Z. Phys. A 359, 99–109 (1997) ZEITSCHRIFTFUR PHYSIK Ac© Springer-Verlag 1997

Probing the origin of the EMC effect via tagged structure functions ofthe deuteronW. Melnitchouk 1, M. Sargsian2, M.I. Strikman 3

1 Department of Physics, University of Maryland, College Park, MD 20742, USA2 School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel, and Yerevan Physics Institute, Yerevan 375036, Armenia3 Department of Physics, Pennsylvania State University, University Park, PA 16802, USA, and Institute for Nuclear Physics, St. Petersburg, Russia

Received: 8 October 1996 / Revised version: 24 February 1997Communicated by W. Weise

Abstract. We demonstrate that measurement of tagged struc-ture functions of the deuteron in (e, e′N ) semi-inclusive re-actions can discriminate between different hypotheses on theorigin of the nuclear EMC effect. By choosing extreme back-ward kinematics for the spectator nucleon to minimize effectsfrom the deuteron wave function and final state interactions,one can isolate the modifications in the structure of the boundnucleon within the impulse approximation. The same reactioncan be used to extract the large-x neutron to proton structurefunction ratio.

PACS: 25.30.-c

1 Introduction

More than a decade after the discovery of the nuclear EMC ef-fect [1] and many fine measurements [2, 3, 4, 5] of the ratios ofstructure functions of nuclei and the deuteron, no consensushas been reached on the origin of the effect. Thex depen-dence of the effect, while non-trivial, is rather smooth andhas the same basic shape for all nuclei, making it is easy tofit in a wide range of models with very different underlyingassumptions. The only extra constraint available so far comesfor measurements of theA-dependence of the sea distribution,which restricts some of the models, but is still not sufficientto allow one to unambiguously identify the origin of the EMCeffect.

In order to move beyond this rather unsatisfactory situa-tion, new experiments involving more kinematical variablesaccessible to accurate measurements are necessary. The aimof this study is to demonstrate that use of semi-inclusive pro-cesses off the deuteron,

e +D → e +N +X, (1)

where a nucleon is detected in the target deuteron fragmen-tation region, may help to discriminate between some classesof models. In particular, one may be able to distinguish be-tween models in which the effect arises entirely from hadronicdegrees of freedom — nucleons and pions (which in the tradi-tional nuclear physics picture are responsible for the binding

of the nucleus), and models in which the effect is attributedto the explicit deformation of the wave function of the boundnucleon itself. By selecting recoil nucleons with small trans-verse momentum in the backward region, effects due to finalstate interactions (FSI) can be minimized, thus allowing oneto probe, within the impulse approximation, the deformationof the structure of the bound nucleon.

It is worth emphasizing that measurement of the reaction(1) in the kinematics of interest will become feasible in thenear future, for example at Jefferson Lab [6]. Recent develop-ment of silicon detectors also make it possible in the jet targetexperiments to measure recoil nucleons with a low momentumthreshold of about 100–150 MeV/c. Plans to implement sucha technique are under discussion at the HERMES detector atHERA [7].

Aside from providing insight into the origin of the nuclearEMC effect per se, the measurements of tagged events mayalso be useful in connection with the problem of extracting theneutron structure function from deuteron data. By selectingonly the slowest recoil protons in the target fragmentationregion, one should be able to isolate the situation wherebythe virtual photon scatters from a nearly on-shell neutron inthe deuteron. In this way one may hope to extract theF2nstructure function while minimizing uncertainties arising frommodeling the nuclear effects in the deuteron.

This paper is organized as follows. In Sect. II we outlinethe basic formalism for semi-inclusive deep-inelastic scatter-ing off the deuteron, and discuss the conditions under whichthe impulse approximation may be valid. Sect. III is devotedto a survey of several models whose predictions for the ratiosof tagged structure functions are compared. Possibilities ofextracting the neutron structure function from semi-inclusiveexperiments are discussed in Sect. IV, and finally some con-clusions are drawn in Sect. V.

2 Basic Formalism

It was observed a long time ago that the nuclear EMC effectfor the deviation from unity of the ratio

R(x,Q2) =2F2A(x,Q2)AF2D(x,Q2)

(2)

100

is approximately proportional to the nuclear density. This isnatural for a dilute system, and indicates that most of the EMCeffect is due to two-nucleon interactions. Based on this obser-vation one should expect that the EMC effect in the deuteronis much smaller than that in heavy nuclei. However, by virtueof the uncertainty principle, one may try to enhance the ef-fect by isolating the configuration where the two nucleonsin the deuteron are close together. For example, it is easy tocheck that the main contributions to the deuteron wave func-tion for nucleon momentap & 300 MeV/c come from dis-tances. 1.2 fm.

To make our discussion quantitative, let us begin by writingthe electromagnetic tensorWµν

D of the deuteron in terms ofthe matrix element of the electromagnetic current:

WµνD =

∑spin,X

〈D | JµD(q) | X,N〉〈X,N | Jµ†D (0) | D〉. (3)

The tensorWµνD can be expanded in terms of four possible

Lorentz structures, with the coefficients given by the invariantstructure functionsFDL , F

DT , F

DTL andFDTT (T = transverse,

L = longitudinal). These functions can in general depend onfour variables, constructed from the four-momenta of the tar-get deuteron (P ), virtual photon (q) and spectator nucleon (ps)(or equivalently the momentum of the struck nucleonp, wherep = −ps in the deuteron rest frame). In terms of the invari-ant structure functions the differential cross section for thesemi-inclusive reaction (e, e′N ) can then be written:

dσ

dxdQ2d3ps/Es=

2α2em

xQ4

(1− y − x2y2M2

Q2

)×[FDL +

(Q2

2q2+ tan2(θ/2)

)ν

MFDT

+

(Q2

q2+ tan2(θ/2)

) 12

cosφ FDTL + cos(2φ) FDTT

], (4)

where the structure functions are related to the components ofthe electromagnetic tensorWµν

D by:

FDL = ν(1 + cosδ)2 ·W−−D , (5a)FDT = M (W xx

D +W yyD ), (5b)

FDTL = 2ν(1 + cosδ) ·W−xD , (5c)FDTT = ν

2 sin2 δ · (W xxD −W

yyD ). (5d)

The kinematic variables in (4) and (2) arex = Q2/2Mν,whereν is the energy of virtual photon in the target rest frameandM the nucleon mass,Q2 = 4E(E − ν) sin2(θ/2) is thesquared four-momentum transfer to the target,E is the beamenergy andy = ν/E. The angleφ is the azimuthal angle for thespectator nucleon, andEs =

√M2 + p2

s the spectator nucleonenergy, and sin2 δ = Q2/q2 in (2). We define the photon three-momentumq to be in the +z direction.

