent expression@39#

^vacu~ cucu1cdcd!uvac&

528NcE d4pE

~2p!4

M

pE21M2U

reg

. ~49!

Two subtractions analog to Eq.~45! are required to regularize Eq. ~49!. In this way the free parametersM1 and M2are fixed.4 It is important to stress that the parametersM1andM2 are fixed in the vacuum and in the meson sectorthe effective theory~10!. In this sense thexQSM, i.e. thebaryon sector of the effective theory~10!, producesparameter-free results.

It is interesting to observe that the coefficientC can di-rectly be expressed in terms of the quark condensate~49!,such that

C5Bsol vacu~ cucu1cdcd!uvac& ~50!

with Bsol defined in Eq.~36!. In Eq. ~50! any details on theregularization have disappeared. However, the explicit de

4For f p2 in Eq. ~48! a single subtraction, f p

2 [ f p2 (M )

2(M /M1)2f p2 (M1) with M15556 MeV is required. For the quark

condensate~49! one needs two subtractions analogous to Eqs.~45!,~46! with M25(9.165.7) GeV in order to reproduce the phenomenological value@(280630) MeV#3 @44#. The first Pauli-VillarsmassM1 is of the order of magnitude of the ‘‘natural cutoff’’rav

21

'600 MeV of the effective theory~10!. The much larger secondPauli-Villars massM2—in some sense merely introduced as‘‘technical device’’ to remove the ‘‘residual~logarithmic! diver-gence’’ left after the subtraction of the leading~quadratic!divergence—has no physical meaning. In non-renormalizable thries ~such as thexQSM) the cutoff~hereM1;rav

21) has a physicalmeaning and shows up in final expressions. In renormalizable thries the result for the quadratically divergent integral~44! would beproportional toM2, see e.g.@45#.

0-7


FIG. 1. ~a! The oversimplified partonic interpretation ofeq(x). ~b! The symbolic diagrammatic interpretation of thed(x) contribution toea(x) as suggested by Eq.~54!. The quark lines ‘‘carry the fractionx50 of the nucleon momentum,’’ see text.

on

rs

ion

t.iva-re

nu-erx

in

onsu-s.

um

on

l

thethe

he-

g, aiesnent1a.areodde-

uonrect-

onstration of the existence of a regularization prescriptiwhich preserves the properties in Eq.~47!, was a crucial stepin the derivation of Eq.~50!.

~h! Gradient expansion: Higher orders.Taking the traceover g and flavor matrices in the expression for the fiorder term in the expansion~34!, (eu1ed)(x)cont

(1) , one ob-tains

~eu1ed!~x!cont(1) 5ImNcMN~2 i8M !

3E d4p

~2p!4

d~p01p32xMN!

~p22M21 i0!2

3pk^pu¹kcosP~ r ,mp!up&ureg50, ~51!

since ^pu“kcosP(r,mp)up&5*d3x¹kcosP(r,mp)50. Alengthy calculation yields for the second order contributin Eq. ~34! the result

~eu1ed!~x!cont(2) 5Im

NcMNM

i8p2 E dn

2peinxMNE

0

1

da

3E0

a

dbE d3xtrF A,

A5U†~x2ane3!@“kU~x2bne3!#

3@¹kU†~x!#1~U↔U†!, ~52!

which does not depend on how contours are closed, i.e.two ways, Eqs.~32! and ~33!, yield the same result. In Eq~52! terms are neglected which contain three or more dertives acting on theU fields. In a strict expansion in the number n of U-field gradients these terms have to be considein higher ordersn>3. The result~52! is real and UV finite.Still, it has to be regularized according to the prescriptio~45!, ~46!. In an exact evaluation of the continuum contribtion it would be, of course, not possible to pick up the divgent contribution from the zeroth order in the gradient epansion~34! and regularize only that. Since (eu1ed)(x)cont

(2)

}M , the application of the regularizations~45!, ~46! to Eq.~52! yields

~eu1ed!~x!cont(2) 50. ~53!

