Historically, all oæ-mass-shell eæects in the DIS on the nucleons are divided into two. x Nm. 2pq = p 0 + p z

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1 Preprint Alberta-Thy On the Oæ-Mass-Shell Deformation of the Nucleon Structure Function. A.Yu. Umnikov and F.C. Khanna Theoretical Physics Institute, Physics Department, University of Alberta, Edmonton, Alberta T6G 2J1, and TRIUMF, 44 Wesbrook Mall, Vancouver, B.C. V6T 2A3, Canada L.P. Kaptari Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 14198, Dubna, Russia Abstract NUCL-TH The oæ-mass-shell behavior of the nucleon structure function, F2 N, is studied within an approach motivated by the Sullivan model. Deep inelastic scattering on the nucleon is considered in the second order in the pion-nucleon coupling constant, corresponding to the dressing of the bare nucleons by the mesonic cloud. The inclusive and semiinclusive deep inelastic processes on the deuteron involving oæ-shell nucleons are considered. A deformation of the mesonic cloud for the oæ-mass-shell nucleon, compared to the free one, generates observable eæects in deep inelastic scattering. In particular, it leads to the breakdown of the convolution model, i.e. the deuteron structure functions are not expressed through the free nucleon structure function. Analysis of the semi-inclusive deep inelastic scattering on the deuteron, in the spectator approximation, shows that this reaction opens new possibilities to study the role of the oæ-shell eæects in determining in detail the nucleon structure function. 1

2 1 Introduction Studying of the deep inelastic scattering èdisè of leptons by nuclei has created a special problem, description of the DIS on the oæ-mass-shell nucleons. By itself this problem is not something unique or new in nuclear physics, since the same problem of the oæ-mass-shell amplitudes arises in investigations of the reactions of any kind èsee e.g. ë1ëè. Historically, all oæ-mass-shell eæects in the DIS on the nucleons are divided into two classes. First, these are the so-called "binding eæects" ë2, 3ë. It was noticed that compared to the free nucleon structure function F N 2 èx N ;Q 2 è, the structure function of the oæ-mass-shell nucleon, the "bound nucleon", has a shift in the kinematical variables: x = Q2 2pq = x Nm p + p z ; è1è where Q 2 =,q 2, q is the 4-momentum transfer in the reaction, m is nucleon mass, p is the virtual nucleon momentum and x N =,Q 2 =è2q mè is the Bjorken scaling variable for free nucleon at rest. Since p émand averaging on the nucleus results in hp + p z i ém, the structure function of the bound nucleon is "shifted" to a smaller value of x N. The exception is the region of high x N, x é N ç 1, where due to the admixture of the p + p z ém, the "Fermi motion", the bound nucleon structure function is extended beyond the single nucleon kinematics. When only such kinematical eæects are taken into account the bound nucleon contribution to the nuclear structure function is given as a convolution of the free nucleon structure function with an eæective distribution function for the nucleons ë4, 5ë. Taking account of the binding eæects in this distribution function leads to a description of the EMC-eæect in the DIS ë2, 3ë. Second, the oæ-mass-shell eæects are all possible phenomena, other than the binding eæects, which make the oæ-mass-shell structure function diæerent from the on-mass-shell one. Such eæects are related to the fact that the nucleon has an internal structure and the structure of the oæ-mass-shell nucleon requires a diæerent description than just a kinematical shift in x. A systematic study of these oæ-mass-shell eæects started recently ë6, 7ë, though some eæects which do not reduce to the binding eæects were discussed earlier ë8ë. Following the tradition of these papers ë6, 7ë we call, through out this paper, the eæects of the second kind as "oæ-shell" eæects, distinguishing them from the binding eæects. 2