A convenient choice of the four independent variables forthe structure functions is the two inclusive deep-inelastic scat-tering variables,x andQ2, and the transverse momentum,psT ,and light-cone momentum fraction,αs, of the spectator nu-cleon:

αs =Es − pszM

, (6)

wherepsz is the longitudinal momentum of the detected nu-cleon (to simplify the expressions we have neglected here the

deuteron binding energy,MD ≈ 2M ). In the Bjorken limitthe variableαs then satisfies the condition [8]:

αs ≤ 2− x. (7)

Having defined the relevant cross sections, we must nowestablish the kinematical range in which the nuclear modifica-tions of the bound nucleon structure function will be accessibleexperimentally. From (4) and (2) the restrictions:

Q2/q2 ∼ 4M2x/Q2¿ 1, (8a)Q2¿ E(E − ν), (8b)

enhance the contribution of the longitudinal structure functionFL, which is expressed through the “good” component of theelectromagnetic current, Eq.(2). Another simplification canbe achieved by considering the situation where the detectednucleon in the deuteron is in the spectator kinematics, namely:

αs ≥ 1− x. (9)

In this kinematical region the contribution of the direct processwhere a nucleon is produced at theγ∗N interaction vertex isnegligible [8].

2.1 Factorization and the impulse approximation

In formulating deep-inelastic scattering from the deuteron thesimplest approach adopted has been the impulse approxima-tion for the nuclear system. We will consider two formula-tions of the impulse approximation: one based on the co-variant Feynman, or instant-form, approach, where one nu-cleon is on-mass-shell and one off-mass-shell, and the non-covariant/Hamiltonian light-cone approach, in which both nu-cleons are on-mass-shell (but off the light-cone energy shell).In the next subsection we shall consider corrections to the im-pulse approximation, in the form of final state interactions, butfor now let us review briefly the basic impulse approximationassumptions and results.

2.1.1. Covariant Instant-Form ApproachIn order to constructcovariant amplitudes in the instant-form of quantization, sum-mation over all possible time-orderings of intermediate statesis essential. Incorporation of negative energy configurationsinto the total Lorentz-invariant amplitude is done on the basisof introducing intermediate state particles which are off theirmass shells. In the impulse approximation for the scatteringof a virtual photon,γ∗, from the deuteron, the invariant am-plitude for the complete process factorizes into a product ofamplitudes forγ∗–off-mass-shell nucleon scattering, and forforward nucleon–deuteron scattering. In this case the hadronictensor of the deuteron can be written [9]:

WµνD (P, p, q) = Tr

[SND(P, p) Wµν

N (p, q)], (10)

where the nucleon–deuteron scattering amplitudeSND (forspin-averaged processes) contains scalar and vector compo-nents:

SND = S0 + γαSα1 . (11)

The operatorWµνN in Eq.(10) is the truncated hadronic tensor

for the off-shell nucleon, which describes theγ∗N interaction.

101

BecauseWµνN is a matrix in both Lorentz and Dirac spaces,

its structure is necessarily more general than that for a freenucleon. In the Bjorken limit, it was shown in Ref.[9] that outof a possible 13, only three independent structures contribute:

WµνN (p, q) = −

(gµν +

qµqνQ2

)×

×(W0(p, q)+ 6p W1(p, q)+ 6q W2(p, q)

)+ O

(1Q2

). (12)

Note that the full expression forWµνN contains both posi-

tive and negative energy pieces. The on-shell nucleon tensoris obtained fromWµν

N by projecting out the positive energycomponents:

WµνN (p, q) =

14

Tr[(6p +M ) Wµν(p, q)

](13a)

= −(gµν +

qµqνQ2

)×

×(MW0 +M2W1 + p · qW2

), (13b)

where the functionsW0···2 here are evaluated at their on-shellpoints.

In terms of the off-shell truncated functionsW0···2 thedeuteron tensor can then be written:

WµνD (P, p, q) = −4

(gµν +

qµqνQ2

)×

×(S0 W0 + S · p W1 + S · q W2

), (14)

so that in general the right-hand-side does not factorize into asingle term with separate nuclear and nucleon components. Asexplained in Ref.[9], factorization can be recovered by project-ing from S0,1 only the positive energy (on-shell) components,in which case:

S0 ∝ S · p ∝ S · q. (15)

This proportionality can also be obtained by taking the non-relativistic limit for theND amplitudes to orderp2/M2 [10],since all negative energy contributions enter at higher orders[11, 12]. For the relativisticND interaction, the negative en-ergy components enter through theP -state wave functions ofthe deuteron. However, in most realistic calculations of these,in the context of relativistic meson-exchange models of thedeuteron [13], the contribution of theP -state wave functionsis only a fraction of a percent, and so long as one avoids re-gions of extreme kinematics which are sensitive to the verylarge momentum components of the deuteron wave function,dropping the negative energy components is a reasonable ap-proximation. Indeed, the explicit calculations of the inclusivestructure function of the deuteron with relativistic wave func-tions shows that the factorization-breaking corrections amountto. 1–2% for all values ofx . 0.9 [11, 14].

Finally, given these approximations one can write thedeuteron hadronic tensor as:

WµνD (P, p, q) ≈ SIF (P, p) Wµν eff

N (p, q), (16)

whereSIF is the nucleon spectral function within the instant-form impulse approximation [11]:

SIF (αs, pT ) = (2− αs)EsMD

2(MD − Es)|ψD(αs, pT )|2, (17)

normalized such that∫d2pT

dαsαs

SIF (αs, pT ) = 1 (18)

to ensure baryon number conservation [15, 16]. Note that theapproximation for the spectral function in Eq.(17) is validonly for non-relativistic momenta, and for large momenta thefull (non-convolution) expression for the hadronic tensor inRef.[11] should be used. The effective nucleon hadronic tensorWµν effN is defined as:

Wµν effN = −

(gµν +

qµqνQ2

)(MW0 + p2W1 + p · qW2

)≡ −

(gµν +

qµqνQ2

)F eff1N

(x

2− αs, p2, Q2

)+ · · · ,

(19)

where the effective nucleon structure functionF eff1N is now afunction of the momentum fractionx/(2− αs) and the virtu-ality p2 of the bound nucleon, as well asQ2. The kinematicsof the spectator process gives rise to the relation:

p2 = −2p2T + (2− αs)M2

αs+

12

(2− αs)M2D , (20)

using which one can equivalently expressF eff1N as a functionof the transverse momentumpT of the interacting nucleon,rather thanp2.

