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MKPH-T-93-04, IU/NTC 92-20

arXiv:hep-ph/9303306v1 25 Mar 1993MKPH-T-93-04, IU/NTC 92-20March 24, 1993Nuclear Shadowing in a Parton Recombination ModelS. Kumano ∗Institut f¨ur Kernphysik, Universit¨at Mainz6500 Mainz, GermanyandNuclear Theory Center, Indiana UniversityBloomington, Indiana 47408, U.S.A.ABSTRACTDeep inelastic structure functions F A2 (x) are investigated in a Q2 rescaling model withparton recombination effects.

We find that the model can explain experimentally measuredF A2 (x) structure functions reasonably well in the wide Bjorken−x range (0.005 < x < 0.8).In the very small x region (x < 0.02), recombination results are very sensitive to inputsea-quark and gluon distributions. * address after April 1, 1993: Department of Physics, Saga University, Saga 840, Japan.submitted to Phys.

Rev. CPACS numbers: 13.60.Hb, 25.30.-c

1. IntroductionSeveral years have passed since the discovery of nuclear modifications in structurefunctions F2(x) by the European Muon Collaboration (“old” EMC effect) [1].

In spiteof initial expectation of an explicit quark signature in nuclear physics, it is still not clearwhether the effect originates in nucleon substructures or just in nuclear physics. There areseveral proposed models for explaining the EMC effect.

Some models tried to interpret itby explicit quark effects, e.g. Q2 rescaling models [2] and six-quark bag models.

Othersinvestigated models based on conventional nuclear physics, such as nuclear binding andpion excess. For details of these models, we suggest that interested readers look at summarypapers in Ref.

3. Although there are still experimental activities [4] for investigating thesedifferent models, it is rather difficult to discriminate among these models.Most of these investigations discuss a “global” EMC effect in the sense that the effectis averaged over all constituents in a nucleus.However, it is interesting to investigatepossible semi-inclusive or semi-exclusive processes for finding a “local” EMC effect [5] andpossible relations between a local gluonic EMC effect and the J/ψ suppression [6].Considering the fact that the average nucleon separation in nuclei is 2.2 fm and thenucleon diameter is 1.8 fm, we expect that nucleons in nuclei could overlap strongly.

Ifa multiquark cluster is formed in a nucleus due to this overlap, the confinement radiusfor such a quark should be different from the one in a free nucleon.Using this kindof simple picture in a small Q2 region and the Q2-evolution equation to compare withexperimental data at large Q2, we obtain a simple prescription for the nuclear structurefunction F A2 (x, Q2) [2]. It is related to the nucleon structure function by a simple Q2rescaling, F A2 (x, Q2) = F N2 (x, ξAQ2), where ξA is called as rescaling parameter.

Althoughthe simple size change could be too simple to explain many details of nuclear physics [7],it is a useful effective model in explaining deep inelastic data in the medium x region.The New Muon Collaboration (NMC) [8] and the Fermilab E665 collaboration [9]recently measured accurately the structure functions F2(x) at very small x. These dataas well as EMC data [10] provide an opportunity for investigating physics details in theshadowing region (x < 0.1).

The shadowing means that central constituents are shieldeddue to surface constituents, hence the cross section behaves like Aα (α < 1). There are2

different ideas for explaining the shadowing phenomena: vector meson dominance [11],parton recombination [12−17], q¯q fluctuations of the virtual γ [18], pomeron dominance[19] (with Pauli blocking [20]), and hybrid models [21].At first, there are models based on the traditional idea, the vector meson dominancemodel [11]. A virtual photon transforms into vector meson states (ρ, ω, φ), which theninteracts with a target nucleus.

The propagation length of the hadronic (H) fluctuationsis given by λ ≈1/|EH −Eγ| = 2ν/(M 2H + Q2) ≈0.2/x fm. For x < 0.1, the propagationlength (>2 fm) exceeds the average nucleon separation (2.2 fm) in nuclei and the shadowingtakes place due to multiple scatterings.

For example, the vector meson interacts elasticallywith a surface nucleon and then interacts inelastically with a central nucleon. Because thisamplitude is opposite in phase to a one-step amplitude for an inelastic interaction withthe central nucleon, the nucleon sees a reduced hadronic flux (namely the shadowing).The parton recombination model [12−17] has been investigated as a mechanism forexplaining the shadowing within the framework of a quark-parton model.

In an infinitemomentum frame, the average longitudinal nucleon separation in a Lorentz contractednucleus is L = (2.2 fm)MA/PA = (2.2 fm)mN/pN, and the longitudinal localizationsize of a parton with momentum xpN is ∆L = 1/(xpN). If the parton dimension (∆L)exceeds the average nucleon separation (L), partons from different nucleons could interact(we call this parton recombination or parton fusion) significantly and the shadowing couldoccur due to processes in Figs.

1b, 1c, 1e, and 1f. Partons with momentum fraction x arelost in these processes, so that their contributions are negative ∆F2(x) < 0 (shadowing).However, ∆F2(x) depends much on input sea-quark and gluon distributions as we find insections 3 and 4.

A relevant x region for the shadowing is obtained by using L < ∆L andwe find x < 0.1. Even though the recombinations could produce shadowing type effects inthe small x region, they are very small compared with experimental shadowings if they arecalculated at Q2=5 GeV2.

On the other hand, modifications of F2(x) due to recombinationeffects on gluon distributions are large [17]. Details of numerical results are discussed insection 3.Because the recombination results by Close, Qiu, and Roberts [17] are not comparedin detail with the recent experimental data, we investigate whether the model can explain3

the EMC, NMC, and E665 shadowing data in this investigation. Furthermore, we studywhether the Q2 rescaling model combined with parton recombination effects [14,15] canexplain the F2(x) structure functions in the whole Bjorken−x range [16].Earlier comparisons with experimental data have been made in similar parton pictures[15].

Let us clarify differences from and advantages over the previous investigations. (1)Covolan, Predazzi, and others [15] used the Qiu’s parametrization [13] for the shadowingand combined its results with the rescaling.We note that the Q2 evolution is wellinvestigated in Ref.

13; however, the x dependence is rather assumed. Therefore, we cannothave a dynamically consistent picture by using the shadowing-parametrization with therescaling.