3 In ref. ë6ë the general form of the truncated nucleon tensor èsee Fig. 1èaèè is studied. The structure functions of physical states are then presented as Feynman diagrams with insertion of the truncated nucleon tensor. For the transverse unpolarized structure functions the form of the insertion is found to be: m ^WT èp; qè =^qç 1 èp; qè+^pç 2 èp; qè+mç 3 èp; qè; è2è where ç i are three diæerent scalar functions, while for the scattering on the point-like fermion only the term ç ^q exists ë5ë. Calculation of the nuclear structure functions, keeping only the ç ^q term leads to the existence only of the binding eæects and the convolution formula ë9, 1ë, which is a reasonable approximation, since the entire non-relativistic nuclear physics works with point-like nucleons. Non-uniform deformation of each of the three terms in eq. è2è oæ the mass-shell and the non-trivial spin structure of ^WT èp; qè lead to new eæects in the processes involving the oæ-mass-shell nucleons. 3

4 Figure 1. The truncated nucleon tensor in the pion-nucleon model. Graph for the dressed nucleon tensor èaè is the sum of graphs èbè-ècè, where virtual photon scatters on the bare nucleons and mesons. The investigations of refs. ë6, 7, 8ë are based on models for the nucleon structure, motivated by the constituent quark model, and they involve unknown quark-nucleon amplitudes which are parametrized to æt the nucleon structure function on the mass-shell. The oæ-shell 4

5 eæects are found to be only a few percent in the absolute value of the structure function of nuclei. Apparently the magnitude and behavior of these corrections are dependent onthe form of the parametrizations of the quark-nucleon vertices with parameters being æxed to describe the observables èstructure functionsè for the nucleon on the mass shell. Whereas the continuation to oæ-mass-shell region must be ruled by dynamics èthe equations of motion for participating æeldsè and æxed by some constraints, such as the Ward-Takahashi relations, etc, so as to provide consistency of the model with the calculation of any other observables. On the other hand, the smallness of the corresponding corrections makes it diæcult to test models experimentally in the usual deep inelastic experiments on nuclei. The circumstances mentioned above motivate us to continue investigations of the oæ-shell eæects in the nucleon structure function. In particular, we aim èiè to consider the dynamical model for the nucleon structure function other than the quark models of refs. ë6, 7, 8ë, so as to compare results of independent approaches to the problem and èiiè to discuss possible experiments, other than inclusive DIS, which can open new perspectives to study oæ-shell eæects. In the present paper we consider the pion-nucleon model for the structure function of the nucleon, which is motivated by the well-known Sullivan model ë12, 13, 14ë. We argue that the model is relevant for the case and discuss model ambiguities involved in the calculations. Then we calculate oæ-shell eæects in the inclusive and semi-inclusive DIS on the deuteron. The second reaction provides new opportunities for an experimental investigation of the oæ-shell eæects in the nucleon structure function. 2 The Pion-Nucleon Model for the Nucleon Structure Function The role of the pion cloud or, more generally, mesonic cloud in the formation of the peripheral structure of the nucleon structure function has been discussed widely ë12, 13, 14, 15ë. We will discuss only the pion cloud, since its contribution signiæcantly dominates over the contributions of the heavier mesons. The conventional analysis is based on corrections to the scattering on the nucleon, calculated in the lowest order on the pion-nucleon coupling 5