2.1.2. Light-cone approachIn the light-cone formalism onecan formally avoid the problems associated with negative en-ergy solutions in (12), however, to obtain (16) one must on theother hand consider contributions arising from instantaneousinteractions [17]. To take these effects into account one has touse gauge invariance to express the contribution of the “bad”current components of the electromagnetic tensor through the“good” components:

JA+ = − q+

q−JA− , (21)

and include the contribution of the instantaneous exchangesusing the prescription of Brodsky and Lepage [18]. In the ap-proximation when other than two-nucleon degrees of freedomin the deuteron wave function can be neglected, one can un-ambiguously relate the light-cone deuteron wave functions tothose calculated in the rest frame in non-relativistic instant-form calculations [19, 20]. The final result for the spectralfunction in the light-cone approach can be written similarly to(16), only nowS is replaced by the light-cone density matrix:[8, 17],

SLC(αs, pT ) =

√M2 + k2

2− αs|ψD(k)|2, (22)

where

k ≡ |k| =√

M2 + p2T

αs(2− αs)−M2 (23)

102

is the relative momentum of the two nucleons on the light-cone, andWµν eff

N in Eq.(16) is now the bound nucleon elec-tromagnetic tensor defined on the light-cone. Note that themain operational difference between the instant-form (17) andlight-cone (22) impulse approximations is the different rela-tion between the deuteron wave function and the scatteringamplitude. Numerical studies have demonstrated that in thekinematical region of interest,|ps| . 0.5 GeV/c, the differ-ence between the results of the two approximations (for thesame deuteron wave function) is quite small — see Sec.3.4and [8].

The structure functions for the scattering from the off-“+”-shell bound nucleons may depend on the variables of thisnucleon similarly to the case of the Bethe-Salpeter, or covari-ant Feynman, approach. In another language this dependencecan be interpreted as the presence of non-nucleonic degreesof freedom in the deuteron. With this is mind, we shall useEq.(16) as the basis for the results discussed in the follow-ing sections. Before focusing on specific model calculationsof the semi-inclusive deuteron structure functions, however,let us first turn our attention to the validity of the impulseapproximation, and the problem of final state interactions inparticular.

2.2 Final state interactions

In the kinematical region defined by (9) the contribution ofdirect processes, where a nucleon is produced in theγ∗N in-teraction, is negligible [8]. Therefore within the framework ofthe distorted wave impulse approximation (DWIA)1 the totaldeuteron tensorWµν

D can be expressed through the nucleonelectromagnetic currents as:

WµνD (x, αs, pT , Q

2) ≈∣∣∣∑〈D|pn〉〈XN | OIA +OFSI | XN〉∣∣∣2 ×

×WµνN (x, αs, pT , Q

2) ≡ SDWIA(αs, pT ) ××Wµν eff

N (x, αs, pT , Q2), (24)

whereOIA is the impulse approximation operator, whileOFSI describes the soft final state interactions between the fi-nal hadronic products and the spectator nucleon. The functionSDWIA(αs, pT ) now represents the spectral function distortedby FSI effects.

Analysis [21] of the recent high energy deep-inelastic scat-tering data on slow neutron production [22] is rather indicativethat even in heavy nuclei final state interactions are small, sothat the average number of hadrons which reinteract in thetarget does not exceed unity. This indicates that the systemwhich is produced in theγ∗N interaction is quite coherentand interacts at high energies with a relatively small effectivecross section,

σeff ¿ σNN . (25)

Such a situation allows one to factorize theγ∗N interactionand use the calculation of FSI ineD → e p n processes as aconservative upper limit.

1 Note that the DWIA approach works best at extreme backward kinemat-ics, Eq.(26) below, where final state interactions have a small contribution,Eq.(27)

The final simplification which we can gain is to considerthe extreme backward kinematics where, in addition to (9), wealso require:

pT ≈ 0. (26)

In this case it can be shown [23, 24] that:

SDWIA(αs, pT ≈ 0)∼ S(αs, pT ≈ 0)×

×[

1− σeff (Q2, x)8π < r2

pn >

|ψD(αs, < pT >)ψD(αs,0)|S(αs, pT ≈ 0)/

√Es Es(< p2

T >)

],

(27)

where< r2pn > is the average separation of the nucleons

within the deuteron,Es is the spectator nucleon energy, andEs(< p2

T >) =√M2 + ps 2

z + < p2T > is the energy evalu-

ated at the average transverse momentum< p2T >

1/2∼ 200–300 MeV/c transferred for the hadronic soft interactions witheffective cross sectionσeff . The steep momentum depen-dence of the deuteron wave function,|ψD(αs, < pT >)| ¿|ψD(αs, pT ≈ 0)|, ensures that FSI effects are suppressed inthe extreme backward kinematics, in which case the originalimpulse approximation expression forS(αs, pT ) can be usedrather thanSDWIA(αs, pT ). Finally, expressing the electro-magnetic tensor of the nucleon,Wµν

N , through the effectivenucleon structure functionF eff1N andF eff2N , we can then writefor the deuteron tensor:

WµνD ≈ S(αs, pT )

{−(gµν +

1Q2

qµqν

)1MF eff1N ×

×(

x

2− αs, pT , Q

2

)+

(pµ +

p · qQ2

qµ

)×

×(pν +

p · qQ2

qν

)1

νM2F eff2N

(x

2− αs, pT , Q

2

)}. (28)

DefiningFNL,T,TL,TT to be the semi-inclusive structure func-tions in (2) with the spectral function factored,

FDL,T,TL,TT = S(αs, pT ) FNL,T,TL,TT , (29)

one can express the semi-inclusive nucleon functions in termsof the effective structure functions of the nucleon as:

FNL (x, αs, pT , Q2) = − sin2 δ

ν

MF eff1N

(x

2− αs, pT , Q

2

)+ (1 + cosδ)2

(αs +

p · qQ2

αq

)2ν

νF eff2N

(x

2− αs, pT , Q

2

),

(30a)

FNT (x, αs, pT , Q2) = 2F eff1N

(x

2− αs, pT , Q

2

)+p2T

M2

M

νF eff2N

(x

2− αs, pT , Q

2

), (30b)

FNTL(x, αs, pT , Q2) = 2(1 + cosδ)

pTM

(αs +

p · qQ2

αq

)×

× ν

νF eff2N

(x

2− αs, pT , Q

2

), (30c)

FNTT (x, αs, pT , Q2) =

sin2 δ

2p2T

M2

ν

νF eff2N

(x

2− αs, pT , Q

2

),

(30d)

103

where

αq ≡ν − |q|M

, (31a)

ν ≡ p · qM

= |q|1 + cosδ2

αs + αqM2 + p2

T

2αsM. (31b)

Note that within the impulse approximation the Callan-Grossrelation between theF eff1N andF eff2N structure functions is pre-served. Therefore the experimental verification of this relationcould serve as another way to identify FSI effects.