11401

,

t

he

-

d

s

--

All higher ordersn>3 in the gradient expansion~34! are UVfinite.

~i! Intermediate summary.In @29# it was shown that—if itoccurs—the non-equivalence of Eqs.~32! and ~33! at thelevel of unregularized model expressions only shows upthe lowest UV-divergent order~s! of the expansion~34!.Thus, the results of this section show that the regularizati~45!, ~46! ~i! consistently regularize the continuum contribtion to (eu1ed)(x), and~ii ! ensures the equivalence of Eq~32! and ~33!.

The final regularized model expression for the continucontribution consists of ad function atx50 with a coeffi-cient proportional to the quark condensate and the factorBsolin Eq. ~36! which encodes the information on the nucle~i.e. soliton! structure,

~eu1ed!~x!cont5Cd~x!,

C5Bsol vacu~ cucu1cdcd!uvac&. ~54!

The existence of thed function is a feature of the mode~with the Pauli-Villars regularization method!. The fact thatthe continuum contribution consists of no regular part butd function only has to be considered as a peculiarity ofapproximation~interpolation formula! used.

~j! How to interpret ad function?A d(x) contribution toa distribution function has no partonic interpretation. Tobservation in Eq.~54!, however, suggests an intuitively appealing ‘‘interpretation.’’

Oversimplifyingly ea(x)dx can be interpreted as takinout of the nucleon in the infinite momentum frame, e.g.left-handed good light-cone quark component which carrbetweenx and x1dx of the nucleon momentum, and thereinserting a right-handed bad light-cone quark componwith the same momentum back into the nucleon, see Fig.@The good and bad quark light-cone degrees of freedomstrictly speaking defined in the light-cone quantization. Gomeans independent degrees of freedom, the bad quarkgrees of freedom are composites of good quark and gldegrees of freedom. Considering this one obtains the corpartonic interpretation ofea(x) as a quark-gluon-quark correlation function@1,2#.#

Does it make sense to pick up hereby a quark~or anti-quark! which carries the fractionx50 of the nucleon mo-

0-8


FIG. 2. ~a! The regular part of the chirally odd twist-3 flavor-singlet quark and antiquark distribution functions, (eu1ed)(x) and (eu

1ed)(x), at a low normalization point of the model of about 600 MeV vsx. At x50 there is ad(x) contribution.~b! The chiral quark-solitonmodel results for (eu1ed)(x) ~computed here! and (f 1

u1 f 1d)(x) ~from @27#! at a low scale of about 600 MeV vsx. ~c! The same as Fig. 2b

but for antiquarks.

in

inssThloura

taE

isveis

ria

heit

de

rld-

n-the

on-

–

lueto

toe

mentum, i.e. which is at rest with respect to the fast movnucleon? Equation~54! suggests that such a quark~or anti-quark! is picked up from the vacuum, which to a certaextent is present also inside the nucleon. It should be strethat one does not deal with a disconnected diagram.factor Bsol shows that the nucleon line and the vacuum bare connected by the exchange of a resonance with the qtum numbers of the sigma meson, see the symbolic diagin Fig. 1b.

It would be interesting to see whether such an interpretion could be confirmed by observations analogous to~54! in other models.

C. Discussion of the results for„eu¿ed…„x…

The final result for (eu1ed)(x) from the interpolationformula is the contribution of the discrete level~31! ~alreadythe total result forxÞ0) and the continuum contribution~54!consisting of ad(x) function. It should be noted that thereno freedom to also regularize the UV-finite discrete lecontribution in the Pauli-Villars regularization method. Thcontribution must not be regularized for otherwise the vational problem of minimizing the soliton energy in Eq.~16!has no solution, i.e. no soliton exists@27#.

Figure 2a shows the final results for (eu1ed)(x) and(eu1ed)(x) ~no effort is made to indicate thed function atx50). It is instructive to compare (eu1ed)(x) to ( f 1

u

1 f 1d)(x) in the model. Both are of the same order in t

large-Nc limit and become equal in the non-relativistic lim~see Sec. V!. For the quarks one observes that (f 1

u1 f 1d)(x) is

about 2–3 times larger than (eu1ed)(x), while the corre-sponding antiquark distributions are of a similar magnitusee Figs. 2b and 2c.