The x dependence of the shadowing was later investigated in Ref. 17.

Using thisshadowing picture combined with the rescaling, we could possibly obtain a dynamically-consistent parton model in the wide x-range. This is the purpose of our investigation.

(2) We study Q2 evolution effects in the combined model. (3) We investigate effects ofinput sea-quark and gluon distributions on the shadowing.

(4) Improved experimentaldata became available by the NMC and E665.2. Parton RecombinationsIf a nucleon is in a nucleus, parton distributions are modified because of the existenceof neighboring nucleons.

Partons from different nucleons could interact with each other,and the interactions become important, especially in the shadowing region.Processescontribute to modifications in a quark distribution q(x) are shown in Fig. 1.

For example,Fig. 1a indicates that a quark from the nucleon 1 fuses with a gluon from the nucleon 2and produces a quark with momentum x.

Because the process creates a quark with themomentum x, this is a positive contribution to the quark distribution at x. There are fiveother contributions as shown in Fig.

1. In general, a modification of a parton distributionp3(x3), due to the process of producing the parton p3 with the momentum x3 by a fusionof partons p1 and p2, is given by [17]∆p3(x3) = KZdx1dx2 p1(x1) p2(x2) Γp1p2→p3(x1, x2, x3 = x1 + x2) δ(x, x1, x2) , (1)where K is given by K = 9A1/3αs/(2R20Q2) [22].

Nuclear radius R is R = R0A1/3 with4

R0=1.1 fm [23] and the strong-interaction coupling constant is αs(Q2) = 4π/[β0ln(Q2/Λ2)]with β0 = 11 −2Nf/3, Nf=3.The momentum conserving δ function is given byδ(x −x1 −x2) for Figs. 1a and 1d and δ(x −x1) for Figs.

1b, 1c, 1e, and 1f. The partonfusion function Γ(x1, x2, x3) is a probability for producing a parton p3 with momentumx3 by a fusion of partons p1 and p2 with momenta x1 and x2 respectively.

Four possibleparton fusion processes are shown in Fig. 2.

It is related to a splitting function Pp1p3(z)in the Altarelli-Parisi equation [24] byΓp1p2→p3(x1, x2, x3) =x1x2x23Pp1←p3(x1x3) Cp1p2→p3. (2)The splitting function Pp1p3(z) is the probability of finding the parton p1 with fraction zof the parent momentum in the parton p3.

Cp1p2→p3 is the ratio of color factors in pro-cesses p1p2 →p3 and p3 →p1p2. For example, CqG→q =−X(l,a),k(takl)∗takl/−X(k),l,a(talk)∗talk =[CF /(N2C −1)]/CF, where averages are taken over initial color indices (k in the denomina-tor, l and a in the numerator).

talk is given by the SU(3)c Gell-Mann matrix by talk = λalk/2,and CF is CF = (N2C −1)/(2NC) [25]. In this way, the color factors are calculated as:CqG→q =1N2C −1 = 18,Cq¯q→G =CFNCTF= 89,CGq→q =TFNCCF= 18,CGG→G =1N2C −1 = 18,(3)where TF = 1/2 and NC = 3.

Using the above results and splitting functions in Ref. 24,we obtain the parton fusion functions:ΓqG→q′(x1, x2, x3) =16 z (1 + z2),Γq¯q→G(x1, x2, x3) =49 z (1 −z) [z2 + (1 −z)2],ΓGq→q′(x1, x2, x3) =16 (1 −z) [1 + (1 −z)2],(4)ΓG1G2→G3(x1, x2, x3) =34 z (1 −z) [z1 −z + 1 −zz+ z(1 −z)],where x3 = x1 + x2 and z ≡x1/x3.5

Using these fusion functions, we calculate modifications of quark distributions in nucleifor processes shown in Fig. 1.

Explicit expressions for the modifications ∆qi(x) are shownin Appendix (Eq. (A1)).

From Eq. (A1) and a similar expression for ∆¯qi(x), we obtainrecombination contributions to the structure function F2(x) in Eqs.

(A2.1)−(A2.4). Directrecombination effects on the EMC effect are given by the ratioRA(recombination) = 1 + ∆F A2 (x, Q2)F D2 (x, Q2),(5)where ∆F A2 (x, Q2) is calculated by using Eqs.

(A2.1)−(A2.4) with input parton distribu-tions at Q2. The most important physics for the shadowing in the recombination model isthe effect due to modifications in gluon distributions.

Recombination effects on the gluondistribution are given in Eqs. (A3.1)−(A3.4).

Calculating these equations, we find thatthe recombination mechanism produces large shadowing effects on the gluon distribution[17].3. Gluon-shadowing effects on FA2 (x)3.1 An approximate way to take into account the gluon shadowingGluons and quarks are coupled to each other and their distributions are related bythe Altarelli-Parisi equation.

Because of the large modifications in G(x < 0.1) due tothe recombinations, we expect that sea-quark distributions are also affected by the gluonmodifications. In the very small x region, the Altarelli-Parisi equation is dominated bygluon dynamics, and it can be solved analytically.

As a result, there is a relationshipbetween a sea quark and gluon distributions [26]: xqseai(x) = −x12∂∂x[xG(x)], where i=u,d, or s. Using this relation, we obtain a contribution to F2(x) from the modification in thegluon distribution:δF2(x, Q2) = −θ(x0 −x) 43x12∂∂x[x∆G(x, Q2)],(6)where x∆G(x, Q2) is given in Eqs. (A3.1)−(A3.4), and a step function θ(x0 −x) isintroduced because the relation is valid only at very small x.

It is defined by θ(x0 −x) = 1(for x < x0) or 0 (for x > x0). We should note that this is a crude estimate of the gluonic6

contribution. To be precise, the recombination calculations are done at small Q2, thenthe results are evolved to the Q2 region where experiments were done, by using a nuclear-modified Altarelli-Parisi equation [13].