Equations (30) and (31) show that at fixedx andQ2→∞,whenαq → 0 the longitudinal structure function,FNL , doesnot depend explicitly on the transverse momentum of the nu-cleon. It containspT dependence only in the argument of thebound nucleon structure functions, which arises from the pos-sible nuclear modifications of the nucleon’s parton distribu-tions. The above argument, and the fact that FSI of hadronicproducts withpX & 1 GeV/c practically conserveαs, allowsone to conclude that the FSI effect onFNL is minimal.

On the other hand the functionsFT , FTL andFTT doexplicitly depend on the spectator transverse momentum andtherefore the hadronic reinteractions in the final state maystrongly affect these structure functions. Such a situation sug-gests that a separate study of the complete set of the structurefunctions will allow one to investigate the effects of final stateinteractions in the deep-inelastic (e, e′N ) reactions. Note thatfor the production of spectators withpT > 0, FSI effects arenot likely to depend strongly onx for x > 0.1. This is becauseatx > 0.1 the essential longitudinal distances in deep-inelasticscattering are small. Therefore for these kinematics one ex-pects thatσeff (Q2, x) ≈ σeff (Q2). The effect of FSI will beinvestigated in the dedicated deep-inelastic scattering experi-ments at HERMES which will measure theA-dependence offorward produced hadrons.

These observations enable us to conclude that in the kine-matic region defined by Eqs.(2), (9) and (26), where the con-tribution of FNL is enhanced and FSI effects are small, thedifferential cross section (4) can be written:

dσeD→epX

dxdW 2d(logαs)d2pT

≈ 2α2em

Q4(1− y)

S(αs, pT )2− αs

[FNL +

Q2

2q2

ν

MFNT

]=

2α2em

Q4(1− y)S(αs, pT )F eff2N

(x

2− αs, pT , Q

2

)(32)

where we have made the transformationdQ2/x→ dW 2, withW 2 = −Q2 + 2Mν +M2.

Based on the expectation that FSI effects should notstrongly depend onx, from Eq.(32) it may be advantageousto consider the ratio of cross sections relative to a givenx, inthe range 0.1–0.2, where the observed EMC effect in inclusivescattering is small [17, 25]:

G(αs, pT , x1, x2, Q2)

≡ dσ(x1, αs, pT , Q2)

dxdW 2d(logαs)d2pT

/dσ(x2, αs, pT , Q

2)dxdW 2d(logαs)d2pT

=F eff2N (x1/(2− αs), pT , Q2)

F eff2N (x2/(2− αs), pT , Q2). (33)

In our analysis we will consider only production of backwardnucleons,αs ≥ 1, to suppress contributions from the directprocesses where a nucleon is produced in theγ∗N interac-tion vertex. The more liberal condition, Eq.(9), is in realitysufficient [8].

Note also that for heavier nuclei the FSI becomes muchmore important in the limit of largex. As was demonstratedin [17], in this limit rescattering of hadrons produced in theelementary deep-inelastic scattering off the short-range cor-relation is dynamically enhanced, since the average value ofthe Bjorken-variable for this mechanism is≈ x, as opposedto x/(2− αs) for the spectator mechanism. In this sense thedeuteron target provides the best way of looking for the EMCeffect for bound nucleons.

3 Models

In this section we briefly summarize several models of theEMC effect which we use in our analysis and present theirpredictions for the tagged structure functions. The differencesbetween the models stem from dynamical assumptions aboutthe deformation of the bound nucleon wave functions, andfrom the fraction of the EMC effect attributed to non-baryonic(mesonic) degrees of freedom in nuclei — from the dominantpart in some versions of the binding model, to the modelswhere non-baryonic degrees of freedom play no role, as inthe color screening or QCD radiation models. (Other modelswhich have been used in studies of semi-inclusive DIS includethe six-quark cluster models discussed in Refs.[26, 27].)

3.1 Binding models

One of the simplest of the early ideas proposed to explainthe nuclear EMC effect was the nuclear binding model, inwhich the main features of the EMC effect could be under-stood in terms of conventional nuclear degrees of freedom— nucleons and pions — responsible for the binding in nu-clei [17, 28, 29, 30, 31, 32, 33, 34, 35]. Within the formal-ism of Sec.2.1, the inclusive nuclear structure function in theEMC ratio, Eq.(2), is expressed through a convolution of thenuclear spectral function and the structure function of thebound (off-shell) nucleon (c.f. Eq.(16)). Contributions fromDIS from the pionic fields themselves, which are needed tobalance overall momentum conservation, were considered inRefs.[28, 30, 31, 32, 34], however, their role is most evidentonly at smallx (x . 0.2).

The bulk of the suppression of the EMC ratio (2) atx ∼ 0.6in the binding model can be attributed to the fact that the aver-age value for the interacting nucleon light-cone fraction is lessthan unity. For the case of the deuteron, this corresponds to theaverage spectator light-cone fraction〈αs〉 > 1, which is con-trary to what one would have from Fermi motion alone, wherethe averageαs is < 1. A relatively minor role is played bythe structure function of the bound nucleon itself — the onlyrequirement is that it be a monotonically decreasing functionof x [8, 28, 35]. This is clearly the case for the on-shell struc-ture function, and since the off-shell behavior of the boundnucleon structure function is unknown, most early versions ofthe binding model simply neglected the possible dependence

104

onp2 with the expectation that it is not large for weakly boundsystems. Only very recently has the issue of off-shell depen-dence in the bound nucleon structure function been addressed[9, 10], where the first attempts to construct models for thep2

dependence ofF eff2N were made. Note that a consequence ofassuming the absence of any off-shell effects inF eff2N is thatthe tagged structure function ratioG in Eq.(33), normalizedto the corresponding ratio for a free proton, would be unity(see Fig.5 below). Any observed deviation of this ratio fromunity would therefore be a signal of the presence of nucleonoff-shell effects.