In order to compute the coefficientC of thed(x) functionin Eq. ~54!, one has to evaluateBsol ~36!. For the physicalsituation withmp5140 MeV one obtains

Bsol51

2E d3x trFS U1U†

221D

5E d3x@cosP~r ,mp!21#5217.2Rsol3 . ~55!

11401

g

ede

ban-m

-q.

l

-

,

In the chiral limit mp→0 the integral in the expression foBsol in Eq. ~36! can be evaluated in an elementary way yieing Bsol522A2p2Rsol

3 5227.9Rsol3 . I.e. the coefficientC

is by about 40% increased in the chiral limit. This demostrates the importance of considering this quantity inphysical situation with finitemp . Numerically one obtainsfor Rsol5M 215(350 MeV)21 and mp5140 MeV the re-sult

C5~9.162.8!, ~56!

where the error is due to the uncertainty of the vacuum cdensate (2280630)3 MeV3 @44#. For the first moment of(eu1ed)(x) one thus obtains

E21

1

dx~eu1ed!~x!5E21

1

dx~eu1ed!~x! lev

1E21

1

dx~eu1ed!~x!cont

51.61~9.162.8!5~10.762.8!

~57!

in good agreement with Eq.~9!. In order to obtainspN fromEq. ~57! it is convenient to use the Gell-Mann–OakesRenner relation

mp2 f p

2 52m^vacu~ cucu1cdcd!uvac&, ~58!

which holds in the effective theory~10! @39# and allows toeliminate the uncertainty from the phenomenological vaof the vacuum condensate in the continuum contributionspN ,

~spN!cont5mE21

1

dx~eu1ed!~x!cont5mC

52mp2 f p

2 Bsol567.8 MeV. ~59!

In order to obtain the contribution of the discrete levelspN one can use Eq.~58! to obtain a consistent value for thcurrent quark massm5(8.262.5)MeV. This yields

0-9

inrp

ltsith

ly

a

of

et

on

ioe-

ule

care

he

ly

ec

e

r

n

l-

al

e

nly

n-hes.

f

t

alswer

lts

ed.u-bag

f

.6


(spN) lev5(13.264.1)MeV and the total result isspN5(8164)MeV. There is no point in keeping track of the errorthis case, since it is smaller than the accuracy of the intelation formula~which was found to be about610% when-ever it was checked quantitatively!. Thus one obtains

spN581 MeV. ~60!

The result~59! is about 30% larger than former exact resufrom thexQSM which, however, have been calculated wa different~proper-time! regularization@42,43#. Consideringthat spN is quadratically divergent and thus rather strongsensitive to regularization, the result in Eq.~60! is in goodagreement with the results of Refs.@42,43#. Worthwhile men-tioning is that all model numbers—from Eq.~60! and fromRefs. @42,43#—are consistent with the phenomenologicvalue forspN in Eq. ~8! within (10–30)%.

In QCD—as mentioned in Sec. II—the first momentea(x) is due to thed(x) function only. In thexQSM thed(x) function provides the dominant~more than 80%) butnot the only contribution to the first Mellin moment of (eu

1ed)(x). The second moment of (eu1ed)(x) receives nocontribution from the continuum and is due to the discrlevel contribution only,

E21

1

dxx~eu1ed!~x!50.20. ~61!

This result would imply that the quarks bound in the solitfield have an effective massbM;78 MeV ~in the chirallimit !, see the discussion below the Eqs.~29!, ~30!.

