Generalized parton distributions and composite constituent quarks
Sergio ScopettaDipartimento di Fisica, Universita` degli Studi di Perugia, via A. Pascoli 06100 Perugia, Italy
and INFN, Sezione di Perugia, Perugia, Italy
Vicente VentoDepartament de Fisica Teo`rica, Universitat de Vale`ncia, 46100 Burjassot, Vale`ncia, Spain
and Institut de Fı´sica Corpuscular, Consejo Superior de Investigaciones Cientı´ficas, Vale`ncia, Spain~Received 10 July 2003; published 13 May 2004!
An approach is proposed to calculate generalized parton distributions~GPDs! in a constituent quark model~CQM! scenario, considering the constituent quarks as complex systems. The GPDs are obtained from thewave functions of the nonrelativistic CQM of Isgur and Karl, convoluted with the GPDs of the constituentquarks themselves. The latter are modeled by using the structure functions of the constituent quark, the doubledistribution representation of GPDs, and a recently proposed phenomenological constituent quark form factor.The present approach permits us to access a kinematical range corresponding to both the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi and the Efremov-Radyushkin-Brodsky-Lepage regions, for small values of the momen-tum transfer and of the skewedness parameter. In this kinematical region, the cross sections relevant to deeplyvirtual Compton scattering could be estimated by using the obtained GPDs. As an example, the leading twist,unpolarized GPDH has been calculated. Its general relations with the nonrelativistic definition of the electricform factor and with the leading twist unpolarized quark density are consistently recovered from our expres-sions. Further natural applications of the proposed approach are addressed.
Generalized parton distributions~GPDs! @1# parametrizethe nonperturbative hadron structure in hard exclusive pcesses~for comprehensive reviews, see, e.g.,@2–7#!. Themeasurement of GPDs would provide information whichusually encoded in both the elastic form factors and the uparton distribution functions~PDFs! and, at the same time,would represent a unique way to access several crucialtures of the structure of the nucleon@8,9#. By measuringGPDs, a test of the angular momentum sum rule of the pton @10# could be achieved for the first time, determining tquark orbital angular momentum contribution to the protspin @9,11#.
In addition, the possibility of obtaining, by means of GPmeasurements, information on the structure of the protothe impact parameter@12,13# and position@14–16# spaces isbeing presently discussed. Therefore, relevant experimeefforts to measure GPDs, by means of exclusive elecdeep inelastic scattering~DIS! off the proton, are likely totake place in the next few years@17–19#.
In this scenario, it becomes urgent to produce theoretpredictions for the behavior of these quantities. Severalculations have already been performed by using differentscriptions of hadron structure: bag models@20,21#, solitonmodels @4,22#, light-front @23# and Bethe-Salpeter approaches@24#, and phenomenological estimates based onrametrizations of PDFs@25,26#. In addition, an impressiveeffort has been devoted to studying the perturbative Qevolution@27,28# of GPDs and the GPDs at twist 3 accura@29#.
Recently, calculations have been performed also in cstituent quark models~CQMs! @30,31#. The CQM has a long
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history of successful predictions in low energy studies ofelectromagnetic structure of the nucleon. In the high enesector, in order to compare model predictions with data tain DIS experiments, one has to evolve, according to perbative QCD, the leading twist component of the physicstructure functions obtained at the low momentum scalesociated with the model, the so-called ‘‘hadronic scale’’m0
2.Such a procedure, already addressed in@32,33#, has
proven successful in describing the gross features of stanPDFs by using different CQMs~see, e.g.,@34#!. Similar ex-pectations motivated the study of GPDs in Ref.@30#. In thatpaper, a simple formalism has been proposed to calculatequark contribution to GPDs from any nonrelativistic or reltivized model and, as an illustration, results obtained inIsgur and Karl~IK ! model @35# have been evolved fromm0
2
up to DIS scales, to next to leading order NLO accuracy.Ref. @31# the same quark contribution to GPDs has beevaluated, atm0
2, using the overlap representation of GPD@6# in light-front dynamics, along the lines developed in@36#.
Here, the procedure of Ref.@30# is extended and generaized. As a matter of fact, the approach of Ref.@30#, whenapplied in the standard forward case, has been provereproduce the gross features of PDFs@34# but, in order toachieve a better agreement with data, it has to be improIn a series of papers, it has been shown that unpolarized@37#and polarized@38# DIS data are consistent with a low energscenario, dominated by complex constituent quarks insthe nucleon, defined through a scheme suggested bytarelli, Cabibbo, Maiani, and Petronzio~ACMP! @39#, up-dated with modern phenomenological information. The saidea has been recently applied to demonstrate the evidencomplex objects inside the nucleon@40#, analyzing interme-diate energy data of electron scattering off the proton.
S. SCOPETTA AND V. VENTO PHYSICAL REVIEW D69, 094004 ~2004!
addition, a similar scenario has been extensively usedother groups, starting from the concept of the ‘‘valon,’’ itroduced more than 20 years ago@41# ~for recent develop-ments, see@42#!.
We here generalize our description of the forward c@37# to the calculation scheme of Ref.@30#, in order to obtainmore realistic predictions for the GPDs and, at the satime, explore kinematical regions not accessible before.
In particular, the evaluation of the sea quark contributbecomes possible, so that GPDs could be calculated, in pciple, in their full range of definition. Such an achievemewould permit us to estimate the cross sections that areevant for actual GPD measurements, providing us withimportant tool for planning future experiments. Actually,will be shown, the proposed approach will be applied herea nonrelativistic ~NR! framework, which allows one toevaluate the GPDs only for small values of the foumomentum transferD2 ~corresponding toDW 2!m2, wheremis the constituent quark mass! and small values also for thskewedness parameterj. The full kinematical range of definition of GPDs will be studied in a follow-up, introducinrelativity in the scheme.
The paper is structured as follows. After the definitionthe main quantities of interest, an impulse approximat~IA ! convolution formula for the current quark GPDsterms of the constituent quark off-diagonal momentum dtributions and constituent quark GPDs is derived in the thsection. Then, the constituent quark GPDs are built infourth section, according to the ACMP philosophy and usthe double distribution~DD! representation@3,26,43# of theGPDs. In the fifth section, as an illustration, results obtainby using CQM wave functions of the IK model and the otained constituent quark GPDs will be shown. Conclusiowill be drawn in the last section.