6 constant, ç g 2, èsee Fig. 1èbè-èdèè with pseudoscalar coupling: L int =,ig ç èèxèæ 5 ççèèxè; è3è Then the diagrams ècè and èdè in the pion-nucleon model are logarithmically divergent. The formal logic of the æeld theory requires renormalization by introducing structure functions of the "bare" nucleons and mesons and some counter-terms in such away astoprovide the correct value of the calculated structure functions where the A natural normalization point for the counter-terms is the nucleon mass shell. This is enough if we are going to consider oæ-mass-shell behavior of the nucleon structure function. However, this is not suæcient if we are intending to reach conclusions about contributions of the mesons to the structure function of the free nucleon. The physically motivated Sullivan model gives the possibility to estimate the contribution of the pion cloud to the free nucleon structure function. In the modern form ë13, 15ë the model is based on the graphs èbè-èdè in Fig. 1 and ingredients of the model include elementary structure functions of bare nucleons and mesons and the meson-nucleon vertex formfactors in the diagrams ècè and èdè. The vertex formfactors cut the "unphysical" high momenta of the pions and make the contribution of the diagrams ècè and èdè ænite. Physical picture corresponding to the DIS on the nucleon in the Sullivan model is the following. The nucleon is presented as a superposition of two states, the bare nucleon state and the nucleon plus one pion state. As a result, the physical nucleon is the point-like bare nucleon è"the nucleon core"è surrounded by the extended pion cloud. The quark-gluon degrees of freedom in this picture are "hidden" in the eæective hadron degrees of freedom and their presence is displayed through the elementary structure functions of the bare nucleons and mesons. This model is relevant to a study of the nuclear eæects in the nucleon structure function, since the NNpotential can be succesfully deæned in the same g 2 -approximation, the one-boson-exchange potential, as the dressing diagrams on Fig. 1. At the same time the oæ-mass-shell behavior of the structure functions is governed by the meson-nucleon dynamics and, therefore, is consistent with dynamics of the deuteron, which we assume to describe also in the mesonnucleon model ë17, 18, 16, 19ë. The vertex formfactor plays the crucial role in the Sullivan model of the nucleon structure function ë13, 15ë. Since without formfactors the one-loop diagrams ècè and èdè are divergent, 6

7 the cut-oæ parameters control the magnitude of the contribution of these diagrams to the nucleon structure function. On the other hand, for our purpose, to analyse the oæ-shell behavior of the nucleon structure function, we can work without formfactors, attributing all divergences to the renormalization of the structure functions of the bare nucleons and mesons. However, intending to make connection with other calculations, we perform all numerical estimates within the Sullivan model with formfactors. The form of vertex formfactor can be chosen as ë16ë: F èp; p è=fèèp, p è 2 èhèp 2 èhèp 2 è; è4è where p and p are incoming and outgoing nucleon momenta in the vertex, respectively, èp, p è is the pion momentum. The formfactors are normalized so that: fèç 2 è=1; hèm 2 è=1; è5è where ç is the pion mass. So for the on-mass-shell nucleon structure function the diagram ècè is regularized by single formfactor hèp è and diagram èdè by fèèp, p è 2 è. We accept the point-like behavior of the bare nucleons, while the extended structure is generated by the pion cloud, dressing diagrams ècè and èdè. Calculation of the diagrams èbè-èdè gives èwe consider the structure function F 2 è: ^F 2 èx; p 2 è= ^f N èy;p 2 è= ^f ç èy;p 2 è= Z 1 dy è ^f N èy;p 2 è ~ F N 2 è! x + y ^f ç èy;p 2 è F ~ ç 2 è!! x ; è6è y ^q ç æè1, yè+f N 1 2pq è ç +^pf N 2 èy;p2 è+mf N m èy;p2 è; è7è ^q 2pq f ç 1 èy;p 2 è+^pf ç 2 èy;p 2 è+mfmèy;p ç 2 è; è8è where ^q = q ç æ ç, etc, ~ F N èçè 2 is the bare nucleon èpionè structure function. The term æè1, yè arises from the diagram èbè, f N i from ècè and f ç i from èdè èi =1:::3è. The explicit form of the functions f N;ç i èy;p 2 è is presented in the Appendix A. If neglect the formfactors, these functions satisfy the condition f N i èy;p 2 è=f ç i è1, y;p 2 è; è9è 7