A comment is in order on an exact numerical evaluatof (eu1ed)(x). The distribution functions computed in thxQSM so far were all either UV finite or at most logarithmically divergent. In the latter case always a single PaVillars subtraction was sufficient. The regularization prscriptions ~45!, ~46! precisely state how (eu1ed)(x) canpractically be regularized. However, an exact numerievaluation meets the problem to evaluate the model expsions for a Pauli-Villars massM2' several GeV~see foot-note 4!. In the numerical calculation the spectrum of tHamiltonian~14! is discretized and made finite~see e.g.@26#and references therein!. For the latter step one considers onquark momenta below some large numerical cutoffLnumchosen much larger than any other~physical or numerical!scale involved in the problem. So farLnum'(8 –10)GeVwas sufficient, but this is the order of magnitude of the sond Pauli-Villars massM2. To compute (eu1ed)(x) one hasto chooseLnum much larger thanM2, which would result inan uneconomically large increase of computing time.

Interestingly, the singulard(x) contribution would con-ceptually cause no problem for the numerical method of R@26#. The descretized spectrum of the Hamiltonian~14!yields discontinuous~distribution! functions ofx. In @26# itwas proposed to smear the distribution functionsq(x), i.e. toconvolute them with a narrow Gaussian with an appropately chosen widthg as q(x)smear5(1/gp1/2)*dx8exp@2(x2x8)2/g2#q(x8). ~The smearing can be removed by a decovolution procedure.! This trick would turn thed(x) contri-

11401

o-

l

e

n

i--

ls-

-

f.

i-

-

bution in (eu1ed)(x) into a narrow Gausssian with the weldefined widthg. In this way the coefficient of thed(x)contribution could be well determined from the numericresult.

Finally, a comment is in order on Eq.~26! which relatesthe logarithmically divergent nucleon massMN and the qua-dratically divergentspN . MN requires a single Pauli-Villarssubtraction, whilespN requires two subtractions. Thus thquantities on the left-hand and right-hand sides in Eq.~26!are regularized differently. In the Pauli-Villars regularizatioscheme the relation~26! has to be considered as formalcorrect modulo regularization effects.~Some other ambigu-ities in the Pauli-Villars regularization scheme were metioned in@46#.! In other regularization methods—such as tproper time regularization—there are no such ambiguitie

D. Comparison to the bag model

Studies ofea(x) were also performed in the framework othe bag model in Refs.@2,21#. In Fig. 3 thexQSM result forthe regular part of (eu1ed)(x) is compared to the resulfrom the MIT bag model from Ref.@2#. „For that the flavor-independent results for ‘‘e(x)’’ from Ref. @2# are multipliedby the factorNc53 in order to compare to (eu1ed)(x) ob-tained here.… The comparison of the position of the maximof the curves for quark distributions from the two modeindicates that the bag model results refer to a somehow loscale than 0.6 GeV, the scale of thexQSM. ~In @21# thevalue of 0.4 GeV was quoted.! Taking this into account oneconcludes that both models give qualitatively similar resufor quark distributions.

Concerning antiquarks the difference is more pronouncHowever, the bag model description of antiquark distribtions cannot be considered as reliable. A drawback of the

model in this context is that it yields negativef 1q(x) in con-

tradiction to the positivity requirement.

V. ea„x… IN THE NON-RELATIVISTIC LIMIT

In the limit of the soliton sizeRsol→0 the expressions othe xQSM go into the results of the non-relativistic~‘‘na-

FIG. 3. The twist-3 distribution function (eu1ed)(x) from thexQSM ~solid line, obtained here! and from the bag model~dashedline, from Ref.@2#! vs x. The meaning of the curves at negativex isexplained in Eq.~1!. ThexQSM result refers to a scale of about 0GeV. The bag model result refers to a scale of about 0.4 GeV.

0-10

of

thofitie

ft

e

-fi-

elqo

cr

c-

e-

for

ce

ce

ed in

on-

ry,n-3

of-

e

e-

ully

rsity


ive’’ ! quark model formulated for an arbitrary numbercolorsNc @47#. In this sense the limitRsol→0 corresponds tothe non-relativistic limit in thexQSM. This was studied indetail in Ref.@48#.