II. QUARK MODEL CALCULATIONS OF GPD
We are interested in hard exclusive processes. The abstion of a high energy virtual photon by a quark in a hadrtarget is followed by the emission of a particle to be ladetected; finally, the interacting quark is reabsorbed binto the recoiling hadron. If the emitted and detected partis, for example, a real photon, the so called deeply virtCompton scattering~DVCS! process takes place@8,9,11#. Weadopt here the formalism used in Ref.@2#. Let us think of anucleon target, with initial~final! momentum and helicityP(P8) and s(s8), respectively. The GPDsHq(x,j,D2) andEq(x,j,D2) are defined through the expression
Fs8sq
~x,j,D2!
51
2E dl
2peilx^P8s8ucqS 2
ln
2 Dn”cqS ln
2 D uPs&
5Hq~x,j,D2!1
2U~P8,s8!n”nU~P,s!
1Eq~x,j,D2!1
2U~P8,s8!
ismnnmDn
2MU~P,s!, ~1!
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where D5P82P is the four-momentum transfer to thnucleon,cq is the quark field, andM is the nucleon mass. Iis convenient to work in a system of coordinates wherephoton four-momentumqm5(q0 ,qW ) and P5(P1P8)/2 arecollinear alongz. The j variable in the arguments of thGPDs is the so called ‘‘skewedness,’’ parametrizing tasymmetry of the process. It is defined by the relatj52n•D/2, wheren is a lightlike four-vector satisfying thecondition n• P51. As explained in@9,11#, the GPDs de-scribe the amplitude for finding a quark with momentufractionx1j ~in the infinite momentum frame! in a nucleonwith momentum (11j) P and replacing it back into thenucleon with a momentum transferD. In addition, when thequark longitudinal momentum fractionx of the averagenucleon momentumP is less than2j, the GPDs describeantiquarks; when it is larger thanj, they describe quarkswhen it is between2j and j, they describeqq pairs. Thefirst and second cases are commonly referred to asDokshitzer-Gribov-Lipatov-Altarelli-Parisi~DGLAP! regionand the third as the Efremov-Radyushkin-Brodsky-Lepa~ERBL! region@2#, following the pattern of evolution in thefactorization scale. One should keep in mind that, in additto the variablesx,j, and D2 explicitly shown, the GPDsdepend, like the standard PDFs, on the momentum scaleQ2
at which they are measured or calculated. For easy presetion, this latter dependence will be omitted in the rest of tpaper, unless specifically needed. The values ofj that arepossible for a given value ofD2 are
0<j<A2D2/A4M22D2. ~2!
The well known natural constraints ofHq(x,j,D2) are asfollows.
~i! The so-called ‘‘forward’’ or ‘‘diagonal’’ limit P85P,i.e., D25j50, where one recovers the usual PDFs
Hq~x,0,0!5q~x!. ~3!
~ii ! The integration overx, yielding the contribution of thequark of flavorq to the Dirac form factor~ff ! of the target:
E dxHq~x,j,D2!5F1q~D2!. ~4!
~iii ! The polynomiality property@2#, involving higher mo-ments of GPDs, according to which thex integrals ofxnHq
and ofxnEq are polynomials inj of ordern11.In @30#, the IA expression forHq(x,j,D2), suitable to
perform CQM calculations, was obtained.Now, it will be shown that the same basic formula can
derived as the NR reduction of the definition~1! of GPDs,analyzed initially in the noncovariant framework of lighcone quantization, involving partons on their mass shell.
Using the light-cone spinor definitions as given in Appe
dix B of @5#, and definingk15(k01k3)/A2,kW'5(k1 ,k2),for the light-cone helicity combinations8s5 1
212 511, one
obtains
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GENERALIZED PARTON DISTRIBUTIONS AND . . . PHYSICAL REVIEW D69, 094004 ~2004!
F11q ~x,j,D2!5A12j2Hq~x,j,D2!2
j2
A12j2Eq~x,j,D2!,
~5!
so that, forj2!1,
F11q ~x,j,D2!5Hq~x,j,D2!2j2S 1
2Hq~x,j,D2!
1Eq~x,j,D2! D1O~j4!, ~6!
i.e.,
F11q ~x,j,D2!5Hq~x,j,D2!1O~j2!. ~7!
The reader should be aware that the so called ‘‘Munsymmetry’’ for double distributions excludesO(j) contribu-tions to GPDs@44#, so that the accuracy of the above eqution is worse than it appears.
According to the latter equation, in order to obtain tGPDHq(x,j,D2) for j2!1 one has to evaluateF11
q , start-ing from its definition Eq.~1!. In the left-hand side~LHS! ofthe latter, using light-cone quantized quark fields, whoseation and annihilation operatorsb†(k) and b(k) obey thecommutation relation
$b~k8!,b†~k!%5~2p!32k1d~k812k1!d2~kW'2kW'8 !,~8!
and using properly normalized light-cone states
^P8uP&5~2p!32P1d~P812P1!d2~PW'2PW'8 !, ~9!
one obtains, forx.j @2#,
F11q ~x,j,D2!5
1
2P1VE d2k'
2Aux22j2u~2p!3
3^b†~k1D!b~k!&, ~10!
where
^b†~k8!b~k!&5(l
^P81ubl†@~x2j!P1,kW'8 #
3bl@~x1j!P1,kW'#uP1&, ~11!
and V is a volume factor. We are interested here in thex.j region, since we want to obtain only the quark contribtion to H, the only one that can be evaluated in a CQM wthree valence, pointlike quarks.
Equation~10! can be written
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51
2P1VE d2k'
2Aux22j2u~2p!3^b†~k1D!b~k!&
51
2P1VE d2k'dk1
2Aux22j2u~2p!3
3d„k12~x1j!P1…^b†~k1D!b~k!&
51
2P1VE d2k'dk1
2A~x1j!P1~x2j!P1~2p!3
3 P1d„k12~x1j!P1…^b†~k1D!b~k!&
51
2P1VE d2k'dk1
2Ak1k81~2p!3
3dS k1
P12~x1j!D ^b†~k1D!b~k!&. ~12!