This approach is discussed in section 4. Combiningall the contributions, we obtain nuclear structure functions F A2 (x) calculated in our partonmodel (the Q2 rescaling model with parton recombination effects):F A2 (x, Q2)F D2 (x, Q2)=˜F A2 (x, Q2) + ∆F A2 (x, Q2) + δF A2 (x, Q2)F D2 (x, Q2),(7)where ˜F A2 is given by ˜F A2 (x, Q2)= F N2 (x, ξAQ2) in the rescaling model.3.2 Results by using the analytical solution at small xWe calculate nuclear structure functions and investigate whether our theoretical resultsare compatible with experimental data [27] by a SLAC group [28], EMC [10,29], NMC [8],and E665 [9].

It is interesting to investigate not only the shadowing region at small x butalso the large x region, which is traditionally called “nucleon-Fermi-motion” part.In evaluating equations in Appendix, we first assume that a leak-out parton is asea quark or a gluon and that the momentum cutofffunction [17,30] for this parton istaken as w(x) = exp(−m2Nz20x2/2), namely q∗(x) = w(x)qsea(x), ¯q ∗(x) = w(x)¯q(x), andG∗(x) = w(x)G(x). Input parton distributions are given by a recent parametrization byMartin, Roberts, and Stirling (MRS) [31] or by Kwiecinski, Martin, Roberts, and Stirling(KMRS) [32].Q2 = 5 (or 4) GeV2 is used in the input parton distributions and forcalculating K in Eq.

(1). The cutofffunction w(x) supposedly takes into account effectsof bound partons and it is shown as the function of x in Fig.

3. As discussed in theintroduction, x = 0.1 corresponds to the length scale 2 fm, the average nucleon separationin nuclei.Hence, we expect that only partons p∗with x<∼0.1 could participate in therecombinations.

We then find in Fig. 3 that an appropriate choice is z0=2−3 fm.

Weshow theoretical results with z0=2 fm in this section; some z0 dependencies are discussedin section 4.3. x0 in Eq. (6) is taken as x0 = 0.1.

We do not show the ratio in Eq. (7) inthe x range 0.1 < x < 0.2 because physics is not well described by the crude estimate inEq.

(6) for taking into account the gluonic modifications. This x region is investigated by7

a more-complete Q2-evolution picture in section 4. The QCD scale parameter Λ in αs(Q2)is taken as Λ=0.2 GeV.In our theoretical analysis, Ag, Sn, and Xe targets are assumed as 107Ag, 118Sn, and131Xe nuclei.

We take the rescaling parameters in Table II of the Close, Jaffe, Roberts, andRoss paper [2]. They are ξA=1.43 (4He), 1.60 (12C), 1.86 (40Ca), 2.17 (107Ag), and 2.24(118Sn).

ξA=2.24 is assumed for the 131Xe nucleus. The above parameters are obtained atQ2=20 GeV2.

Even though the model can explain the data, it is known that the originalfits are no longer valid if we include Fermi-motion effects [33]. Furthermore, the rescalingparameters are calculated in a “semi-classical” way in the sense that overlapping volumesare estimated simply by the geometrical ones.

Considering these points, we do not thinkthat the parameters are not well defined in their overall magnitudes. According to thedynamical rescaling model by Close, Roberts, and Ross [2] in 1988, nuclear dependence isthe result of a scale change.

So, we could in principle set a rescaling parameter by fittingF A2 (x) data of a medium-size nucleus, and then use the density dependence of the rescalingmodel in calculating ξA for other nuclei. However, we find in our numerical analysis thatthe above ξA explain the SLAC data reasonably well, so that we simply keep the originalrescaling parameters in Table II of Ref.

2.We discuss results for the calcium nucleus in Figs. 4a and 4b.

In Fig. 4a, theoreticalresults in Eqs.

(5) and (7) are compared with the SLAC data [28] in the linear x scale. Forsimplicity, the deuteron structure functions are assumed as F D2 (x) = [F p2 (x) + F n2 (x)]/2 ≡F N2 (x) in our theoretical analysis.

Two sets of results are shown: the solid curves areobtained by using the MRS-1 distributions [31] and the dotted curves are by the KMRS-B0 [32]. A, B, C, and C′ in the figure show(A)[F N2 (x, Q2) + ∆F2(x, Q2)]/F N2 (x, Q2),(B)[F N2 (x, ξAQ2) + ∆F2(x, Q2)]/F N2 (x, Q2),(C)[F N2 (x, Q2) + ∆F2(x, Q2) + δF2(x, Q2)]/F N2 (x, Q2),(C′)[F N2 (x, ξAQ2) + ∆˜F2(x, Q2) + δ ˜F2(x, Q2)]/F N2 (x, Q2),where ∆˜F2(x, Q2) and δ ˜F2(x, Q2) are recombination results with Q2-rescaled input dis-tributions.

Direct recombination contributions, shown by the curves with A, are positive8

in the medium and large x regions. These come from the quark-gluon fusion processes inFigs.

1a and 1d. Combined results with the Q2 rescaling are shown by the curves with B.The C and C′ curves are explained in the following paragraph.

Fig. 4a indicates that ourmodel results are in overall agreement with the experimental data.

The EMC effect in themedium x region is mainly due to the Q2 rescaling in our model, because the recombinationcontributions are rather small. It is noteworthy in these figures that the recombinationcan explain nuclear structure functions in the large x region without explicit Fermi-motioneffects.

Physics in this region is usually attributed to the nucleon Fermi motion in thenucleus. In our model, a quark with x<∼1 could be pushed by a gluon from a differentnucleon and becomes a quark with x > 1.

This is the reason why the ratio goes to infinityat x →1 in Fig. 4a.

Perhaps, there exists a correlation between these two apparentlydifferent descriptions: the parton recombination and the nucleon Fermi motion.In Fig.4b, theoretical results are compared with EMC [10] and NMC [8] data inthe logarithmic x scale. In the small x region, the processes in Figs.

1b, 1c, 1e, and 1fsignificantly contribute and produce negative (shadowing) results in the case of KMRS-B0.However, they are too small to explain the whole shadowing data if the recombinations arecalculated at Q2=5 GeV2. The important point in our model is the shadowing which isproduced through the gluon distributions.

Combined results are shown by the curves withC or C′ in Fig. 4b.

The figure indicates that the shadowing produced through modificationsin gluon distributions can explain the EMC and NMC shadowing data fairly well, eventhough the direct recombination effects on F2(x) (shown by A) are very small. The curvesC are obtained without rescaling and C′ is with the Q2-rescaled inputs, where the sameξA is used in all parton distributions.

