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We close this introduction with a remark on references to the literature. As usual, aiming for completeness is an impossible task. What we hope to be a reasonable compromise is a combination of references to previous reviews in which earlier references can be found, together with selected original references to data and theory whenever they are of direct relevance in the text. Deep-inelastic scattering: kinematics and structure functions Consider the scattering of an electron or muon with four-momentum kI" E, k and invariant mass m from a nucleon carrying the four-momentum PI" E , P and mass M.

Neglecting weak interactions which are relevant at very high energies only, the di! We denote IJ the nucleon spin by S. Inclusive deep-inelastic lepton-nucleon scattering.

This is the basis of the naive parton model which gives a simple interpretation of nucleon structure functions. In this picture the nucleon is composed of free point-like constituents, the partons, identi"ed with quarks and gluons. Virtual Compton scattering The hadronic tensor 2. By comparison of Eqs. They have values 1,! For the spin-dependent structure functions one "nds: 2.

Due to their interaction with gluons, quarks receive momentum components transverse to the photon direction. Then they can absorb also longitudinally polarized photons. This leads to RO0. These scaling violations can be described within the framework of the QCD-improved parton model which incorporates the interaction between quarks and gluons in the nucleon in a perturbative way see e.

The scale at which this interaction is resolved is determined by the momentum transfer. They are di! Typical examples of non-singlet combinations are the di! The di! Similar interpretations hold for the remaining splitting functions. For further details we refer the reader to one of the many textbooks on applications of QCD, e. Light-cone dominance of deep-inelastic scattering The QCD analysis of deep-inelastic scattering has generated its own terminology and specialized jargon.

In this section we summarize some of the basic notions. They are grouped according to the order of their singularity. Each operator has a characteristic dimensionality, d, in powers of mass or momentum. For example, the symmetric traceless Lorentz tensors of rank n with minimum dimensionality and p"n are the operators 2. The operators OO and OE have dimensionality d"3 n! Comparing dimensions in Eq. Matrix elements of the operators O between nucleon states are of genuinely non-perturbative origin.

For spin-averaged quantities they must be of the form 2. Trace terms have been subtracted in I I Eqs. We can now make contact with observables. We mention that, in general, the representation of a given structure Q function in terms of separate quark and gluon contributions is a matter of de"nition.

The Q measured structure functions are, of course, free of such ambiguities.

Revisiting Gundersen: Nuclear 101 - Advantages & Disadvantages of Splitting Atoms to Boil Water

Facts about free nucleon structure functions In this section we brie y review the present experimental status on free nucleon structure functions as measured in deep-inelastic lepton scattering. We focus on those aspects which are of direct relevance for our further discussion of nuclear deep-inelastic scattering. Reviews can be found e. This is di! The data summarized in Figs. This behavior is commonly interpreted in terms of the dominant role of gluons at small x, the density of which rises strongly with decreasing x.

Here only a minor x-dependence has been observed in "xed target experiments, which is nevertheless enhanced at very small x;0. It is often parametrized using Regge phenomenology [21,22]. In Regge theory the dependence of cross sections on the center-of-mass energy is determined by the t-channel exchange of families of particles permitted by the conservation of all relevant quantum numbers. Bjorken scaling must break down in this kinematic regime.

In particular, at small x 0. Then the scattering from parton constituents in the target takes over and leads to Bjorken scaling. This can be understood within the framework of perturbative QCD. In the limit xP1, a single valence quark struck by the virtual photon carries all of the nucleon momentum. The only way for such a con"guration to evolve from a bound state wave function which is centered around low parton momenta, is through the exchange of hard gluons. In this region R is small. New data from the NMC collaboration are available for 0. A rise of R with decreasing x has been observed as shown in Fig.

This behavior can be understood within the framework of perturbative QCD [31]. In the QCD-improved parton model such transverse quark momenta result from gluon bremsstrahlung which is important for low parton momenta, i. A "rst analysis gives RK0. Spin-dependent structure functions In recent years polarized deep-inelastic scattering experiments have become a major activity at all high-energy lepton beam facilities. They aim primarily at the exploration of the spin structure of Fig. A compilation of data of the proton, deuteron, and neutron spin structure functions g from Refs.

We thank U. Stoesslein for the preparation of this "gure. In the data analysis such corrections have commonly been done in terms of e! They account for the fact that bound nucleons carry orbital angular momenta. As a consequence their polarization vectors need not be aligned with the total polarization of the target. At the present level of accuracy the use of e! In Fig. This is in contrast to the unpolarized case where proton and neutron structure functions show a qualitatively similar behavior. They can be decomposed in terms of proton matrix elements of SU 3 axial currents, as follows for a review see e.