If in a dilute system such as the deuteron the nucleon off-shell effects do not play a major role (at least atx . 0.7),one could expand the effective nucleon structure function in aTaylor series aboutp2 = M2:

F eff2N (x, p2, Q2) = F2N (x,Q2) + (p2−M2) ×

× ∂F eff2N (x, p2, Q2)∂p2

∣∣∣∣∣p2=M2

+ · · · . (34)

Here the off-shell dependence is determined, to orderp2/M2,entirely by the derivative ofF eff2N with respect top2. In orderto model this correction a microscopic model of the nucleonstructure is required [9, 10, 12]. In any generic quark-partonmodel, the effective nucleon structure function can be writ-ten as an integral over the quark momentumpq of the quarkspectral functionρ:

F eff2N (x, p2, Q2) =∫dp2q ρ(p2

q, p2, x,Q2). (35)

To proceed from (35) requires additional assumptions aboutthe quark spectral function. The simplest is to assume that thep2 andp2

q dependence inρ is factored [10, 12], which then

leads to an explicit constraint on thep2 derivative ofF eff2Nfrom baryon number conservation in the deuteron:∫ 1

0

dx

x

∂F eff2N (x, p2, Q2)∂p2

∣∣∣∣∣p2=M2

= 0. (36)

This then allows∂F eff2N (x, p2, Q2)/∂p2 to be determinedfrom the x-dependence of the on-shell structure functionF2N (x,Q2) in terms of a single parameter, the squared massof the intermediate state “diquark” system that is spectator tothe deep-inelastic collision, (p− pq)2. One can obtain a goodfit to the on-shell nucleon structure function data in terms ofthis model if one restricts the spectator “diquark” mass to bein the range (p− pq)2 ≈ 2− 4 GeV2 [10].

A more microscopic model which does not rely on the fac-torization of thep2 andp2

q dependence inρ was discussed in[9]. The quark spectral function there was determined entirelyfrom the dynamics contained in the nucleon–quark–spectator“diquark” vertex function,Γ (p, pq). Within the approxima-tion discussed in Sect. 2.1, taking the positive energy nucleonprojection only,

ρ(p2q, p

2, x,Q2)→ Tr[(6p +M ) Γ (p, pq) (6pq −mq)

−1×× 6q (6pq −mq)

−1 Γ (p, pq)], (37)

wheremq is the quark mass. Angular momentum conservationallows two forms for the vertex functionΓ , namely scalar and

pseudo-vector. In Refs.[9, 11, 14] it was found that the on-shelldata could be well described in terms of only a few of the manypossible Dirac structures forΓ . In particular, the vertices werechosen to be∝ I andγαγ5. The momentum dependence of thevertex functions, on the other hand, is more difficult to derive,and must in practice be either parameterized or calculated bysolving bound state Faddeev equations in simple models of thenucleon [36]. In order to obtain realistic static properties of thenucleon, and to account for the bound nature of the nucleonstate, the vertex functions must have the form:

Γ (p, pq) ∝(m2q − p2

q

)(Λ2− p2

q

)n , (38)

where the parametersΛ andn are fixed by comparing withthe quark distribution data, and the overall normalization isfixed by the baryon number conservation condition for boththe nucleon and deuteron structure functions [9, 11, 14]. Inthe comparisons in Sec.3.4 we use the parameters from theanalysis of [9].

3.2 Color screening model of suppression of point-likeconfigurations in bound nucleons

A significant EMC effect in inclusive (e, e′) reactions occursfor x ∼ 0.5–0.6 which corresponds to the high-momentumcomponent of the quark distribution in the nucleon. Thereforethe EMC effect in thisx range is sensitive to a rather rare com-ponent of the nucleon wave function where 3 quarks are likelyto be close together[17, 25]. It is assumed in this model that forlargex the dominant contribution toF2N (x,Q2) is given bythe point-like configurations (PLC) of partons which weaklyinteract with the other nucleons. Note that due to scaling viola-tionF2N (x,Q2) atx & 0.6,Q2 & 10 GeV2, is determined bythe nonperturbative nucleon wave function atx & 0.7. Thus itis actually assumed that in the nonperturbative nucleon wavefunction point-like configurations dominate atx & 0.7. Thesuppression of this component in a bound nucleon is assumedto be the main source of the EMC effect in inclusive deep-inelastic scattering [17, 25]. Note that this suppression doesnot lead to a noticeable change in the average characteristicsof nucleons in nuclei [25].

To calculate the change of the probability of a PLC in abound nucleon, one can use a perturbation series over a smallparameter,κ, which controls corrections to the description of anucleus as a system of undeformed nucleons. This parameter istaken to be the ratio of the characteristic energies for nucleonsand nuclei:

κ =| 〈UA〉/∆EA |∼ 1/10, (39)

where 〈UA〉 is the average potential energy per nucleon,〈UA〉 |AÀ1≈ −40 MeV, and∆EA ≈M∗ −M ∼ 0.5 GeV isthe typical energy for nucleon excitations within the nucleus.Note that∆ED & 2(M∆ −M ) ∼ 0.6 GeV, since theN −∆component in the deuteron wave function is forbidden due tothe zero isospin of the deuteron.

To estimate the deformation of the bound nucleon wavefunction we consider a model where the interaction betweennucleons is described by a Schr¨odinger equation with potentialV (Rij , yi, yj) which depends both on the inter-nucleon dis-tances (spin and isospin of nucleons) and the inner variablesyi

105

andyj , whereyi characterizes the quark-gluon configurationin thei-th nucleon [17, 25, 37]. The Schr¨odinger equation canbe represented as:− 1

2mN

∑i

∇2i +

′∑i,j

V (Rij , yi, yj) +∑i

H0(yi)

××ψ(yi, Rij) = Eψ(yi, Rij). (40)

HereH0(yi) is the Hamiltonian of a free nucleon. In the nonrel-ativistic theory of the nucleus the inter-nucleon interaction isaveraged over allyi. Thus the nonrelativisticU (Rij) is relatedto V as:

U (Rij) =∑

yi,yj ,yi,yj

〈φN (yi)φN (yj) ×

× | V (Rij , yi, yj , yi, yj) | φN (yi)φN (yj)〉, (41)

whereφN (yi) is the free nucleon wave function. The unpertur-bated wave function is the solution of (40) with the potentialVreplaced byU . Treating (U−V )/(Ei−EN ) as a small param-eter, whereEi is the energy of an intermediate excited nucleonstate, one can calculate the dependence of the probability tofind a nucleon in a PLC to the momentum of a nucleon insidethe nucleus. One finds that this probability is suppressed ascompared to the similar probability for a free nucleon by thefactor [25]:

δA(k2) ≈ 1− 4(k2/2M + εa)/∆EA, (42)

where∆EA = 〈Ei − EN 〉 ≈ M∗ −M , in first order of theperturbation series. An estimate of higher order terms gives[17]:

δA(k2) = (1 +z)−2, z = (k2/M + 2εA)/∆EA. (43)

Thex dependence of the suppression effect is based on theassumption that the PLC contribution in the nucleon wavefunction is negligible atx . 0.3, and gives the dominantcontribution atx & 0.5 [25, 38]. We use a simple linear fit todescribe thex dependence between these two values ofx [38].One can then obtain an estimate forRA in Eq.(2) for largeAatx ∼ 0.5,

RA(x) |x∼0.5∼ δA(k2) ≈ 1 +4〈UA〉∆EA

∼ 0.7− 0.8, (44)

since here Fermi motion effects are small. The excitation en-ergy∆EA for the compressed configuration is estimated as∆EA ∼ (M (1400)−M )−(M (1680)−M ) ∼ (0.5–0.8) GeV,while 〈UA〉 ≈ −40 MeV. Since〈UA〉 ∼ 〈ρA(r)〉 the modelpredicts also theA dependence of the EMC effect, which isconsistent with the data [17].