We find that the differences between the C and C′curves are rather small considering the fact that the Q2 rescaling produces large sea-quarkmodifications. This is because the Q2-rescaled sea-quark distributions and contributions(Eq.

(6)) from Q2-rescaled gluon distributions almost cancel each other.We have shown that our model can explain the NMC shadowing in Fig. 4b.

However,the results are very sensitive to input gluon distributions in the small x region. Becausethe E665 collaboration measured F Xe2(x)/F D2 (x) at very small x recently, we compare ourresults with the E665 data.

In order to find details of input gluon distribution effects, weuse parton distributions with “hard”, “soft”, and “1/√x” gluon distributions given by the9

MRS [31]. Analytical distributions at Q2=4 GeV2 are used as the input distributions.

TheMRS soft, hard, and 1/√x gluon distributions are given by δG = 0, ηG = 5, γG = 0,δG = 0, ηG = 4, γG = 9, and δG = −1/2, ηG = 4, γG = 9 respectively for thenotation xG(x) = AGxδG(1 −x)ηG(1 + γGx).These gluon distributions are shown inFig. 5a and they are very different especially at x < 0.01.

Our shadowing results areshown in Fig. 5b.

Because the experimental Q2 of the E665 data becomes very small(Q2 << 1 GeV2) at x < 0.001, our perturbative results cannot be compared with theE665 data. This is the reason why curves in the region x < 0.001 are shown by the dashedones.

As shown in the figure, our calculations produce widely-different shadowing resultsdepending on the input gluon distribution. If it is the 1/√x distribution, shadowings aretoo large and are contradictory to the experimental data.

If it is the hard one, theoreticalresults underestimate the shadowing. Our theoretical results are nearly consistent withexperimental data if the input distributions are the soft MRS distributions at Q2=4 GeV2.We found that our model with the recombination and the rescaling works in the wide-x region, if an appropriate input gluon distribution is taken.Nevertheless, it is not acomplete description in the sense that F A2 (x) in the x ∼0.1 region cannot be calculated.Furthermore, a curve in the x > 0.2 region and one in the x < 0.1 are not smoothlyconnected in Fig.

4b. It implies that the analytical solution Eq.

(6) at x < 0.1 is usedbeyond the applicable region. In order to have detailed comparisons with experimentaldata and to have a dynamically consistent picture, Q2 evolution effects should be fullyinvestigated.

We address this point in the following section.10

4. Q2 evolution of nuclear structure functions4.1 Q2 evolutionThe approximate description with Eq.

(6) is good enough for rough magnitudeestimates. For detailed comparisons with the experimental data in the x ≈0.1 region,we should solve Q2-evolution equations.

Namely, we calculate the recombination and therescaling at small Q2 (≡Q 20 ), then obtained distributions are evolved to those at largerQ2 where experimental data are taken. However, this is not an easy work by the followingreasons.First, the shadowing data are mostly taken in the Q2 region of a few GeV2.

We needto assign a very small value to Q 20 , for example Q 20 =1 GeV2, in order to study evolutioneffects by comparing our results with the existing data. The perturbative QCD, especiallyin the leading order, would become questionable in such small Q2 region.

Nevertheless, wepresent the evolution effects in this report because the gluon shadowing connected with theQ2 evolution is the essential part of our model in explaining the F2 shadowing, as we foundin the previous section. Next-to-leading (and higher) order corrections are important inthe small Q2 and small x region.

Hence, our investigation should be considered as a firststep for understanding F A2 (x) dynamically in the wide x region within the framework of aquark-parton model.Secondly, the evolution equation for parton distributions in a nucleus is given inRef.13. Finding a numerical solution for the equation is by itself a significant researchproblem.

So, instead of solving the nuclear evolution equation, we use the Altarelli-Parisiequation for our Q2 evolution. Both evolution results are qualitatively similar; however, thenuclear-evolution shows smaller Q2-dependence for large nuclei [13].

Because our resultsindicate small Q2 dependence for large nuclei in section 4.2, this problem is not consideredto be very serious.We calculate F A2 (x) in the following way. Valence-quark distributions in the nucleonare modified according to the Q2 rescaling mechanism at Q 20 : ˜V (x, Q 20 ) = V (x, ξVAQ 20 ).In order to satisfy the momentum conservationZdxx[˜uv(x) + ˜dv(x) + ˜S(x) + ˜G(x)] = 1,sea-quark and gluon distributions are simply modified as ˜S(x, Q 20 ) = CsgS(x, Q 20 ) and˜G(x, Q 20 ) = CsgG(x, Q 20 ).

(It is not clear whether the rescaling picture could be used11

also for sea-quarks and gluons. This issue is discussed in section 4.3.) Obtained partondistributions are then used for calculating the recombination effects in Appendix, and weget nuclear parton distributions at Q 20 .

These distributions are evolved to those at largerQ2 by using the ordinary Altarelli-Parisi equation [24]. For the flavor-singlet case, it is∂∂tqs(x, t) =1Zxdyy [ qs(y, t) Pqq(xy ) + G(y, t) PqG(xy ) ],∂∂tG(x, t) =1Zxdyy [ qs(y, t) PGq(xy ) + G(y, t) PGG(xy ) ],(8)where qs is the flavor-singlet distribution given by qs =Xi(qi + ¯qi), Pij is the splittingfunction, and t is defined by t = −(2/β0)ln[αs(Q2)/αs(Q 20 )].

We used a solution of thisintegrodifferential equation in the leading order [34]. In the flavor-nonsinglet case, thegluon does not enter into the evolution equation:∂∂tqns(x, t) =1Zxdyy qns(y, t)Pqq(xy ),(9)where qns= qi −¯qi.We use the distributions at Q 20calculated by the rescalingand the recombinations in the above equations: qns(x, Q 20 ) = ˜uv(x, Q 20 ) + ˜dv(x, Q 20 ),qs(x, Q 20 ) = ˜uv(x, Q 20 ) + ˜dv(x, Q 20 ) + ˜S(x, Q 20 ), and G(x, Q 20 ) = ˜G(x, Q 20 ).