8 which heuristically can be obtained as a consequence of the probabilistic interpretation of the structure function ë13ë. However, inclusion of the formfactors èdiæerent for the diagrams ècè and èdè!è breaks the relation è9è, which leads to a violation of the charge andèor momentum conservation in the process ë13ë. The underlying reason is that in the covariant calculations it is impossible to introduce "symmetric" formfactor with respect to the nucleon and meson momenta. Other topic is the choice of the cut-oæ parameters in the formfactors. It is known ë13, 15ë that to have a reasonable physical interpretation of the calculated structure functions cut-oæ masses in the formfactor è4è should be signiæcantly smaller than it is found from an analysis of the meson-exchange potentials ë17, 18, 16, 19ë. To regulate the divergent diagram ècè, the formfactor is chosen ë13ë: è! æ 2 fèk 2 è= ç, ç 2 2 ; æ æ 2 ç, ç =1 GeV; è1è k 2 which corresponds to the formfactor for diagram èdè: è! æ 2 hèp 2 è= N, m 2 2 ; æ æ 2 N, N =1:475 GeV; è11è p 2 where parameter æ N is æxed to preserve the baryon charge ènot the momentum!è conservation. On the other hand, there is no reason to put a smaller cut-oæ mass in the formfactor corresponding to the external nucleon line for the diagram èc-dè, where cut-oæ mass should be ç 1:5, 2: GeV ë16ë. We have to stress once again that for the analysis of the oæ-mass-shell eæects in the nucleon structure functions we do not necessarily need the regulating formfactors in the divergent diagrams. We are introducing these formfactor only to relate to the well-known and widely discussed Sullivan model for the nucleon structure function. The dependence of our results on the choice of the cut-oæ mass, æ çèn è, in the loops in diagrams ècè and èdè is weak compared to the æ çèn è -dependence of the pion ènucleonè contribution to the free nucleon structure function. Moreover, to restore symmetry between nucleon and pion contributions into eq. è6è-è8è we use eq. è9è to deæne the nucleon contribution through the pion one. This way to proceed is supported by analysis ë13ë using the time-ordered perturbation theory, where it is shown that eq. è9è is valid and introducing the formfactors in the covariant calculations damages signiæcantly only the nucleon contribution. 8

9 The free nucleon structure function is deæned by inserting operator eq. è6è between Dirac spinors. The ænal expression coincides with the the result of the Sullivan model ë13, 15ë: F N 2 èx; p 2 = m 2 è=n N Z 1 dy è f N èy;m 2 è ~ F N 2 è! x + f ç èy;m 2 è F y ~ ç 2 è!! x ; è12è y where N N is the normalization factor deæned by the conservation of baryon number: Z1 N,1 N =1+ dyf N èy;m 2 è: è13è A direct calculation shows that è13è actually preserves the baryon number in the one-loop approximation, with or without èæ ç!1è formfactors. As a basic set of parameters for the, 1 ç :24. Therefore, for calculations we take the formfactor è1è ë13ë. In this case N N,1 bare structure functions of the nucleons and mesons we take a æt of the empirical structure functions of the free nucleons and pions. This gives a reasonable agreement of the calculated structure function è12è with the experimental data, since admixtures of the corrections ècè and èdè are not too large. 3 The Nucleon Contribution to the Deuteron Structure Functions Now we are in a position to calculate the nucleon contribution to the deuteron structure function. We start with a consideration of the inclusive DIS on the deuteron èfig. 2èaèè. The structure function for the deuteron, F D 2 èx D è, is deæned as a matrix element of the operator eq. è6è. A consistent way is to use the covariant amplitude for the deuteron, the Bethe- Salpeter amplitude ë19, 11ë or relativistic wave functions ë16ë, with the normalization of the amplitude, based on the deuteron charge, in the one-loop approximation. This will be done elsewhere. Here to test the method we utilize the usual non-relativistic wave function of the deuteron. The non-relativistic reduction of the covariant operator è6è is similar to earlier approaches ë9, 1, 7ë. Here we reduce only the 4 æ 4, Dirac structure of the operator è6è to the 2 æ 2, Pauli operators, keeping all kinematics in the relativistic form. 9