For the deuteron target we can deduce from Eq.(44) using〈UD〉/〈UFe〉 ∼ 1/5,

RD(x,Q2) |x∼0.5≈ 0.94− 0.96. (45)

This number may be somewhat overestimated because, as dis-cussed above, due to the isoscalarity ofD, low-energy exci-tations in the two-nucleon system are forbidden, leading to∆ED > ∆EA.

This model represents one of the extreme possibilitiesthat the EMC effect is solely the result of deformation ofthe wave function of bound nucleons, without attributing any

extra momentum to be carried by mesons. A distinctive fea-ture of this explanation of the EMC effect is that the defor-mation of the nucleon should vary with inter-nucleon dis-tances in nuclei (with nucleon momentum in the nucleus). For|k| ∼ 0.3− 0.4 GeV/c the deviation from the conventionalquantum-mechanical model of a deuteron is expected to bequite large (factor∼ 2). Actually, the size of the effects maydepend not onk2 only but onpT andαs separately, becausethe deformation of a bound nucleon may be more complicatedthan suggested by this simple model.

3.3 QCD radiation, quark delocalization.

It was observed in [39, 40] that the original EMC data couldbe roughly fitted as:

1AF2A(x,Q2) =

12F2D

(x,Q2ξA(Q2)

), (46)

with ξFe(Q2) ≈ 2 for Q2 ≈ 20 GeV2, the so-called dynam-ical rescaling. The phenomenological observation has beeninterpreted as an indication that gluon radiation occurs moreefficiently in a nucleus than in a free nucleon (at the sameQ2)due to quark delocalization, either in a bound nucleon (or intwo nearby nucleons) [39, 41, 42, 43] or in the nucleus as awhole [41].

TheQ2 dependence ofξ(Q2) follows from the require-ment that both sides of (46) should satisfy the QCD evolutionequations. In the leading logarithmic approximation one has:

ξA(Q2) = ξA(Q20)αQCD(Q2

0)/αQCD(Q2). (47)

Experimentallyd lnF2D(x,Q2)/d lnQ2 is positive ifx > x0and negative ifx < x0, wherex0 = 0.15± 0.05. So Eq.(46)predicts that the EMC effect should vanish atx ≈ x0.

If the confinement size in a nucleus,λA, is larger than thatin a free nucleon, one may expect that theQ2 evolution of theparton distributions in nuclei (the bremsstrahlung of gluonsand quarks) may start atQ2

0(A) < Q20(N ). To reproduce (46)

one should have:

Q20(A)/Q2

0(N ) = [ξ(Q20(A))]−1. (48)

Assuming on dimensional grounds that the radii of quark lo-calization in a nucleus,λA, and in a nucleon,λN , are relatedvia:

Q20(A)λ2

A = Q20(N )λ2

N (49)

to the scale for the onset of evolution,Q20, one finally obtains:

λA/λN ≈ ξ(Q20(A))

1/2. (50)

A fit to the original EMC data using (46) leads toξFe(20GeV2)=2. Forµ2

Fe = 0.67 GeV2 andΛMS ≈ 250 MeV thiscorresponds to:

λFe/λD ≈ 1.15 λA∼200/λD ≈ 1.19. (51)

Since in this model the delocalization is approximately pro-portional to the nuclear density, one can expect that the ef-fect is also proportional to thek2 of a bound nucleon. Fixingthe parameters of the model to fit the Fe data, and assum-ing λ(k)/λ = 1 +ak2 we obtaina ≈ 0.4/ < k2 >Fe, where< k2 >Fe∼ 0.08 GeV2/c2. Using this expression forλ(k) one

106

can calculateξd(Q2, k) according to (47)–(50) and express theeffective structure function of a bound nucleon as:

F eff2N (x, αs, pT , Q2) = F2N

(x,Q2ξ(Q2, k)

), (52)

wherek is defined throughαs andpT according to (23).

3.4 Numerical estimates

Comparison of predictions of the models for the nuclear EMCeffect considered in the preceding Sections is most meaningfulin the kinematic range where, firstly, FSI effects are small, andsecondly, the instant-form and light-cone prescriptions for thedeuteron spectral function lead to similar results.

Direct calculation of the FSI contribution to the cross sec-tion would require knowledge of the full dynamics of the finalN–X system, which is a practically impossible task given thepresent level of understanding of nonperturbative QCD. How-ever, it is possible to estimate the uncertainty which would beintroduced through neglect of FSI, by using the calculation ofFSI effects in the high-energyd(e, e′p)n (break-up) reactionin [23, 24], and replacing thep–n rescattering cross section byan effective cross section for thep(n)–X interaction, Eq.(27).For the effective cross sectionσeff one can use the results ofthe recent analysis [21] of soft neutron production in the high-energy deep-inelastic scattering of muons from heavy nuclei[22], which yielded an upper limit ofσeff ≈ 20 mb. To be onthe conservative side, in the following estimates we thereforeuse the value ofσeff ≈ 20 mb suggested by string models ofFSI (for a recent summary see [44]). Furthermore, by retainingonly the imaginary part of the spectator nucleon rescatteringamplitude, one obtains an upper limit of the FSI effect, sincethe real part will contribute to elastic rescattering only, effec-tively suppressing the value ofσeff . In Fig.1 we illustrate theαs dependence of the ratio of the (light-cone) spectral functionincluding FSI effects within the DWIA, Eq.(27), to that cal-culated without FSI effects. At extreme backward kinematics(pT ≈ 0) one sees that FSI effects contribute less than∼ 5%to the overall uncertainty of thed(e, e′N )X cross section forαs . 1.5. As mentioned above, this number can be consid-ered rather as an upper limit on the uncertainties due to FSI.At largerpT (& 0.3 GeV/c), and smallαs (≈ 1), the doublescattering contribution (which is not present for the extremebackward case in Eq.(27)) plays a more important role in FSI[23]. Because its sign is positive, it tends to cancel some ofthe absorption effects of FSI at largepT (for a detailed discus-sion of the double scattering contribution in FSI see Ref.[23]).Note also that the FSI effects do not change significantly withenergy for fixedQ2. Thus, for the ratios discussed, where thecross sections are compared for the sameQ2 but differentx,the changes due to FSI effects are even smaller.