We thenobtain evolved distributions by solving the above equations, and F2 at Q2 is given byF2(x, Q2) = [xqns(x, Q2) + 4xqs(x, Q2)]/18.The use of the above equations insteadof the nuclear evolution equations causes a problem in the large x region, because therecombinations produce distributions at x > 1. However, it is not a serious problem incomparison with the experimental data discussed in the next section.4.2 Q2 evolution resultsStructure functions F A2 (x) are calculated in the following way.

Because the partondistributions in the MRS have simpler analytical forms than those in the KMRS, we firstemploy the MRS-1 (soft gluon) as our parton distributions in the nucleon.Using the12

analytical form of the MRS-1 at Q2=4 GeV2 and the evolution subroutine in Ref. 34,we calculate distributions at Q 20 (≈1 GeV2).

Then, the Q2 rescaling is used for valencedistributions and the constant Csg is determined so that the momentum conservationis satisfied.Input distributions at Q 20are parametrized by a simple analytical form,xp(x) = Axδ(1 −x)η(1 + γx). Constants A, δ, η, and γ are obtained by fitting numericalresults.

The recombinations are calculated by using the modified input distributions atQ 20 , and obtained results are again fitted by xp(x) = Axδ(1 −x)η(1 +γx). For the valencequarks, we used xV (x) = Axδ(1 −βx)η(1 + γx), where β < 1, so that we could take intoaccount the fusion effects at large x (=0.8−0.9).

The recombinations produce distributionsat x > 1; however, we simply neglected such distributions in our Q2 evolution. This causesslight inconsistency, but the neglected effect on the momentum conservation is of the orderof 10−4%, which is insignificant.We discuss results for the nuclei He, C, Ca, Ag, Sn, and Xe in Figs.

6-9. In Figs.6a-9a, our theoretical results are compared with SLAC [28] and EMC [29] (E665 [9]) datain the linear x scale.

In order to see the shadowing due to the Q2 evolution, we shouldchoose Q2 ∼1 GeV2 because the data were taken in the region of a few GeV2. The value ofQ 20 is adjusted so that our results agree with the shadowing data for 40Ca, and we obtainQ20=0.8 GeV2.

We fixed the cutoffat z0=2 fm as it was used in section 3. We discussdependence of our results on these parameters in the next section.

Another problem ishow we should choose the rescaling parameters at very small Q2. As we did in section 3,we may simply determine ξA for a medium-size nucleus and then use the A-dependence forother nuclei.

However, it turns out that an original rescaling parameter for a medium-sizenucleus is not a bad choice, so we simply use the parameters in Ref. 2.

Three curves withQ2=0.8, 5.0, 20.0 GeV2 are shown in Figs. 6-9.

Figures 6a-9a indicate that our modelcan explain the experimental data well except for the x ≈0.7 region of large nuclei. Theresults in this region depend much on the input valence distribution and the cutoff(seesection 4.3).

If we change the input to the KMRS-B0 distributions, our calculations workvery well for large nuclei, but the EMC effect for 4He is overestimated.In Figs. 6b-9b, theoretical results are compared with EMC [10], NMC [8], and E665[9] data in the logarithmic x scale.

Our results are not shown at x < 0.001 in Fig. 9bbecause the used evolution subroutine [34] does not have good accuracy in such region and13

our perturbative calculations cannot be compared with the data with very small Q2. Wefind in Figs.

6b-9b that our results agree quite well with the experimental data. In thesmall nuclei (Figs.

6b and 7b), there are some Q2 dependence at Q2 = 0.8 −2 GeV2, butit becomes Q2 independent at Q2 > a few GeV2. We have very small Q2 dependence inthe larger nuclei as shown in Figs.

8b and 9b. We note that rather large Q2 dependencein Figs.

6b and 7b should not be taken seriously at this stage, because small variations ininput sea-quark and gluon distributions change the dependence drastically as we find insection 4.3.4.3 Dependence on the parametersWe discuss how our results depend on the parameters, the input distributions, and therescaling assumption. We first check the choice of Q 20 .

We choose Q 20 =2.0 GeV2 insteadof 0.8 GeV2 in Fig. 10.

Distributions at 2 GeV2 are calculated and are then evolved tothose at 5 and 20 GeV2. It is obvious that our calculations underestimate the experimentalshadowings and that the Q2 dependence is very different from the one in Fig.

8b. Thedifferences are due to the Q2 factor in K (see Eq.

(1)) and due to input sea-quark andgluon distributions. In fact, we find in the following that slight modifications in thosedistributions change our shadowing results at Q 20 significantly.In order to check input-distribution effects, we take the KMRS-B0 distributions whichare slightly different from those of the MRS-1.

These distributions are shown in Fig. 11.Even though two distributions do not look very different, Q2 dependency of the KMRS-B0results for the carbon nucleus in Fig.

12a is quite different from that in Fig. 7b.

TheKMRS-B0 results for the calcium in Fig. 12b are rather similar to those of the MRS-1 inFig.

8b. Because the parton distributions in the small-x region are not well known, weinevitably have such uncertainty in our model.Effects of a momentum cutofffor leak-out partons are shown in Figs.

13a and 13b.From Fig. 3, we decided to take z0=2 fm and we have been using this cutoffso far.

Resultswith z0=3 fm are shown by the dashed curves. Our model with larger z0 produces smallerrecombination effects at large x and larger effects at small x.

The agreement with thedata becomes better in Fig. 13a in the medium-x region; however, it becomes worse in14

the small-x region in Fig. 13b.

Because the cutoffis one of the important factors in ourmodel, efforts should be made to estimate it in an independent way.The last issue is the rescaling. The original rescaling model is intended for valencequarks.

It is not clear whether sea-quarks and gluon follow the same rule of the scalechange in nuclei. In our analysis of this section, we used the rescaling only for valencequarks.Sea-quark and gluon distributions are simply increased by a constant amount(Csg) so that they carry a momentum deficit produced by the valence rescaling.

This isan arbitrary assumption. We may choose other scenario, for example the Q2 rescaling forall partons.

Using this rescaling picture, we obtain the results in Fig. 14.

The shadowingsare much underestimated in this case. Although numerical values do not agree completelywith those in Fig.