10 The deuteron structure function then is deæned by: F D 2 èx N è=n D Z è h^qi 2pq ~ F N 2 d 4 p è xn ç! MD=m Z ç dçæ ç, p + m + h^qi 2pq F è1è 2 è xn ç ;p2 ç jæ D èpèj 2 ç! + h^pif è2è 2 è! è!è xn ç ;p2 + hmif è3è xn 2 ç ;p2 ; è14è where N D is renormalization constant preserving the conservation of the baryon number in the deuteron, M D is the deuteron mass, p = M D, p m 2 + p 2 is the oæ-mass-shell energy of the nucleon, p + = p + p z and F èiè 2 è! Z 1 xn ç ;p2 = è dy ~F N 2 è xn yç! f N i èy;p 2 è+ ~ F ç 2 è xn yç! f ç i èy;p2 è è ; i =1:::3: è15è The brackets h:::i denotes the non-relativistic expression for the subsequent operators: ç h^qi 2pq = ç 1+ p z m ç è! ; çh^pi =2p + p + p2 ; çhmi =2mp + : è16è m It is anticipated that the oæ-shell eæects are small in the deuteron. This is a consequence of the fact that, on the average, the shift from the mass shell for the nucleon in the deuteron is small: hp 2 içm 2 è1, è3 æ 4è æ 1,2 è. In heavier nuclei this shift is apparently larger, up to ç 1è for nucleus like iron, which can lead to a more signiæcant eæect. However, it is interesting to ænd other possibilities to investigate the oæ-shell behavior of the nucleon structure functions. 1

11 Figure 2. The deep inelastic scattering on the deuteron: inclusive èaè and semiinclusive èbè. Let us consider now the semi-inclusive DIS on the deuteron. The spectator mechanism ë2ë for this reaction is presented in Fig. 2èbè. Even for non-relativistic momenta of the spectator nucleon the shift from the mass shell of the interacting nucleon can be much larger than in heavy nuclei! For instance, ç 2è for p s ç 3MeV=c and ç 4è for p s ç 5MeV=c.Thus semi-inclusive DIS provides a unique possibility to study the oæ-shell 11

12 behavior of the nucleon structure function. In particular, the oæ-shell eæects can be studied as a function of the shift from the mass shell èor spectator momentumè. In the non-relativistic approach, with disregard of the oæ-shell eæects, the structure function of the oæ-shell neutron, F n 2 èx; p sè, measured with the detection of the protonspectator has the form: ç F n 2 èx; p s è= 1+ p z m ç jæè,p s èj 2 F n 2 è! x ; è17è ç where ç = p + =m, p = M D, p m 2 + p s2, p z =,èp s è z. Dividing data by the æux-factor, è1 + p z =mè, which isknown from kinematics, and plotting F n 2 èx; p s è as a function of z = x=ç èz 2 è; 1èè we should have a result proportional to the free neutron structure function, F n 2 èzè, with coeæcient jæèp sèj 2, i.e. at any æxed p s the ratio of the measured structure function to the free neutron structure function should be constant. This conclusion has to be changed if there is a non-trivial p 2 -dependence of the nucleon structure function, i.e. if the oæ-shell eæects exist. The structure function F n 2 èx; p s è in this case is deæned by the equation similar to eq. è14è only without integration over p and with the correspondent isospin modiæcations. 4 Numerical Results and Discussion Prior to calculating the deuteron structure functions, let us qualitatively discuss the possible phenomena in the structure functions of the oæ-shell nucleon. Omitting details of the pseudoscalar coupling of the pions and nucleons we can estimate the amplitude of the nucleon to emit the virtual meson as where kinematics is deæned by Mèp; p è è 1 k 2, ç 2 ; p =èp ;pè; p =èe ; p 2, kè; E = rm 2 +è p 2, kè2 ; è18è k =èk ; p 2 + kè; k = p, E : è19è 12