In a NR framework, states and creation and annihilationerators have to be normalized according to
respectively. As a consequence, in order to perform areduction of Eq.~12!, one has to consider that@45#
uP&→A2P1uP&W , ~15!
b~k!→A2k1b~k1,kW'!, ~16!
so that in Eq.~12!, in terms of the new states and fields, ohas to perform the substitution
^b†~k1D!b~k!&5(l
^P81ubl†@~x2j!P1,kW'1DW '#
3bl@~x1j!P1,kW'#uP1&
→2A12j2P1A2k812k1
3(l
^PW 8ubl†~k11D1,kW'1DW '!
3bl~k1,kW'!uPW &, ~17!
where use has been made of the relation 2A12j2P1
5A2P1A2P81.
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Now, inserting Eq.~17! in Eq. ~12!, one gets
F11q ~x,j,D2!
51
VE d2k'dk1
2Ak1k81~2p!3dS k1
P12~x1j!DA2k812k1
3(l
^PW 8ubl†~k11D1,kW'1DW '!bl~k1,kW'!uPW &
1O~j2!
51
VE d2k'dk1
~2p!3 dS k1
P12~x1j!D
3(l
^PW 8ubl†~k11D1,kW'1DW '!bl~k1,kW'!uPW &
1O~j2!. ~18!
Now, since the constituent quarks with massm are taken to
be on shell, so thatk05AkW21m2, one has
d2k'dk1→S k1
k0DdkW , ~19!
so that, from Eq.~18! and Eq.~7!, one gets
Hq~x,j,D2!5E dkWdS k1
P12~x1j!D F 1
~2p!3V
k1
k0
3(l
^PW 8ubl†(k11D1,kW'1DW '!bl~k1,kW'!uPW &G
1O~j2! ~20!
5E dkdW S k1
P12~x1j!D
3Fnq~kW ,kW1DW !1OS kW2
m2 ,~kW1DW !2
m2 D G1O~j2!. ~21!
In the last step, the definition of the~nondiagonal! momen-tum distributionnq(kW ,kW1DW ), has been used, together withe fact that a NR momentum distribution describes the prability of finding a constituent of momentumkW in a givensystem up to terms of orderkW2/m2 @46#.
Summarizing, we find that, in a NR CQM, the GPHq(x,j,D2) can be calculated, forj2!1, kW2!m2, and (kW
1DW )2!m2 ~which means, in turn,DW 2!m2), through thefollowing expression:
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Hq~x,j,D2!5E dkWdS k1
P12~x1j!D nq~kW ,kW1DW !
1OS j2,kW2
m2 ,DW 2
m2D . ~22!
The above equation, corresponding to Eq.~8! in @30#, per-mits the calculation ofHq(x,j,D2) in any CQM, and it natu-rally verifies some of the properties of GPDs. In fact, tunpolarized quark densityq(x), as obtained by analyzing, inthe IA, DIS off the nucleon~see, e.g.,@45#!, assuming thatthe interacting quark is on shell, is recovered in the forwalimit, whereD25j50:
q~x!5Hq~x,0,0!5E dkWnq~kW !dS x2k1
P1D , ~23!
so that the constraint Eq.~3! is satisfied. In the above equation, nq(kW ) is the momentum distribution of the quarks in thnucleon:
nq~kW !5E eik•W (rW2rW8)rq~rW,rW8!drWdrW8. ~24!
In addition, integrating Eq.~22! over x, one obtains
E dxHq~x,j,D2!5E drWeiDW •rWrq~rW !,
where rq(rW)5 limrW8→rWrq(rW8,rW) is the contribution of thequarkq to the charge density. The RHS of the above eqtion gives the IA definition of the charge FF:
E drWeiDW •rWrq~rW !5Fq~D2!, ~25!
so that, recalling thatFq(D2) coincides with the nonrelativ-istic limit of the Dirac FFF1
q(D2), Eq. ~4! is verified.The polynomiality condition is also formally satisfied b
the GPD defined in Eq.~22!, although the present accuracof the model, explicitly written in the latter equation, donot allow one to really check polynomiality, due to the aready mentioned effects of the Munich symmetry@44#.
The definition of Hq(x,j,D2) in terms of CQM wavefunctions can be generalized to other GPDs, and the relaof the latter quantities with other form factors~for example,the magnetic one! and other PDFs~for example, the polar-ized quark density! can be recovered. Therefore the proposscheme allows one to calculate the GPDs by usingCQM.
With respect to Eq.~22!, a few caveats are necessary.~i! One should keep in mind that Eq.~22! is a NR result,
holding forj2!1, under the conditionskW2!m2, DW 2!m2. Ifone wants to treat more general processes, the NR lshould be relaxed by taking into account relativistic corretions. In this way, at the same time, an expression to evaluEq(x,j,D2) could be obtained. Since our main aim here isdescribe our approach, rather than to obtain realistic e
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GENERALIZED PARTON DISTRIBUTIONS AND . . . PHYSICAL REVIEW D69, 094004 ~2004!
mates, we postpone to a later publication the discussionrelativistic model, which will permit us to study the fullD2
andj ranges, together with the GPDEq(x,j,D2).~ii ! The constituent quarks are assumed to be pointlik~iii ! If use is made of a CQM containing only constitue
quarks~and also antiquarks in the case of mesons!, only thequark~and antiquark! contribution to the GPDs can be evalated, i.e., only the regionx>j ~and alsox<2j for mesons!can be explored. In order to introduce the study of the ERregion (2j<x<j), so that observables like cross sectiospin asymmetries, and so on can be calculated, the modeto be enriched.
~iv! In actual calculations, the evaluation of Eq.~22! re-quires the choice of a reference frame. In the following,Breit frame will be chosen, where one hasD252DW 2 and, in
the NR limit we are studying one findsA2P1→M . It hap-pens therefore that, in the argument of thed function in Eq.~22!, thex variable for the valence quarks is not defined innatural support, i.e., it can be larger than 1 and smaller tj. Several prescriptions have been proposed in the paovercome such a difficulty in the standard PDFs c@33,34#. Although the support violation is small for the caculations that will be shown here, it has to be reported adrawback of the approach.