4b, we find a similar tendency in the sense that calculated shadowingsbecome smaller in the x region (0.01 < x < 0.1) if the rescaling is used for all partons.We learned the following from the above analyses. There are a few parameters andassumptions in our model and obtained results depend inevitably on their choice.

Themajor factors are (1) Q 20 , (2) input S(x) and G(x) in the small x, (3) z0, and (4) Q2rescaling for (valence) partons. We find that magnitude of our shadowing and the Q2dependence are very sensitive to the above factors.

Nevertheless, we find that it is possibleto choose a set of the factors within our initial-expectation ranges (e.g. Q2 ∼1 GeV2,z0 ∼2 fm) and we explain many existing data.

Obviously, much efforts should be donefor investigating these factors and also next-to-leading order effects.4. ConclusionsWe investigated nuclear structure functions F2(x, Q2) from small x(≈10−3) to largex(≈0.9) in a parton model based on a Q2 rescaling model with parton recombinationeffects in order to compare them with the recent experimental data.As a result, weobtained reasonably good agreement with the experimental data in the region (0.005

In the large x region, the ratio (F A2 (x)/F D2 (x) > 1) is explained by quark-gluonrecombinations, which produce results similar to those by the nucleon Fermi motion. Inthe medium x region, the EMC effect is mainly due to the Q2 rescaling mechanism inour model.

In the small x region, shadowing effects are obtained through modifications in15

gluon distributions. However, our shadowings at very small x(< 0.02) are very sensitive tothe input gluon distribution.

We have a few parameters in our model; however, we couldchoose a set of the parameters and explain many existing experimental data. Q2 variationsof our shadowing results depend much on the input sea-quark and gluon distributions, andalso on the parameters.AcknowledgmentsThis research was supported by the Deutsche Forschungsgemeinschaft (SFB 201) andby the US-NSF under Contract No.

NSF-PHY91-08036. S.K.

thanks Drs. F. E. Close,J.

Qiu, and R. G. Roberts for helpful suggestions and for patiently answering questionsabout parton recombinations; Drs. M. van der Heijden, Q. Ingram, and G. van Middelkoopfor discussion about NMC experiments; Drs.

C. W. Salgado and H. M. Schellman forinformation about the E665 experimental results; Dr.W. J. Stirling for sending theircomputer programs for calculating the parton distributions in Ref.

32.16

AppendixAlthough detailed formalisms are given in Ref.17, explicit equations used in thisinvestigation are shown in the following. Using Eqs.

(1)−(4) and changing the integrationvariable x1 ↔x2 for the process in Fig. 1e, we obtain modifications of quark distributionsin nuclei due to the parton recombination mechanism:x · ∆qi(x) = +K6xZ0dx2x2x2G∗(x2)(x −x2)qi(x −x2)1 + (x −x2x)2−xqi(x)xx + x21 + (xx + x2)2 −K61Zxdx2x2xqi(x) x2G∗(x2)xx + x21 + (xx + x2)2−4K9 x1Z0dx2 xqi(x) x2¯q∗i (x2)x2 + x22(x + x2)4+K6xZ0dx1x1x1G(x1)(x −x1)q∗i (x −x1)1 + (x −x1x)2−xq∗i (x)xx + x11 + (xx + x1)2 −K61Zxdx1x1x1G(x1) xq∗i (x)xx + x11 + (xx + x1)2−4K9 x1Z0dx2 xq∗i (x) x2¯qi(x2)x2 + x22(x + x2)4,(A1)where ∗indicates a leak-out parton.

For example a gluon, which leaks out from the nucleon2, interacts with a quark in the nucleon 1 and produces a quark with momentum x in thefirst term of Eq. (A1).

Explicit Q2 dependence in parton distributions is not shown inthis appendix in order to simplify the notations. Modifications in antiquark distributionsare obtained in the similar way by changing qi ↔¯qi in Eq.

(A1).Using the above17

expression, we obtain parton recombination effects on the structure functions F2(x) by∆F2(x) =Xie2i x[∆qi(x) + ∆¯qi(x)]∆F2(x) = ∆F (1)2 (x) + ∆F (2)2 (x) + ∆F (3)2 (x)(A2.1)∆F (1)2 (x) = +K6xZ0dx2x2x2G∗(x2)F2(x −x2)1 + (x −x2x)2−F2(x)xx + x21 + (xx + x2)2 −K61Zxdx2x2F2(x) x2G∗(x2)xx + x21 + (xx + x2)2(A2.2)∆F (2)2 (x) = +K6xZ0dx1x1x1G(x1) −Xie2i(x −x1)q∗i (x −x1) + (x −x1)¯q∗i (x −x1)×1 + (x −x1x)2−−Xie2ixq∗i (x) + x¯q∗i (x)xx + x11 + (xx + x1)2 −K61Zxdx1x1x1G(x1)−Xie2ixq∗i (x) + x¯q∗i (x)xx + x11 + (xx + x1)2(A2.3)∆F (3)2 (x) = −4K9 x1Z0dx2−Xie2ixqi(x)x2¯q∗i (x2) + x¯qi(x)x2q∗i (x2)x2 + x22(x + x2)4−4K9 x1Z0dx2−Xie2ixq∗i (x)x2¯qi(x2) + x¯q∗i (x)x2qi(x2)x2 + x22(x + x2)4(A2.4)where−Xindicates that the summation is averaged over partons in the proton and theneutron. For example, they are−Xie2i qi = 518(u + d) + 218sproton and−Xie2i qi¯qi =18

518(u¯u + d ¯d) + 218s¯sproton.In the similar way, modifications in a gluon distribution in a nucleus are obtained as[35]x∆G(x) = x∆G(1)(x) + x∆G(2)(x) + x∆G(3)(x)(A3.1)x∆G(1)(x) = +3K4 xxZ0dx1 x1G(x1) (x −x1)G∗(x −x1)× 1x2x1x −x1+ x −x1x1+ x1(x −x1)x2−3K4 x1Z0dx2 xG(x) x2G∗(x2)1(x + x2)2 xx2+ x2x +xx2(x + x2)2−3K4 x1Z0dx2 xG∗(x) x2G(x2)1(x + x2)2 xx2+ x2x +xx2(x + x2)2(A3.2)x∆G(2)(x) = +4K9 xxZ0dx1−Xix1qi(x1)(x −x1)¯q∗i (x −x1)+ x1¯qi(x1)(x −x1)q∗i (x −x1) 1x4x21 + (x −x1)2(A3.3)x∆G(3)(x) = −K61Z0dx2 xG(x)−Xix2q∗i (x2) + x2¯q∗i (x2)1x + x21 + (x2x + x2)2−K61Z0dx2 xG∗(x)−Xix2qi(x2) + x2¯qi(x2)1x + x21 + (x2x + x2)2(A3.4)where−Xiqi =Xi(qi)proton and−Xiqi¯qi =Xi(qi¯qi)proton. Using these expressions, we findthat the momentum conservation is explicitly satisfied,ZdxXix∆qi(x) + ∆¯qi(x)19