To extract unambiguous information from the semi-inclusive cross section ratios on the medium modificationsof the nucleon structure discussed in this Sect. requires oneto establish the regions of kinematics where the differencesbetween the various prescriptions for the deuteron spectralfunction are minimal. In Fig.2 we illustrate theαs and pTdependence of the ratio of spectral functions calculated inthe light-cone (22) and instant-form (17) approaches. ForpT ≤ 0.1 GeV/c the light-cone and instant-form predictionsdiffer up to the 20% for the entire range ofαs ≤ 1.5. However,

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1 1.1 1.2 1.3 1.4 1.5 1.6αs

SDW

IA/S

pT=0.0

pT=0.1

pT= 0.2

pT= 0.3

pT= 0.4

Fig. 1. Theαs dependence of the ratio of cross sections calculated with FSIeffects within the DWIA, and without FSI effects. The curves correspond todifferent values of the spectator nucleon transverse momentum (in GeV/c)

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1 1.1 1.2 1.3 1.4 1.5 1.6αs

SLC

/SIF

pT= 0.4

pT= 0.3

pT= 0.2

pT=0.1

pT= 0

Fig. 2.Theαs dependence of the ratio of cross sections calculated within thelight-cone and instant-form approaches. The curves correspond to differentvalues of the spectator nucleon transverse momentum (in GeV/c)

choosing the isolated values ofαs ≤ 1.2 orαs ≈ 1.4 one canconfine the uncertainty in the spectral function to within 10%,which will offer the optimal conditions in which to study thenucleon structure modification.

In Fig.3 we show theαs dependence of the ratio of theeffective proton structure function,F eff2p , for extreme back-ward kinematics,pT = 0, to the structure function of a freeproton. As expected, the suppression of the ratio in the ver-sion of the binding model with explicit nucleon off-shell cor-rections is quite small,. 10% forαs < 1.5, reflecting therelatively minor role played here by off-shell effects in thenucleon structure function. (Note that in versions of the bind-

107

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1 1.1 1.2 1.3 1.4 1.5αs

F2p

eff /F

2p

Q2 = 5 GeV2, x = 0.6

Fig. 3.Theαs dependence ofF eff2p /F2p forx = 0.6 andpT = 0.Dashed lineis the PLC suppression model,dottedis the rescaling model, anddot-dashedthe binding/off-shell model

ing model in which there is nop2 dependence in the effectivenucleon structure function this ratio would be unity.) The ef-fects in the PLC suppression and rescaling models in Fig.3are somewhat larger. For a neutron target one predicts similarresults, however, the neutron structure function is not as wellknown experimentally as the proton due to the absence of freeneutron targets (see Sect. IV). For this reason we restrict ourdiscussion to ratios of proton structure functions only.

In Fig.4 we illustrate the dependence ofF eff2p /F2p on thevariable (p2− 2Mε)/M2, wherep2 is the bound nucleon vir-tuality defined in Eq.(20) andε = −2.2 MeV is the deuteronbinding energy. The comparison in Fig.4 is done for differentvalues ofx andαs. It is noticeable that because of the neg-ligible amount of the PLC component in the nucleon wavefunction atx . 0.3, the PLC suppression model predicts nomodification of the structure function in this region. On theother hand, atx = 0.6 it predicts a maximal effect because ofthe PLC dominance in the nucleon wave function here.

Because the momentum (virtuality) dependent density ef-fect generates the modification of the bound nucleon structurefunctions in the PLC suppression and rescaling models, the ra-tios for these models atx & 0.5–0.6 vary similarly withαs and(p2− 2Mε)/M2, as does also the off-shell model ratio. How-ever the mechanism for such a variation is different, which isclearly seen in Fig.5, where thex dependence of the same ratiois represented at different values ofαs and fixedpT = 0. Thecurves for the off-shell model extend only tox ∼ 0.7 becausefor largerx values the approximations discussed in Sect.2.1involved in obtaining (16) become numerically less justified[9, 11].

To further reduce any uncertainties due to the deuteronspectral function in the model comparisons, we concentrateon the predictions for the ratioG(αs, pT , x1, x2, Q

2), definedin (33), of experimentally measured cross sections at two dif-ferent values ofx [25]. Since the functionG is defined by the

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

-0.5 0 0.5(p2 - 2Mε)/M2

F2p

eff /F

2p

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

-0.5 0 0.5(p2 - 2Mε)/M2

F2p

eff /F

2p

0.2

0.4

0.6

0.8

1

1.2

-0.5 0 0.5(p2 - 2Mε)/M2

F2p

eff /F

2p

0.2

0.4

0.6

0.8

1

1.2

-0.5 0 0.5(p2 - 2Mε)/M2

F2p

eff /F

2p

Q2 = 5 GeV2

αs=1.2, x=0.3 αs=1.4, x=0.3

αs=1.2, x=0.6 αs=1.4, x=0.6

Fig. 4. The (p2 − 2Mε)/M2 dependence ofF eff2p /F2p, for αs = 1.2 and1.4, andx = 0.3 and 0.6. Curves are as in Fig.3

0.8

0.85

0.9

0.95

1

1.05

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9x

F2p

eff /F

2p

0.6

0.7

0.8

0.9

1

1.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9x

F2p

eff /F

2p

Q2 = 5 GeV2

αs = 1.2

αs = 1.4

Fig. 5. Thex dependence ofF eff2p /F2p for αs = 1.2 and 1.4, withpT = 0.Curves are as in Fig.3

ratio of cross sections at the sameαs andpT , any uncertain-ties in the spectral function cancel. This allows one to extendthis ratio to larger values ofαs, thereby increasing the util-ity of the semi-inclusive reactions when analyzed in terms ofthis function. Figure 6 shows theαs andQ2 dependence ofG(αs, pT , x1, x2, Q

2) at pT = 0. The values ofx1 andx2 areselected to fulfill the conditionx1/(2−αs) = 0.6 (large EMCeffect in inclusive measurements) andx2/(2− αs) = 0.2 (es-sentially no EMC effect in inclusive measurements). Again,the PLC suppression andQ2 rescaling models predict a muchfaster drop withαs than does the binding/off-shell model,where theαs dependence is quite weak.