The issue~iii ! will be discussed in the next sections, brelaxing the condition~ii ! and allowing for a finite size andcomposite structure of the constituent quark.
III. GPDs IN A CONSTITUENT QUARK SCENARIO
The procedure described in the previous section, wapplied in the standard forward case, has been proven table to reproduce the gross features of PDFs@34#. In order toachieve a better agreement with data, the approach hasimproved.
In a series of previous papers, it has been shownunpolarized@37# and polarized@38# DIS data are consistenwith a low energy scenario dominated by composite constents of the nucleon. This scenario was obtained usinsimple picture of the constituent quark as a complex sysof pointlike partons, and thus constructing the forward pardistributions by a convolution between constituent quark mmentum distributions and constituent quark structure futions. The latter quantities were obtained by using updaphenomenological information in a scenario first suggesby Altarelli, Cabibbo, Maiani, and Petronzio already in t1970s@39#.
Following the same idea, in this section a model for treaction mechanism of an off-forward process, suchDVCS, where GPDs could be measured, will be proposAs a result, a convolution formula giving the protonHq GPDin terms of a constituent quark off-forward momentum dtribution, Hq0
, and of a GPD of the constituent quarkq0
itself, Hq0q , will be derived.It is assumed that the hard scattering with the virtual p
ton takes place on a parton of a spin 1/2 target, madecomplex constituents. This can be the case of a spinnucleus, such as3He, or of the proton, if this is assumed
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be made of composite constituent quarks. The latter situais the one we are interested in.
The scenario we are thinking of is depicted in Fig. 1 fthe special case of DVCS, in the handbag approximatOne parton~currentquark! with momentumk, belonging to agiven constituent of momentump, interacts with the probeand is afterwards reabsorbed, with momentumk1D, by thesame constituent, without further rescattering with the recing system of momentumPR . We suggest here an analysof the process which is quite similar to the usual IA approato DIS off nuclei @45–47#.
In the class of frames chosen in Sec. II, and in additionthe kinematical variablesx andj already defined, one needa few more to describe the process. In particular,x8 andj8,for the ‘‘internal’’ target, i.e., the constituent quark, havebe introduced. The latter quantities can be obtained by deing the ‘‘1’’ components of the momentumk and k1D ofthe struck parton before and after the interaction, withspect toP1 and p15 1
2 (p1p8)1:
k15~x1j!P15~x81j8!p1, ~26!
~k1D!15~x2j!P15~x82j8! p1. ~27!
From the above expressions,j8 andx8 are immediately ob-tained as
j852D1
2p1, ~28!
x85j8
jx, ~29!
and, sincej52D1/(2P1), if z5p1/P1, one also has
j85j
z~11j!2j. ~30!
These expressions have already been found and used iIA analysis of DVCS off the deuteron@48# and, in general,off nuclei @49#.
In order to derive a convolution formula, a standard pcedure will be adopted@45–47#. In Eq. ~20!, two complete
FIG. 1. The handbag contribution to the DVCS process inpresent approach.
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S. SCOPETTA AND V. VENTO PHYSICAL REVIEW D69, 094004 ~2004!
sets of states, corresponding to the interacting constituand to the recoiling system, are properly inserted in the land right-hand sides of the quark operator:
Hq~x,j,D2!
5^P81u (PW R8 ,SR8 ,pW 8,s8
$uPR8SR8 &up8s8&%$^PR8SR8 u^p8s8u%
3E dkW
~2p!3
k1
k0
dS k1
P12~x1j!D
31
V(l
bl†~kW1DW !bl~kW !
3 (PW R ,SR ,pW ,s
$uPRSR&ups&%$^PRSRu^psu%uP1&,
and since, using the IA,
$^PRSRu^psu%uPS&
5^PRSR ,psuPS&~2p!3d3~PW 2PW R2pW !dS,SRs ,
a convolution formula, valid for any GPDHq of the spin 1/2complex target~the proton in the present case! in terms ofthe GPDHq0q of spin 1/2 structured constituents~the con-
stituent quark, in the present case! q0 is readily obtained:
Hq~x,j,D2!5(q0
E dEE dpW Pq0~pW ,pW 1DW ,E!
3j8
jHq0q~x8,j8,D2!. ~31!
In the above equation,E is the excitation energy of the recoiling system andPq0
(pW ,pW 1DW ,E) the one-body off-
diagonal spectral function for the constituent quarkq0 in theproton:
Pq0~pW ,pW 1DW ,E!5 (
SW R ,s^PW 8Su~PW 2pW !SR ,~pW 1DW !s&
3^~PW 2pW !SR ,pW suPS&d~E2ER!.
If the E dependence ofj8, i.e., theE dependence ofz @cf.Eq. ~30!# is disregarded in Eq.~31!, so that the one-bodyoff-diagonal momentum distribution
nq0~pW ,pW 1DW !5E dEPq0
~pW ,pW 1DW ,E! ~32!
is recovered, Eq.~31! can be written in the form
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q0
E dpW nq0~pW ,pW 1DW !
j8
jHq0q~x8,j8,D2!
5(q0
Ex
1 dz
z E dpW nq0~pW ,pW 1DW !dS z2
j
j8D
3Hq0qS x
z,j
z,D2D . ~33!
Taking into account that
z2j
j85z2@ z~11j!2j#
5z1j2p1
P1~11j!5z1j2
p1
P1, ~34!
Eq. ~33! can also be written in the form
Hq~x,j,D2!5(q0
Ex
1 dz
zHq0
~z,j,D2!Hq0qS x
z,j
z,D2D ,
~35!
where
Hq0~z,j,D2!5E dpW nq0
~pW ,pW 1DW !dS z1j2p1
P1D ~36!
is to be evaluated in a given CQM, according to Eq.~22!, forq05u0 or d0, while Hq0q(x/z,j/z,D2) is the constituentquark GPD, which is still to be discussed and will be moeled in the next section. One should notice that the forwlimit of Eq. ~35! gives the expression which is usually founfor the parton distributionq(x), in the IA analysis of unpo-larized DIS off nuclei@45–47#. In fact, if in Eq. ~35! theindex q0, labeling a constituent quark, is replaced by tindex N, labeling a nucleon in the nucleus, in the forwalimit one finds the well known result:
q~x!5Hq~x,0,0!5(N
Ex
1 dz
zf N~z!qNS x
zD , ~37!
where
f N~z!5HN~z,0,0!5E dpW nN~pW !dS z2p1
P1D ~38!
is the light-cone momentum distribution of the nucleonN inthe nucleus andqN(x)5HNq(x,0,0) is the distribution of thequark of flavorq in the nucleonN.