+x∆G(x)= 0. For numerical analysis, the last integral in ∆G(1)(x) should be evaluatedby separating the integral region1Z0dx2 =xZ0dx2 +1Zxdx2 and by changing variablesx′ = x −x2 in the integralxZ0dx2.

We use the following equations for evaluating ∆G(1)(x)in order to cancel out infinities:x∆G(1)(x) =x/2Z0dx′ [ f(x, x′) + f(x, x −x′) ] −1Zxdx′ g(x, x′)(A4.1)f(x, x′) = +3K4 x x′G(x′) (x −x′)G∗(x −x′) 1x2x′x −x′ + x −x′x′+ x′(x −x′)x2−3K4 x xG(x) x′G∗(x′)1(x + x′)2 xx′ + x′x +xx′(x + x′)2−3K4 x xG∗(x) (x −x′)G(x −x′)1(2x −x′)2xx −x′ + x −x′x+ x(x −x′)(2x −x′)2(A4.2)g(x, x′) = +3K4 x [ xG(x) x′G∗(x′) + xG∗(x) x′G(x′) ]×1(x + x′)2 xx′ + x′x +xx′(x + x′)2(A4.3)Nuclear gluon distributions in our model are discussed in Ref.36and they arecompared with recent NMC measurements of GSn(x)/GC(x).20

References1. J. J. Aubert et al.

(EMC collaboration), Phys. Lett.

123B, 275 (1983).2. F. E. Close, R. G. Roberts, and G. G. Ross, Phys.

Lett. 129B, 346 (1983); Nucl.Phys.

B296, 582 (1988); F. E. Close, R. L. Jaffe, R. G. Roberts, and G. G. Ross,Phys. Rev.

D31, 1004 (1985).3. E. L. Berger and F. Coester, Annu.

Rev. Nucl.

Part. Sci.

37, 463 (1987); R. P.Bickerstaffand A. W. Thomas, J. Phys. G 15, 1523 (1989); R. P. Bickerstaff, M. C.Birse, and G. A. Miller, Phys.

Rev. D33, 3228 (1986).4.

D. M. Alde et al. (E772 collaboration), Phys.Rev.Lett.64, 2479 (1990); A.Magnon, talk given at the workshop on Baryon Spectroscopy and the Structure ofthe Nucleon, Saclay, France, Sept. 23−25, 1991.5.

S. Kumano and F. E. Close, Phys. Rev.

C41, 1855 (1990).6. S. Kumano, Indiana University preprint IU/NTC-92-02; talk given at the 13thInternational Conference on Few Body Problems in Physics, Adelaide, Australia,Jan.

5−11, 1992; in Proceedings of the International Workshop on Gross Propertiesof Nuclei and Nuclear Excitations, Hirschegg, Austria, Jan. 20−25, 1992, edited byH. Feldmeier; to be submitted for publication.7.

T. de Forest and P. J. Mulders, Phys. Rev.

D35, 2849 (1987); S. Kumano and E. J.Moniz, Phys. Rev.

C37, 2088 (1988).8. P. Amaudruz et al.

(NMC collaboration), Z. Phys. C51, 387 (1991); M. van derHeijden, Ph.D thesis, University of Amsterdam (1991); M. A. J. Botje and C. Scholzin Intersections between Particle and Nuclear Physics, edited by W. T. H. van Oers,American Institute of Physics (1991).9.

M. R. Adams et al. (E665 collaboration), Phys.

Rev. Lett.

68, 3266 (1992); D.E. Jaffe, in Intersections between Particle and Nuclear Physics, edited by W. T.H.

van Oers, American Institute of Physics (1991); C. W. Salgado, talk given at theworkshop on High Energy Probes of QCD and Nuclei, University Park, Pennsylvania,March 25-28, 1992.21

10. M. Arneodo et al.

(EMC collaboration), Nucl. Phys.

B333, 1 (1990); Phys. Lett.211B, 493 (1988).11.

For recent investigations on the vector meson dominance model, see C. L. Bilchak,D. Schildknecht, and J. D. Stroughair, Phys.

Lett. 214B, 441 (1988); 233B, 461(1989); G. Shaw, Phys.

Lett. 228B, 125 (1989); G. Piller and W. Weise, Phys.

Rev.C42, R1834 (1990).12. N. N. Nicolaev and V. I. Zakharov, Phys.

Lett. 55B, 397 (1975).13.

A. H. Mueller and J. Qiu, Nucl. Phys.

B268, 427 (1986); J. Qiu, Nucl. Phys.

B291,746 (1987); E. L. Berger and J. Qiu, Phys. Lett.

206B, 141 (1988).14. F. E. Close and R. G. Roberts, Phys.

Lett. 213B, 91 (1988).15.

R. J. M. Covolan and E. Predazzi, Nuovo Cimento, 103A, 773 (1990); W. Zhu andJ. G. Shen, Phys.

Lett. 235B, 170 (1990); J. Phys.

G16, 925 (1990); G. Li, Z. Cao,and C. Zhong, Nucl. Phys.

A509, 757 (1990); M. Altmann, M. Gl¨uck, and E. Reya,Phys. Lett.

285B, 359 (1992).16. L. L. Frankfurt, S. Liuti, and M. I. Strikman, research in progress; S. Liuti, personalcommunication; V. Barone et al., preprint KFA-IKP(TH)-1992-13.17.

F. E. Close, J. Qiu, and R. G. Roberts, Phys. Rev.

D40, 2820 (1989).18. L. L. Frankfurt and M. I. Strikman, Phys.