As Rsol→0 the soliton profile~18! goes to zero, and theUfield approaches unity. Correspondingly, the spectrum ofHamiltonian~14! becomes more and more similar to thatthe free Hamiltonian~15!. Considering vacuum subtractionis clear that the contribution of the continuum vanishesthis limit @48#, and all that remains is the contribution of thdiscrete levelu lev&. More precisely, asRsol→0 the energy ofthe discrete levelElev→M @so the nucleon mass, Eq.~16!,formally MN→NcElev→NcM ], and the lower component othe Dirac-spinor of the discrete level wave function goeszero @48#.

The limit Rsol→0 means thatU→1 i.e. logU!1 whichis the case~iii ! in which the interpolation formula yieldsexact results~see Sec. IV B!. The first feature—in this casthe vanishing of the continuum contribution in Eqs.~32!,~33!—can be observed in the final result (eu1ed)(x)cont5Cd(x) in Eq. ~54!. The factorBsol in the coefficientCvanishes withRsol→0 as can be seen from its definition~36!or Eq. ~55!. Thus, in the non-relativistic limit the contribution of the d function atx50 vanishes because the coefcient C goes to zero.

To study the non-relativistic limit in the discrete levcontribution it is convenient to use the first expression in E~31!. Since only the upper component of the Dirac spinorthe discrete level wave function survives the limit@48#, onecan replaceg0 by the unity matrix, i.e.

limnon-rel

~eu1ed!~x!→NcMN^ levud~xMN2 p32Elev!u lev&.

~62!

Next consider thatElev→M and MN→NcM while the mo-menta of the non-relativistic quarksupi u!M such that thecorresponding operator in thed function in Eq.~62! can beneglected. Using the normalization^ levu lev&51 one obtains

limnon-rel

~eu1ed!~x!→NcdS x21

NcD . ~63!

The first two moments of Eq.~63! read

limnon-rel

E21

1

dx~eu1ed!~x!5Nc ,

limnon-rel

E21

1

dxx~eu1ed!~x!51. ~64!

To see that the relations~64! are the correct non-relativistiresults for the QCD sum rules~4! and~6! one has to considethat in the non-relativistic limit the current quark massm→M5MN /Nc and spN→MN . ~The latter relation followsformally, e.g., from the Feynman-Hellmann theorem~26!with m5M5MN /Nc .)

The twist-2 unpolarized flavor-singlet distribution funtion ( f 1

u1 f 1d)(x) is given in thexQSM by @25#

11401

e

n

o

.f

~ f 1u1 f 1

d!~x!5NcMN (n occ

^nu~11g0g3!

3d~xMN2p32En!un&, ~65!

i.e. the only difference to (eu1ed)(x) is the different Diracstructure (11g0g3) instead ofg0. This difference becomesirrelevant in the non-relativistic limit~where only the uppercomponent of the Dirac spinor of the discrete level wavfunction survives! such that in this limit (eu1ed)(x) and( f 1

u1 f 1d)(x) become equal. This argument holds also

separate flavors and allows to generalize

limnon-rel

eq~x!5 limnon-rel

f 1q~x!5NqdS x2

1

NcD , ~66!

where Nq denotes the number of the respective valenquarks ~i.e. for the protonsNu52 and Nd51 for Nc53colors!. The result~66! means that in the non-relativistilimit eq(x) is given by the mass term contribution in thdecomposition~2! because

limnon-rel

H mq

MN

f 1q~x!

x J 5M

MNlim

non-relH f 1

q~x!

x J 5 limnon-rel

f 1q~x!.

~67!

In the intermediate step in Eq.~67! the d(x21/Nc) functionwas used to replacex in the denominator by 1/Nc . One mayworry that the large valuespN5MN would mean a largestrangeness content of the nucleon. However, as discuss@8# the value spN5MN correctly implies a vanishingstrangeness contribution to the nucleon mass in the nrelativistic limit.

Thus, though phenomenologically it is not satisfactothe non-relativistic picture of the twist-3 distribution functioeq(x) is consistent. In this limit the singular and pure twistcontributions in the decomposition~2! vanish, andeq(x) isgiven by the mass term. Thus the chirally odd natureeq(x) arises from a ‘‘mass insertion’’ into a quark line. Moreover, eq(x) and f 1

q(x) become equal in this limit, and artrivial d-functions concentrated atx51/Nc which means thatthe nucleon momentum is distributed equally among theNcmassive and non-interacting constituent quarks.