IV. A MODEL FOR THE GPDS OF THECONSTITUENT QUARK
The crucial problem now is the definition oHq0q(x/z,j/z,D2), the constituent quark GPD, appearingEq. ~35!.
As usual, we can start modeling this quantity, thinkifirst of all of its forward limit, where the constituent quar
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parton distributions have to be recovered. As we said inprevious section, in a series of papers@37,38# a simple pic-ture of the constituent quark as a complex system of polike partons has been proposed, retaking a scenario suggby Altarelli, Cabibbo, Maiani, and Petronzio@39#.
Let us recall the main features of that idea.The constituent quarks are themselves composite ob
whose structure functions are described by a set of functfq0q(x) that specify the number of pointlike partons of typ
q which are present in the constituent of typeq0, with frac-tion x of its total momentum. We will hereafter call thesfunctions, generically, the structure functions of the constent quark.
The functions describing the nucleon parton distributioare expressed in terms of the independentfq0q(x) and of the
constituent density distributions (q05u0 ,d0) as
q~x,Q2!5(q0
Ex
1 dz
zq0~z,Q2!fq0qS x
z,Q2D , ~39!
where q labels the various partons, i.e., valence qua(uv ,dv), sea quarks (us ,ds ,s), sea antiquarks (u,d,s), andgluonsg.
The different types and functional forms of the structufunctions of the constituent quarks are derived from thvery natural assumptions@39#: ~i! The pointlike partons areQCD degrees of freedom, i.e., quarks, antiquarks, andons; ~ii ! Regge behavior forx→0 and duality ideas;~iii !invariance under charge conjugation and isospin.
These considerations define in the case of the valequarks the following structure function:
fqqv~x,Q2!5
G~A1 12 !
G~ 12 !G~A!
~12x!A21
Ax. ~40!
For the sea quarks the corresponding structure functioncomes
fqqs~x,Q2!5
C
x~12x!D21, ~41!
and, in the case of the gluons, it is taken as
fqg~x,Q2!5G
x~12x!B21. ~42!
The last assumption of the approach relates to the scawhich the constituent quark structure is defined. We chofor it the so called hadronic scalem0
2 @34,50#. This hypothesisfixes all the parameters of the approach@Eqs. ~40! through~42!#. The constantsA, B, G and the ratioC/D are deter-mined by the amount of momentum carried by the differpartons, corresponding to a hadronic scale ofm0
2
50.34 GeV2, according to the parametrization of@50#. C ~orD) is fixed according to the value ofF2 at x50.01@39#, andits value is chosen again according to@50#. We stress that althese inputs are forced only by the updated phenomenol
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through the second moments of PDFs. The values ofparameters obtained are listed in@37#.
We note here that the unpolarized structure functionF2 israther insensitive to the change of the sea (C, D) and gluon(B, G) parameters.
The other ingredients appearing in Eq.~39!, i.e., the den-sity distributions for each constituent quark, are definedcording to Eq.~22!.
Now we have to generalize this scenario to describeforward phenomena. Of course, the forward limit of oGPD formula, Eq.~35!, has to be given by Eq.~39!. Bytaking the forward limit of Eq.~35!, one obtains
Hq~x,0,0!5(q0
Ex
1 dz
zHq0
~z,0,0!Hq0qS x
z,0,0D
5(q0
Ex
1 dz
zq0~z!Hq0qS x
z,0,0D , ~43!
so that, in order for the latter to coincide with Eq.~39!, onemust haveHq0q(x,0,0)[fq0q(x).
In this way, through the ACMP prescription, the forwalimit of the unknown constituent quark GPDHq0q(x/z,j/z,D2) can be fixed.
Now the off-forward behavior of the constituent quaGPDs has to be modeled.
This can be done in a natural way by using the ‘‘a doubledistribution’’ ~DD! language proposed by Radyushkin@3,43#.The DDsF( x,a,D2) are a representation of GPDs, whicautomatically guarantees the polynomiality property. GPcan be obtained from DDs after a proper integration. In cstructing models, the DDs can represent a more approplanguage with respect to GPDs. In fact, the hybrid charaof GPDs, which are something in between parton densiq(x) and distribution amplitudesf(a), is naturally empha-sized when the latter are obtained from DDs. The DDs dodepend on the skewedness parameterj; rather, they describehow the total, P, and transfer, D, momenta are shared between the interacting and final partons, by means of the vablesx and a, respectively. As can be argued from Fig.where the DD representation of GPDs is illustrated schemcally, and as explained in@3,43#, parton densities are recovered in the forward,D50 limit, while meson distributionamplitudes are obtained in theP50 limit of DDs. In somecases, such a transparent physical interpretation, togewith the symmetry properties which are typical of distribtion amplitudes (a→2a symmetry!, allows a direct model-ing, already developed in@26#.
The relation between any GPDH, defined in the manneof Ji, for example, the one we need, i.e.,Hq0q for the con-
stituent quark target, is related to thea DDs, which we call
Fq0q( x,a,D2) for the constituent quark, in the followingway @3,43#:
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FIG. 2. Illustration of the relation betweenaDDs ~a! and GPDs~b!.
o
-
-
en
flyin
rkth
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e-
cu-
PD
ated
ibedbyk
Hq0q~x,j,D2!5E21
1
dxE211uxu
12uxud~ x1ja2x!
3Fq0q~ x,a,D2!da. ~44!
With some care, the expression above can be integratedx and the result is explicitly given in@3#. The DDs satisfy therelation
Fq0q~ x,a,D2!5Fq0q~ x,2a,D2! ~45!
and the polynomiality condition@2#.In @43#, a factorized ansatz is suggested for the DD’s:
Fq0q~ x,a,D2!5hq~ x,a,D2!Fq0q~ x!Fq0~D2!, ~46!
with thea dependent termhq( x,a,D2), which has the char-acter of a mesonic amplitude, satisfying the relation
E211uxu
12uxuhq~ x,a,D2!da51. ~47!