Rep. 160, 235 (1988); Nucl. Phys.

B316,340 (1989); Phys. Rev.

Lett. 65, 1725 (1990); S. J. Brodsky and H. J. Lu, Phys.Rev.

Lett. 64, 1342 (1990); N. N. Nikolaev and B. G. Zakharov, Phys.

Lett. 260B,414 (1991); Z. Phys.

C49, 607 (1991); V. R. Zoller, Z. Phys. C53, 443 (1992); Phys.Lett.

279B, 145 (1992).19. P. Castorina and A. Donnachie, Phys.

Lett. 215B, 589 (1988); Z. Phys.

C45, 141(1989); J. Kwiecinski, Z. Phys. C45, 461 (1990).20.

G. Preparata and P. G. Ratcliffe, preprint MITH91-13.21. J. Kwiecinski and B. Badelek, Phys.

Lett. 208B, 508 (1988); B. Badelek and J.Kwiecinski, Nucl.

Phys. B370, 278 (1992); W. Melnitchouk and A. W. Thomas,preprint ADP-92-192-T120.22

22. See Refs.

13 and 17 for obtaining the expression of K.23. R. C. Barrett and D. F. Jackson, Nuclear Sizes and Structure (Clarendon, Oxford,1977).24.

G. Altarelli and G. Parisi, Nucl. Phys.

B126, 298 (1977); In deriving Eq. (2), weshould note a factor of 2 coming from averages over initial spin states.25.

F. J. Yndur`ain, Quantum Chromodynamics, (Springer-Verlag, New York, 1983).26. J. P. Ralston, Phys.

Lett. 172B, 430 (1986).27.

For a recent summary of experimental data for deep inelastic scatterings, see R. G.Roberts and M. R. Whally, J. Phys. G 17, D1 (1991).28.

R. G. Arnold et al., Phys. Rev.

Lett. 52, 727 (1984).29.

J. Ashman et al. (EMC collaboration), Phys.

Lett. 202B, 603 (1988).30.

C. H. Llewellyn Smith, Nucl. Phys.

A434, 35c (1985).31. A. D. Martin, R. G. Roberts, and W. J. Stirling, Phys.

Rev. D37, 1161 (1988).32.

J. Kwiecinski, A. D. Martin, W. J. Stirling, and R. G. Roberts, Phys. Rev.

D42,3645 (1990).33. R. P. Bickerstaffand G. A. Miller, Phys.

Lett. 168B, 409 (1986).34.

S. Kumano and J. T. Londergan, Comp. Phys.

Commu. 69, 373 (1992).35.

J. Qiu, personal communications (1992); Using K[p1(x1)p2(x2) + p1(x2)p2(x1)] ⇒2Kp1(x1)p2(x2) in Eqs. (A3.3) and (A3.4), we obtain Eq.

(24) of Ref. 17 from Eqs.

(A3.1)−(A3.4) with x1 ↔x2 in the last integral of Eq. (A3.2) .36.

S. Kumano, Phys. Lett.

298B, 171 (1993); This paper is based on the model insection 3.23

Figure Captions1. Schematic pictures of parton recombination processes.2.

Parton fusions for (a) qG →q, (b) q¯q →G, (c) Gq →q, and (d) GG →G.3. Momentum cutofffor leak-out partons, w(x) = exp(−m2Nz20x2/2).4.

Comparisons with (a) SLAC data [28], (b) EMC-90 [10] and NMC [8] data forCa. Solid (dotted) curves are obtained by using the MRS-1 (KMRS-B0) inputdistributions.

Q2=5 GeV2 and z0=2 fm. (A) recombinations, (B) recombinations+ rescaling (ξA = 1.86).

(C) recombinations with gluon-shadowing effects, (C′) thesame as C except for the rescaling. The only KMRS-B0 curve is shown in C′.

Seetext for detailed explanations of A, B, C, and C′.5. (a) MRS gluon distributions.

See Ref. 31and text for details of hard, soft, and1/√x gluon distributions.

(b) Comparisons with E665 data for Xe [9] and EMC-88data for Sn [29]. Theoretical results are obtained (b) for the Xe nucleus (ξA = 2.24).Q2=4 GeV2 and z0=2 fm.6.

Comparisons with (a) SLAC data [28], (b) NMC data [8] for He. Q2=0.8, 5, and 20GeV2 for the dotted, solid, and dashed curves respectively.

z0=2 fm, ξVA = 1.43 andQ 20 =0.8 GeV2.7. Comparisons with (a) SLAC [28] and EMC-88 [29] data, (b) EMC-90 [10] and NMC[8] data for C. Notations for the dotted, dashed, and solid curves are the same inFig.

6. ξVA = 1.60.8. Comparisons with (a) SLAC data [28], (b) EMC-90 [10] and NMC [8] data for Ca.ξVA = 1.86.9.

Comparisons with (a) SLAC data for Ag [28], EMC-88 data for Sn [29], and someof E665 data for Xe [9], (b) E665 data for Xe [9] and EMC-88 data for Sn [29].Calculated results are for Ag (ξVA = 2.17) in (a) and for Xe (ξVA = 2.24) in (b).10. Dependence on Q 20 .

Q 20 =2.0 GeV2 is taken. Q2=2, 5, and 20 GeV2 for the dotted,solid, and dashed curves respectively.

z0=2 fm and ξVA = 1.86.24

11. Sea-quark and gluon distributions of MRS-1 (solid curves) and KMRS-B0 (dottedcurves).12.

Dependence on input distributions (see Fig. 11).

Results by using the KMRS-B0inputs for (a) 12C and (b) 40Ca. Q2=0.8, 5, and 20 GeV2 for the dotted, solid, anddashed curves respectively.

z0=2 fm. ξVA =1.60 for C and 1.86 for Ca.

Q 20 =0.8 GeV2.Compare them with the results in Figs. 7b and 8b.13.

Dependence on the cutoffz0. Only Q2=5 GeV2 curves are shown.

z0=2.0 (3.0) fmfor the solid (dashed) curves. ξVA = 1.86 and Q 20 =0.8 GeV2.14.

Rescaling are used for all partons. Notations and parameters are same in Fig.

7b.25

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