13 For simplicity let us compare amplitudes for nucleon with p 1 ; èp 1 = p émè and "more virtual" nucleon with p 2 ; èp 2 =èp, æè; æ é è, and other components of p 1, p 2 and p are kept the same. The sign of the combination: M,1 èp 1 ;p è,m,1 èp 2 ;p è è,æ 2 + 2æèp o, E è é ; è2è controls relative magnitude of these two amplitudes. Since amplitudes è18è are negative, the relation è2è means that absolute value of the amplitude Mèp 2 ;p è is larger than Mèp 1 ;p è. It means that for a nucleon further from the mass shell an increase in the emission of virtual pions may be expected. In accordance with eq. è9è the role of virtual nucleons is also increased. On the other hand, eq. è13è implies that weight of the bare component is decreased. The maximums of both eæective distributions f N èyè and f ç èyè are at yé1, y ç :2,:3 for pions and y ç :7,:8 for nucleons, therefore contributions of both components are concentrated at smaller x than for the bare component, where f bare èyè è æè1, yè. As result, for the oæ-shell structure function we expect an increase at small x and a decrease at large x, compared to the structure function with smaller virtuality. These conclusions depend on the choice of the pseudoscalar coupling, the vertex formfactors and the Fermi motion in the deuteron. The parametrizations of the nucleon structure functions from ë21ë and pion structure function from ë22ë are used as input. The Bonn potential wave function for the deuteron ë18ë is utilized throughout. The formfactor for the external nucleon line of the diagrams ècè and èdè on Fig. 1, is taken of the form ë16ë: hèp 2 è= 2è ~ æ 2 N, m 2 è 2 2è ~ æ 2 N, m 2 èè ~ æ 2 N, p 2 è+èm 2, p 2 è 2: Since the role of this formfactor in the diagrams for the reaction with external probe is uncertain, results are presented both with è ~ æ 2 N =1:65 GeVè and without è ~ æ 2 N!1è the formfactor. The second case is our choice for basic set of parameters. The results for the deuteron structure function, F D 2 èxè, are presented in Fig. 3. The dotted line presents the calculation with disregard of the oæ-shell eæects, the convolution model. Solid curve is a result of calculations with full formulae è6è-è8è with our basic set of parameters. The dashed curve shows the eæect of the extra formfactor for the external line of the virtual nucleon. The additional formfactor slightly decreases the oæ-shell eæects, since 13 è21è

14 it "holds" the nucleon closer to the mass shell. These results conærm our estimates presented at the beginning of the present section. They are also in qualitative agreement with earlier results ë7, 8ë for xé:3, where the structure function of the nucleon in the deuteron suæers additional suppression in comparison with the usual convolution model. This mechanism, indeed, can be complimentary to the binding eæects in explaining the EMC-eæect. Corresponding eæective distribution functions of the pions are presented in the Fig. 4. "Meanvalue" distribution for the deuteron èdashed curveè diæers only slightly from the free nucleon distribution èsolid lineè. At the same time the deuteron distribution is very similar to the distribution from the semi-inclusive reaction at the spectator momentum p s = 1 MeVèc, the reason is the mean value of the nucleon momentum in the deuteron is ç 1,15 MeVèc, depending on the potential model. 14

15 Figure 3. The ratio of the deuteron structure function to the free nucleon structure function. Curves: solid - basic set of parameters; dashed - with additional formfactor for the oæ-mass-shell nucleon èæ g =1:65 GeVè; dotted -convolution model. Results for the semi-inclusive reaction with the proton spectator are shown in Fig. 5. Calculations for the ratio of the neutron structure function in the semi-inclusive reaction to the free neutron structure function are presented. the This ratio is obtained after exclusion of the èiè æux-factor è1 + p z =mè and èiiè weight jæèpèj 2 from the total structure functions. If 15

16 the ærst one is a procedure well-deæned by kinematics of the reaction, the second is rather ambiguous, since the wave function of the deuteron is strongly model dependent. However, it is worthwhile to compare the relative eæects at diæerent p s. If there is no oæ-shell eæect such a ratio would be just a constant, çjæèpèj 2. Otherwise it will have a slope as shown in Fig. 5. All curves in Fig. 5 are scaled for comparison. Note also that the æux-factors in the three matrix elements è16è are slightly diæerent, so after dividing by the factor è1 + p z =mè, the structure functions remain dependant on angle, ç s, of the spectator momentum relative to the q. This dependence is too weak to discuss in relation to the possible experiments and, furthermore, it is not clear if this angular dependence is just an artifact of the non-relativistic reduction. Here we choose ç s = ç=4. 16