In addition, in Eq.~46! Fq0q( x) represents the forward den
sity and, eventually,Fq0(D2) the constituent quark form fac
tor.One immediately realizes that the GPD of the constitu
quark Eq. ~44! with the factorized form Eq.~46! and thenormalization Eq.~47!, satisfies the crucial constraints oGPDs, i.e., the forward limit, the first moment, and the ponomiality condition, the latter being automatically verifiedthe DD description.
In the following, we will assume for the constituent quaGPD the above factorized form, so that we need to modelthree functions appearing in Eq.~46!, according to the de-scription of the reaction mechanism we have in mind.
For the amplitudehq , use will be made of one of thesimple normalized forms suggested in@43#, on the basis ofthe symmetry properties of DD’s:
hq(1)~ x,a!5
3
4
~12 x!22a2
~12 x!3. ~48!
In addition, since we will identify quarks forx>j/2, pairsfor x<uj/2u, antiquarks forx<2j/2, and, since in our approach the forward densitiesFq0q( x) have to be given by the
standardF functions of the ACMP approach, Eqs.~40!–~42!, one has, for the DD of flavorq of the constituent quark
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Fq0q~ x,a,D2!
5H ~hq~ x,a!Fq0qv~ x!1hq~ x,a!Fq0qs
~ x!!Fq0~D2!
for x>0,
2hq~2 x,a!Fq0qs~2 x!Fq0
~D2! for x,0.
~49!
The above definition, due to Eq.~47!, when integrated overa gives the correct limits@3#
E211 x
12 xFq0q~ x,a,D250!u x.0da5Fq0q~ x! ~50!
and
E211uxu
12uxuFq0q~ x,a,D250!u x,0da52Fq0q~2 x!. ~51!
Eventually, as a FF we will take a monopole form corrsponding to a constituent quark sizer Q.0.3 fm:
Fq0~D2!5
1
12D2r Q2 /6
, ~52!
a scenario strongly supported by the analysis of@40#.By using such a FF and Eq.~48!, together with the stan-
dard ACMPF ’s Eqs.~40! and~41! in Eq. ~49!, and insertingthe obtainedFq0q( x,a,D2) into Eq. ~44!, the constituentquark GPD in the ACMP scenario can eventually be callated.
V. RESULTS AND DISCUSSION
In this section we present the results obtained for the GHq of the proton, forj2!1 andDW 2!m2, according to theapproach described so far. The main equation to be evaluis Eq. ~35!, written again here for the sake of clarity:
Hq~x,j,D2!5(q0
Ex
1 dz
zHq0
~z,j,D2!Hq0qS x
z,j
z,D2D .
In the above equation, the quantityHq0q , the constituentquark GPD, is modeled according to the arguments descrin the previous section. This means that it is obtainedevaluating Eq.~44!, where the DD of the constituent quar
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Fq0q( x,a,D2) is given by Eq. ~49!, calculated in turn
through the FF Eq.~52! and the functionhq Eq. ~48!, to-gether with the standard ACMPF ’s Eqs.~40! and ~41!.
The other ingredient in Eq.~35!, Hq0, has been evaluate
according to Eq.~36!:
Hq0~z,j,D2!5E dpW nq0
~pW ,pW 1DW !dS z1j2p1
P1D .
The calculation has been performed in the Breit frame, wh
one has, in the NR limit studied,A2P1→M . The off-diagonal momentum distribution appearing in the formabove,nq0
(pW ,pW 1DW ), defined in Eq.~32!, has been evaluatewithin the Isgur and Karl model@35#. The calculation is de-scribed in@30# and the main results are listed again herethe reader’s convenience.
In the IK CQM @35#, including contributions up to the2\v shell, the proton state is given by the following admiture of states:
where the spectroscopic notationu2S11XJ& t , with t5A,M ,S being the symmetry type, has been used. Theefficients were determined by spectroscopic properto be @51# aS50.931, aS8520.274, aM520.233, aD520.067.
The results for the GPDH(x,j,D2), neglecting in Eq.~53! the smallD-wave contribution, have been found to b@30#
t
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-s
Hu~x,j,D2!53M
a3S 3
2pD 3/2
eD2/3a2E dkxE dky
3 f 0~kx ,ky ,x,j,D2!
3@ f s~kx ,ky ,x,j,D2!1 f ~kx ,ky ,x,j,D2!#,
~54!
Hd~x,j,D2!53M
a3S 3
2pD 3/2
eD2/3a2E dkxE dky
3 f 0~kx ,ky ,x,j,D2!
3F1
2f s~kx ,ky ,x,j,D2!2 f ~kx ,ky ,x,j,D2!G ,
~55!
for the flavorsu andd, respectively, with
f 0~kx ,ky ,x,j,D2!5k0
k01 kz
f a~Dx ,kx! f a~Dy ,ky! f a~Dz ,kz!,
~56!
f a~D i ,ki !5e(21/a2)(3ki2/21kiD i ), ~57!
kz5M2~x1j!22~m21kx
21ky2!
2M ~x1j!, ~58!
f s~kx ,ky ,x,j,D2!52
3as
21as82 F5
62
k2
a2 11
2
k4
a412
3a2 S 2D2
31DW •kW D S k2
a2 21D G1aM
2 F 5
122
1
2
k2
a211
4
k4
a4 12
9
k
a2A9
4k22D213DW •kW1
1
3a2 S 2D2
31DW •kW D S k2
a2 21D G1aSaS8
2
A3F S 12
k2
a2D 22
3a2 S 2D2
31DW •kW D G , ~59!
f ~kx ,ky ,x,j,D2!52aSaS8
2
A3F S 12
k2
a2D 22
3a2 S 2D2
31DW •kW D G
2aMaS8
1
A2F1
62
k2
a2 11
2
k4
a4 22
3a2 S 2D2
32DW •kW D 1
2k2
3a4 S 2D2
31DW •kW D G , ~60!
re-.
and k05Am21kx21ky
21 kz2, m.M /3 being the constituen
quark mass. Here we have used the notationk25kW2,kW5(kx ,ky ,kz).