The usefulness of results of the kind~66! is best illus-trated by the popularity of the non-relativistic relation btween the twist-2 helicityg1

q(x) and transversityh1q(x) dis-

tribution functions~here forNc53),

limnon-rel

h1q~x!5 lim

non-relg1

q~x!5PqdS x21

3D , Pu54

3, Pd52

1

3,

~68!

which yields for the axial chargesgA(3)5 5

3 and gA(0)51.

Though also these numbers are phenomenologically not fsatisfactory the assumption thath1

a(x)5g1a(x) at some low

scale is a popular guess to estimate effects of transvedistribution, see@49# for a review.

0-11

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nf

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VI. SUMMARY AND CONCLUSIONS

A study of the flavor-singlet twist-3 distribution functio(eu1ed)(x) in the xQSM was presented. It was shown ththe model expressions are quadratically and logarithmicUV divergent and can be regularized by the Pauli-Villamethod. The model expressions for the quark and antiqudistribution functions (eu1ed)(x) and (eu1ed)(x) wereevaluated using an approximation—the interpolation formwhich in general well approximates exact model calcutions.

The remarkable result is that thexQSM expression for(eu1ed)(x) contains ad-function-type singularity atx50as expected from QCD@8#. This result is obtained here froma non-perturbative model calculation. Previously ad(x) con-tribution in ea(x) was observed in a perturbative calculatiin Ref. @9#.

In the xQSM the coefficient of thed function is propor-tional to the quark vacuum condensate. This is natural frthe point of view that in QCD the singular contributioneq(x) and the quark vacuum condensate are both the extation values of the same local scalar operator,cq(0)cq(0),taken respectively in the nucleon and vacuum states. Tobservation allows to make a heuristic but physically appeing interpretation of thed(x) contribution.

At xÞ0 thexQSM yields results for (eu1ed)(x) similarto those obtained in the bag model at a comparably low s@2,21#. Both models suggest thatea(x) is sizable at lowscales. The discription of (eu1ed)(x) in the xQSM is con-sistent in the sense that the sum rules for the first andsecond moment are satisfied. However, in thexQSM thed(x) contribution provides the dominant but not the oncontribution to the first moment of (eu1ed)(x) unlike inQCD, and in the case of the second moment it is necessa

v.

o99

t,

s

11401

tly

rk

a-

m

c-

isl-

le

e

to

interpret the result correspondingly by introducing the notof an effective quark mass. It would be interesting to swhether the failure of the bag model to satisfy the sum rfor the second moment reported in@2# could also be reinter-preted in a similar spirit.

In effective models, such as thexQSM or bag model,equations of motions are altered compared to QCD and this no gauge principle which would allow to cleanly decomposeea(x) into ad(x) contribution, a pure twist-3 part andmass term. Therefore one cannot expect that the QCDrules which are derived by means of the QCD equationsmotion are literally satisfied in such models. Still within thmodels the results are consistent.

The ~large-Nc) non-relativistic limit ofeq(x) was studiedon the basis of thexQSM expressions. It was found that ithis limit eq(x)5 f 1

q(x). The non-relativistic description oeq(x) was shown to be consistent.

The results and interpretations presented here shoulreexamined in the more general framework of the instanmodel of the QCD vacuum, on which thexQSM is founded.This was out of the scope of the study presented herewill be reported elsewhere.

Note added.After this work has been completed the woin @50# appeared, where the authors conclude the existencad(x) contribution in (eu1ed)(x) in the chiral quark-solitonmodel in an independent and complementary way.

ACKNOWLEDGMENTS

I would like to thank A. V. Efremov, K. GoekeP. V. Pobylitsa, M. V. Polyakov, and C. Weiss for manfruitful discussions. This work has partly been performunder the contract HPRN-CT-2000-00130 of the EuropeCommission.

.

l

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in the chiral quark-soliton model