17 Figure 4. The eæective distribution of the pion in the nucleon èthe basic set of parametersè. Curves: solid - free nucleon; dotted - nucleon in the deuteron; dashed - nucleon in the semi-inclusive reaction with diæerent spectator momentum, p s è1 - p s =:1 GeVèc; 2 - p s =:3 GeVèc; 3 - p s =:5 GeVèc; 4 - p s =:9 GeVècè. 17

18 Dependence of the oæ-shell eæects on the spectator proton momentum is shown in Fig. 5èaè. There are two competing mechanisms, the increase of the structure function at small x and decrease at medium x. To understand such a behavior let us consider the eæective distribution functions of the pions in the nucleon in the reaction èfig. 4è. A steady increase of the pion distribution function at small x with an increase of the virtuality of the nucleon is found. At medium and large x and high virtualities these distributions have a tendency to vanish. However, very high virtuality, or large spectator momentum ç 1 GeVèc, are probably beyond, or very close to the boundary, the applicability of the pion-nucleon model and the potential model for deuteron. 18

19 19

20 Figure 5. The ratio of the neutron structure function, measured in the semi-inclusive deep inelastic scattering on the deuteron, to the free neutron structure function. èaè Dependence on the spectator proton momentum. Curves are calculated with a basic set of parameters: dotted -p s =:1 GeVèc; solid - p s =:3 GeVèc; dashed -p s =:5 GeVèc; dot-dashed -p s =:9 GeVèc. èbè Sensitivity to the model assumptions. Curves for p s =:3 GeVèc: solid - basic set of parameters; dashed - with additional formfactor for the oæ-mass-shell neutron èæ g =1:65 GeVè; dotted -calculation with the distributions è23-25è with nucleon formfactor inside the loop èæ N =1:475 GeVè. Fig. 5èbè shows estimates of some of the model ambiguities involved into our calculations. In particular, all calculations give qualitatively the same behavior of the oæ-shell eæects. However, manipulation with the formfactors may lead to a suppression of the magnitude of the eæect. ènote that calculation for dotted curve, Fig. 4èbè, breaks the energy-momentum conservation in the reaction.è 5 Conclusions and Comments We have presented model calculations of the oæ-shell eæects in the deep inelastic scattering on the nucleons. In particular, 1. Truncated nucleon tensor has been calculated in the pion-nucleon model, motivated by the Sullivan model. The formulae explicitly contain the p 2 -dependence and allow an analysis of oæ-shell eæects in the nucleon structure functions. 2. Nucleon contribution in the deuteron structure function, F D 2, has been calculated, using non-relativistic wave function. The oæ-shell corrections are found to be rather small, but they can be complimentary to the binding corrections in the explanation of the EMC-eæect. 3. Semi-inclusive deep inelastic scattering on the deuteron has been considered in the spectator approximation with the proton spectator in the ænal state. It is found that this reaction provides new opportunities to study the oæ-shell eæects in the nucleon structure function. Even at non-relativistic momenta of the spectator, the oæ-shell 2