The harmonic oscillator parametera of the IK model canbe chosen so that the experimental rms of the proton isproduced by the slope, atD250, of the charge form factor
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S. SCOPETTA AND V. VENTO PHYSICAL REVIEW D69, 094004 ~2004!
Such a choice is performed as follows. Integrating Eq.~35!over x, one gets
Fq~D2!5(q0
Fq0~D2!Fq0q~D2!, ~61!
where Fq(D2) is the contribution of thecurrent quark offlavor q to the proton FF;Fq0
(D2) is the contribution of the
pointlike constituent quark of flavorq0 to the proton FF;Fq0q(D2) is the contribution of the current quark of flavorq
to the FF of thecompositeconstituent quark of flavorq0.The latter is given by Eq.~52!, while in the IK model one
has@51#
Fq0~D2!5S 12
a
aD22
b
a2 D4D eD2/6a2, ~62!
with a.0.015 andb.0.0001. Imposing that the slope of thFF Eq. ~61! reproduces atD250 the experimental protonrms, a value ofa51.18 f 21 is obtained. Such a form factoreproduces well the data at the low values ofD2 that areaccessible in the present approach. For higher values ofD2,it would not be realistic@51#.
Results for theu-quarkH distribution, at the scale of themodelm0
2, are shown in Figs. 3–5. In Fig. 3, it is shown foD2520.1 GeV2 andj50.1. One should remember that thpresent approach does not allow us to estimate realisticthe region2D2>m2.0.1 GeV2, so that we are showinghere the result corresponding to the highest possibleD2
value. Accordingly, the maximum value of the skewednesthereforej.0.17 @cf. Eq. ~2!#, satisfying the requiremenj2!1. The dashed curve represents what is obtained inpure Isgur and Karl model, i.e., by evaluating Eq.~54!. Oneshould notice that such a result, obtained in a pure valeCQM, should vanish forx<j and forx>1. The small tailsthat are found in these forbidden regions representamount of support violation of the approach. In particulfor the shown values ofD2 andj, a violation of 2% is found.
FIG. 3. The GPDH for the flavoru, for D2520.1 GeV2 andj50.1, at the momentum scale of the model. Dashed curve: rein the pure Isgur and Karl model, Eq.~54!. The small tails that arefound in the forbidden regionsx<j andx>1 represent the amounof support violation of the approach. Full curve: the complete reof the present approach, Eq.~35!.
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In general, in the accessible region the violation is nelarger than a few percent. The full curve in Fig. 3 represethe complete result of the present approach, i.e., the evation of Eq.~35! following the steps and using the ingrediendescribed in this section and in the previous one. A relevcontribution is found to lie in the ERBL region, in agreemewith other estimates@4#. As already explained, knowledge othe GPDs in the ERBL region is a crucial prerequisite for tcalculation of all the cross sections and the observables msured in the processes where GPDs contribute. We nothat the ERBL region is accessed here, with respect toapproach of Ref.@30# which gives the dashed curve, thanto the constituent structure which has been introduced byACMP procedure.
In Fig. 4, special emphasis is devoted to showing thejdependence of the results. For the allowedj valuesHu(x,j,D2) evaluated using our main formula Eq.~35! isshown for four different values ofx. It is clearly seen thatsuch a dependence is strong in the ERBL region, while irather mild in the DGLAP region, in good agreement wiother estimates@4#. To allow for a complete view of theoutcome of our approach, in Fig. 5 thex andj dependencesare shown together in a three-dimensional plot.
The results shown so far are associated with the low sof the model, the hadronic scalem0
2, fixed to the value0.34 GeV2, as discussed in Sec. IV. As an illustration,
ult
lt
FIG. 4. For thej values that are allowed atD2520.1 GeV2,Hu(x,j,D2), evaluated using our main equation Eq.~35!, is shownfor four different values ofx, at the momentum scale of the modeFrom top to bottom, the dash-dotted line represents the GPDx50.05, the full line atx50.1, the dashed line atx50.2, and thelong-dashed line atx50.4.
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GENERALIZED PARTON DISTRIBUTIONS AND . . . PHYSICAL REVIEW D69, 094004 ~2004!
Figs. 6 and 7 the next to leading order QCD evolution ofnonsinglet~NS!, valenceu distribution, up to a scale ofQ2
510 GeV2, is shown. One should notice that any NS distbution is symmetric inx, due to its definition and in agreement with the conventions used:
HqNS~x,j,D2!5Hq~x,j,D2!2Hq~x,j,D2!
5Hq~x,j,D2!1Hq~2x,j,D2!.~63!
In evolving the model results, the approach of Ref.@28# hasbeen applied and a code kindly provided by A. Freundbeen used. The evolution clearly shows a strong enhament of the ERBL region.
FIG. 5. The x and j dependences ofHu(x,j,D2), for D2
520.1 GeV2, at the momentum scale of the model.
FIG. 6. The nonsingletH GPD for valenceu quarks atj50.1and D2520.1 GeV2, at the momentum scale of the model,m0
2
50.34 GeV2 ~dashed!, and after NLO QCD evolution up toQ2
510 GeV2 ~full !.
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We have therefore developed a scheme that providewith the GPDH in the full x range. This is obtained thanks tthe constituent quark structure, implemented by dressingthree quarks of a CQM, where initially only the DGLAregion of GPDs was accessible. This is an important deopment with respect to previous work, a prerequisite for aattempt to calculate cross sections and asymmetries olated processes. The next step of our studies will be indeeuse the obtained GPDs for the evaluation of cross sectthat have recently been measured@52,53# or will be mea-sured soon. The estimate of cross sections is presentlprogress, together with that of relativistic corrections, whwill permit us to enlarge the kinematical range~basically theD2 and the relatedj ranges! where our results can be applieto predict or to describe the data. For the time being, a coparison with data of our estimates is therefore not possib
Anyway, as already said, the present approach satisseveral theoretical constraints. First of all, the forward limEq. ~39!, provides us with a reasonable description of quadensities~see Ref.@37#, where the IK model together withthe ACMP mechanism has been applied1 to unpolarizedDIS!. Secondly, thex integral of our GPDH turns out to be,formally and numerically,j independent, satisfying thereforthe polynomiality condition for the first moment, and proviing us with a proton form factor Eq.~62! in good agreemenwith the data in the lowD2 region which is studied here. Thmain theoretical drawback is the support violation alrea
1In that work@37#, we showed that more sophisticated quark moels were producing an excellent description of the data@54#.