108

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1 1.1 1.2 1.3 1.4 1.5 1.6αs

Gef

f /Gp

Q2 = 5 GeV2

Fig. 6.Theαs dependence ofG(αs, pT , x1, x2, Q2), withx1 = x/(2−αs) =

0.6 andx2 = x/(2− αs) = 0.2, for pT = 0.Geff (αs, pT , x1, x2, Q2) is

normalized toGeff (αs, pT , x1, x2, Q2) calculated with the free nucleon

structure function. Curves are as in Fig.3

4 Extraction of the neutron/proton structure functionratio

The presence of an EMC effect in the deuteron leads to sub-stantial suppression of the deuteron structure function at largex compared to what one would expect from models withoutnon-nucleonic degrees of freedom. For example, the estimateof [25],

F2D(x,Q2)F2p(x,Q2) + F2n(x,Q2)

≈ 14

2F2A(x,Q2)AF2D(x,Q2)

∣∣∣∣A∼60;0.3<x<0.7

(53)

which is valid for a rather wide class of models in which theEMC effect is proportional to the mean value ofp2 in nuclei,gives a ratio∼ 3–5% below unity atx ∼ 0.6–0.7. Calculationsof the deuteron structure function in models in which bindingeffects are explicitly taken into account also produce similareffects [11].

Inclusion of the EMC effect in the extraction of the neutronstructure function from the inclusiveeD scattering data leadsto significantly larger values forF2n/F2p than the 1/4 valueobtained in analyses in which only Fermi motion is included.The values forF2n/F2p with inclusion of the EMC effect atx ∼ 0.6 [45, 46, 47] are in fact much closer to the expectationof 3/7 from perturbative QCD, predicted by Farrar and Jackson[48]. Therefore observation of a value ofF2n/F2p higher thanthe 1/4 extracted from inclusive data in the early analyseswould by itself serve as another proof of the presence of theEMC effect in the deuteron.

Although other methods to obtain the large-x n/p ratio(or thed/u ratio) have been suggested, none has so far beenable to clearly discriminate between the differentx→ 1 limitsfor F2n/F2p (namely,d/u → 0, which is the minimal possi-ble value allowed in the parton model, which corresponds toF2n/F2p → 1/4, andd/u→ 1/5 in perturbative QCD [48]).

For example, withν andν beams on proton targets one can inprinciple measure theu andd quark distributions separately,however, the statistics inν experiments in general are rela-tively poor. Another possibility would be to extract thed/uratio from charged lepton asymmetries at large rapidities, inW -boson production inpp scattering [49], although here itmay be some time still before a sufficient quantity of large-rapidity events at the CDF at Fermilab are accumulated.

On the other hand, with tagged deuteron experimentsplanned for HERMES, a study of the tagged structure func-tions may allow a resolution of this ambiguity [6, 8, 50]. It isimportant that the ratio of tagged structure functions interpo-lated to the nucleon pole should be exactly equal to the freenucleon ratio — this is the analog of the Chew-Low interpo-lation for the pion case:

F2n(x,Q2)F2p(x,Q2)

≈ F eff2n (x/(2− αs), pT , Q2)

F eff2p (x/(2− αs), pT , Q2)

∣∣∣∣∣αs≈1,pT≈0

. (54)

In practice the data cannot be accumulated for too smallp.However, we observed above that deviations of the ratio fromthe free limit is proportional top2 with a good accuracy. Hence,if one samples the data as a function ofp2 interpolation to thepolep2 −M2 = 2Mε should be smooth. In practice, consid-ering momentum intervals of 100–200 MeV/c and 200–350MeV/c would be sufficient. A potential problem with (54) isthat at very largex (x & 0.7) the factorization approximationitself breaks down and higher order corrections to (16), whichare∝ p4, must be included if one wants accuracy to within afew %. To be on the safe side one should therefore restrict theanalysis to smaller spectator momenta, below 200 MeV/c.

5 Conclusion

Despite the many years of study of the deviations from unityof the ratios of nuclear to deuteron cross sections in inclu-sive high-energy scattering, we have been unable to isolatethe microscopic origin of the nucleon structure modificationin the nuclear environment [1, 2, 3, 4, 5] — the effect canbe described in terms of a number of models based on quitedisparate physical assumptions. In this paper we have arguedthat by allowing greater accessibility to kinematic variables notavailable in inclusive reactions, semi-inclusive deep-inelasticprocesses offer the possibility to make further progress in un-derstanding the origin of the nuclear EMC effect.

In particular, measurements of tagged structure functionsof the deuteron can probe the extent of the deformation of theintrinsic structure of a bound nucleon. Taking ratios of semi-inclusive cross sections at different values ofx enables thecancellation in the tagged structure function ratio, Eq.(33), ofthe dependence on the deuteron wave function, thus permit-ting the nucleon structure to be probed directly. Our resultsshow that possible contamination of the signal due to the finalstate interactions of the spectator nucleon with the hadronicdebris can be minimized by tagging only on the slow back-ward nucleons in the target fragmentation region. This maythen allow one to discriminate between models in which theEMC effect is attributed entirely to the nucleon wave functiondeformation, and ones in which the effect arises from moretraditional descriptions in terms of meson–nucleon degrees offreedom associated with nuclear binding.

109

As a by-product of the semi-inclusive measurements, onemay also be able to extract information on the large-xbehaviorof the neutron to proton structure function ratio, by detectingrecoiling protons and neutrons with small transverse momen-tum in the extreme backward kinematics. Extracting this ratiofrom inclusiveep andeD data is fraught with large uncertain-ties arising from different treatments of the nuclear physics inthe deuteron. Observation of an asymptotic value forF2n/F2pwhich is larger than the ‘canonical’ 1/4 would in itself be proofof the presence of an EMC effect in the deuteron.

First information about the spectra of backward nucleonsin eD scattering is likely to come from the Jefferson Lab ex-periment # 94-102 [6], and from the HERMES experiment atHERA, where the necessary counting rate may be achieved af-ter an upgrade of the detector [51]. Having two energy rangeswould be very useful for checking the basic production mech-anism, and understanding backgrounds and corrections due tofinal state interactions, which are likely to depend substantiallyon the incident energy. Final state interaction effects can alsobe tracked by studying the production of nucleons from heaviernuclei. Probing the quark-gluon structure of short-range cor-relations with heavy nuclei targets should further enable one todetermine whether nucleon deformations predominantly de-pend on the nucleon momentum, or also onA.

We would like to thank L.L. Frankfurt, A.W. Thomas and G. van der Steen-hoven for many useful discussions and suggestions. We thank the Institute forNuclear Theory at the University of Washington for its hospitality and sup-port during recent visits, where part of this work was performed. This workwas supported by the U.S. Department of Energy grants DE-FG02-93ER-40762, DE-FG02-93ER-40771 and by the U.S.A. – Israel Binational ScienceFoundation Grant No. 9200126.

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