21 eæects for the struck nucleon are larger than an averaging in heavy nuclei and order of magnitude larger than averaging in the deuteron. This type of experiments would help to select models relevant to describe the structure functions of nucleons and, therefore, the nuclei. We did not consider here other type of mesonic corrections to the structure functions of deuteron ènucleiè, the contributions of meson exchange currents, which should be part of a consistent analysis of the DIS on nuclei as a system of interacting nucleon and meson æelds ë9, 1, 11, 23ë. However, as soon as the internal degrees of freedom of the nuclear constituents are ëdefrozen", such analysis becomes a non-trivial problem, since it is not clear how internal dynamics of the constituents interferes with the dynamics of the system. Anyhow, the physics here can be extremely interesting and there is much to learn about how to build a composite system from composite constituents. Other type of phenomenon not considered here and which can have an aæect on the deuteron structure functions at very small x, say xé:1, is the so-called nuclear shadowing ë24, 23ë. These corrections would cancel èor partially cancelè the enhancement of the deuteron structure function èsee Fig. 3è, x!, generated by the pions ë23, 25ë. We would like also to make some comments about the pion èmesonè physics in the DIS on nuclei. This topic has as a long history as studies of nuclear eæects in DIS starting from the famous EMC-eæect ë26ë. At some point itwas concluded that there are no excess pions in nuclei. It was based on simple estimates of the pionic contribution to the nuclear structure functions and probably a more correct conclusion has to be that something has been overlooked in these calculations. This point was recently re-examined in an interesting work ë27ë. Not going into details we would like to note that physics here can be even more intricate. For instance, the oæ-shell eæects in the pion structure function can be signiæcant as it was found in ë28ë. The last comment is related to the state of the experiment. There is an interesting potential in the study of the DIS on the deuteron in the semi-inclusive set up. In particular, the possibility to study the oæ-mass-shell behavior of the nucleon structure function is really unique. èother interesting physics could be studied as well ë2ë.è Such experiments, for instance, at CEBAF ë29ë, would be beneæcial both for the theory of nuclear eæects in the DIS and, perhaps, for the more fundamental theories èmodelsè of the structure of the nucleon. 21

22 6 Acknowledgments Authors thank L. Celensa, F. Gross, K. Kazakov, S. Kuhn, S. Kulagin, W. Melnitchouk, C. Shakin, A. Thomas, W. Van Orden and W. Weise for discussions which clarify number of questions. The research is supported in part by the Natural Sciences and Engineering Research Council of Canada. 7 Appendix A Light cone variables: p =èp ;pè=èp + ;p, ;p? è; p æ = p æ p z ; p 2 = p + p,, p?2 ; pp = 1 2 èp +p, + p,p + è, p?p? è22è d 4 p = 1 2 dp +dp, dp? ; dp? = p? dp? dæ: The explicit expressions for functions f N èçè i èy;p 2 è; i =1:::3: f N 1 f N 2 f N 3 èy;p 2 è=, g2 h 2 èp 2 è 16ç 3 èy;p2 è=, g2 h 2 èp 2 è 16ç 3 èy;p2 è=, g2 h 2 èp 2 è 16ç 3 y 1, y y 1, y y 1, y Z 1 dp?p? Z2ç Z 1 Z2ç dp? p? Z1 Z2ç dp? p? dæ h2 èp 2 è èp 2, m 2 è 2 dæ,yh2 èp 2 è èp 2, m 2 è 2 ; dæ h2 èp 2 è èp 2, m 2 è ; 2 è2yp 2 + 1y ëp2, m 2 ë, 2pp è ; è23è è24è è25è f ç 1 èy;p 2 è= g2 h 2 èp 2 è 16ç 3 f ç 2 èy;p 2 è= g2 h 2 èp 2 è 16ç 3 f ç 3 èy;p 2 è=, g2 h 2 èp 2 è 16ç 3 y 1, y y 1, y Z1 Z1 y 1, y Z2ç dk? k? dæ f 2 èk 2 è èk 2, ç 2 è 2 n 2yp2, 2pk o ; è26è Z2ç dk? k? dæ è1, yèf 2 èk 2 è ; è27è èk 2, ç 2 è 2 Z1 Z2ç dk? k? dæ f 2 èk 2 è èk 2, ç 2 è ; 2 22 è28è

23 where hèp 2 è and fèk 2 è are the model formfactors èsee eq. è4è-è11èè, æ-components of p and k are deæned as follows: p + = yp + ; k + = yp + ; p, = k, = 1 h p p + è1, 2, ç 2, p 2?, yp, p + +2p? p? cos æ i ; yè 1 h p2, m p + è1, 2, k? 2, yp, p + +2p? k? cos æ i ; yè è29è 23

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