FIG. 7. The nonsingletH GPD for valenceu quarks atj50.1and D2520.1 GeV2, evolved at NLO from the momentum scaof the model,m0
250.34 GeV2, up toQ2510 GeV2.
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S. SCOPETTA AND V. VENTO PHYSICAL REVIEW D69, 094004 ~2004!
discussed, which in any case affects our findings, in thenematical region under investigation, by a few percentmost. Another theoretical constraint that is satisfied bypresent approach is the inequality
Hq~x,j,D2!<Aq~x1!q~x2!/~12j2!, ~64!
where x15(x1j)/(11j) and x25(x2j)/(12j), provenin @43#. As an illustration, the validity of the above inequaliis shown in Fig. 8 for the u flavor, and for D2
520.1 GeV2 andj50.1.Other theoretical constraints, such as the Ji sum rule@9#,
require the knowledge of the other unpolarized GPDE,which has not yet been calculated in the present schemEarises naturally in a relativistic framework, where we plancalculate it. In fact, a relativistic calculation will permit usovercome the support problem and, at the same time, tolarge the kinematical range of our predictions. Once a retivistic CQM is used to predict GPDs in the DGLAP regiothe structure of the constituent quark can be introducedaccess the ERBL region, so that the Ji sum rule and otheoretical constraints can be checked. Work is being carout in that direction.
VI. CONCLUSIONS
Quark models have been extremely useful for understaing many features of hadrons, even in the DIS regime.previous work, a thorough analysis of both polarized aunpolarized data has shown that constituent quarks cannconsidered elementary when studied with high eneprobes. The first feature found was the need for evoluti.e., the constituent quarks at the hadronic scale have tundressed by incorporating bremsstrahlung in order to rethe Bjorken regime, but this was not all. A second featuwe found, was that the constituent quarks should be endowith soft structure in order to approach the data. Thus,
FIG. 8. Full curve: the GPDHu(x,j,D2), for D2520.1 GeV2
andj50.1, at the momentum scale of the model; dashed curve
quantity Au(x1)u(x2)/(12j2), wherex15(x1j)/(11j) and x2
5(x2j)/(12j) ~see text!.
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constituent quarks appear, when under scrutiny by highergy probes, as complex systems, with a very differenthavior from the current quarks of the basic theory. Thefeatures, which we found in structure functions, have abeen recently discussed in form factors@40#.
The aim of the present work was to generalize the formism of composite constituent quarks to study the generaliparton distributions. We here develop a formalism thatpresses the hadronic GPDs in terms of constituent quGPDs by means of appropriate convolutions. In order toable to predict experimental results we have defined a mowhich incorporates phenomenological features of variousnematical regimes. The model is based on Radyushkin’storization ansatz; thus our constituent quark GPDs arefined in terms of the product of three functions:~i! theconstituent quark structure function, where we useACMP proposal@39#; ~ii ! Radyushkin’s double distribution@43#; ~iii ! the constituent quark form factor as suggestedRef. @40#. Once these GPDs are defined in this way, we hdeveloped a scheme to incorporate them into any nuclmodel by appropriate convolution. In order to show the tyof predictions to which our proposal leads, we have ushere, as an illustration, a naive model of hadron structunamely, the IK model@35#. However, in this latest step of ouscheme, any nonrelativistic~or relativized! model can beused to define the hadronic GPDs.
Looking at our results, we found that the present schetransforms a hadronic model, in whose original descriptonly valence quarks appear, into one containing all kindspartons~i.e., quarks, antiquarks, and gluons!. Moreover, thestarting model produces no structure in the ERBL regiwhile, after the structure of the constituent quark has bincorporated, it does. The completeness of thex range for theallowedD2 andj of the present description is a prerequisfor the calculation of cross sections and other observablea wide kinematical range. In this respect we recall thatfindings hold forD2!m2 and j2!1. Nevertheless, relativistic corrections, which permit one to access a wider kinmatical region, could be included in the approach. Our ahere has been mainly the illustration of our scheme, andinclusion of relativistic corrections, together with the usemore sophisticated models, has been postponed and wishown elsewhere.
We reminded the reader of the calculation for the diagostructure functions and form factors to see how in thecases, where experimental data are available, our schleads, even with a naive quark model, to a reasonablescription of the data. Thereafter, we proceeded to calcuthe GPDs of physical interest to guide the preparationanalysis of future experiments.
This work is the continuation of an effort to constructscheme that describes the properties of hadrons in diffekinematical and dynamical scenarios. Our descriptionnever be a substitute for quantum chromodynamics, but,fore a solution of it can be found, it may serve to guiexperimenters to physical processes where the theory mshow interesting features, worthy of a more fundamentalfort.
e
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GENERALIZED PARTON DISTRIBUTIONS AND . . . PHYSICAL REVIEW D69, 094004 ~2004!
The approach presented here can be extended to lighclei and fragmentation functions, and work is being carrout also in these directions.
ACKNOWLEDGMENTS
We acknowledge useful correspondence with A. Radyukin and helpful discussions with G. Salme`. S.S. thanks theDepartment of Theoretical Physics of the Valencia Univsity, where part of this work was done, for warm hospital
.ar
s.
o,
.
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h-
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and financial support. V.V. thanks the Department of Physof the University of Perugia for hospitality and support, aL. Kaptari for his help and interesting discussions. Sthanks the Department of Energy’s Institute for NucleTheory of the University of Washington, Seattle, WA, for ihospitality during the program ‘‘GPDs and Hard ExclusiProcesses,’’ and the Department of Energy for partial suppduring the completion of this work. This work was supportin part by GV-GRUPOS03/094 and MCYT-FIS2004-0561C02-01 and by MIUR through the funds COFIN03.
.
l.-
J.
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