In Eq. All current studies arrive at the conclusion that the avor singlet contribution to the nucleon spin is small. The missing two thirds probably involve gluon spin contributions and orbital angular momentum of quark, antiquark and gluon constituents. Finally, we note that the Bjorken sum rule 2. We focus here on so-called single di! They are characterized by the proton emerging intact and well separated in rapidity from the hadronic state X produced in the dissociation of the virtual photon see Fig.

Their cross sections drop exponentially with the squared four-momentum transferred by the colliding particles. Furthermore, they generally exhibit a weak energy dependence. Their cross section is parametrized in terms of two structure functions, analogous to the inclusive case. The 6 di! At small x one "nds in analogy with Eq. A reasonably successful description of this behavior has been achieved within Regge phenomenology which assumes that the interaction proceeds in two steps: the emission of a pomeron or subleading reggeon from the proton, and the subsequent hard scattering of the virtual photon from the partons in the pomeron or reggeon, respectively.

This picture leads to a factorization of the di! ZEUS data [53] for the ratio of di! The data show a similar energy dependence of both total 6 and di! ZEUS measurements [54] have investigated the t-dependence of the di! This value is compat "t" 0. Diwractive photoproduction: Di! Around half of these events come from the production of the light vector mesons o, u and. This is contrary to di!

The curves corresponds to a Regge "t [62]. This is in accordance with Regge 6 phenomenology. Observed deviations from the simple behavior 2. Introduction and motivation We now enter into the central topic of this review: an exploration of new phenomena speci"c to deep-inelastic lepton scattering from nuclear rather than free nucleon targets. Nuclei represent systems with a natural, built-in length scale. The baryon density in the center of a typical heavy nucleus is o K0. Such modi"cations are expected to arise, for example, from the mean "eld that a nucleon experiences in the presence of other nucleons, and from its Fermi motion inside the nucleus.

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Such e! A typical example of a coherence e! It turns out, as we will demonstrate, that incoherent scattering takes place primarily in the range 0. Strong coherence e! Cooperative phenomena in which several nucleons participate can also occur at x'1. In fact, the Bjorken variable can extend, in principle, up to x4A in a nucleus with A nucleons. The aim of this section is to prepare the facts and phenomenology of nuclear DIS. A subtopic in this context deals with the deuteron. While this is not a typical nucleus, it serves two purposes: "rst, as a convenient neutron target, and secondly, as the simplest prototype system in which coherence e!

For this purpose we need to introduce the hadronic tensor and structure functions for spin-1 targets as well.

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Once the nuclear structure functions are at hand we will present a survey of nuclear DIS data and give "rst, qualitative interpretations. The more detailed understanding is then developed in subsequent sections. Nuclear structure functions The deep-inelastic scattering cross sections for free nucleons and nuclei have basically the same form as given by Eq. All information about the target and its response to the interaction is included in the corresponding hadronic tensor.

Furthermore, we have used the? The "rst four-structure functions in Eq. Data on nuclear structure functions In this section we summarize the existing experimental information on nuclear e! Their systematic investigation for light and heavy nuclei has been carried out so far only in unpolarized scattering experiments. Most of the data come from deep-inelastic lepton scattering.

Modi"cations of nuclear parton distributions have also been studied in other highenergy processes. In charged lepton scattering from unpolarized nuclear targets these structure functions are de"ned by the di! Since then many experiments dedicated to a study of nuclear e! The primary aim was to explore the di! In the absence of nuclear e! Neglecting small nuclear e! Several distinct regions with characteristic nuclear e! A small enhancement is seen at 0. The dip at 0. For x'0. Finally, note again that nuclear structure functions can extend beyond x"1, the kinematic limit for scattering from free nucleons.

Nuclear targets ranging from He to Pb have been used. As an example we show in Fig. Shadowing systematically increases with the nuclear mass number A. A, The shadowing e! The most precise investigation of this issue has been performed for the ratio of Sn and carbon structure functions presented in Fig. The rate of this decrease becomes smaller with rising x.

At x'0. The band indicates the size of the systematic errors. Shadowing has also been observed in deep-inelastic scattering from deuterium, the lightest and most weakly bound nucleus. E Enhancement region: The NMC data have established a small but statistically signi"cant enhancement of the structure function ratio at 0. The observed enhancement is of 26 G. Data from E [92] and NMC [93]. The magnitude of this depletion grows approximately logarithmically with the nuclear mass number.

The observed e! E Fermi motion region: At x'0. Clearly, even minor nuclear e! Both quasielastic scattering from nucleons as well as inelastic scattering turns out to be important here. Note that in Eq. In the kinematic range covered by these experiments, 3. Ratios of longitudinal and transverse cross sections Investigations of the di! In addition we present the average values from the NMC measurement for R! All measurements are consistent with only marginal nuclear dependence of R. This implies that nuclear e! The average values for R! Other measurements of nuclear parton distributions Nuclear deep-inelastic scattering is sensitive only to the sum of valence and sea quark distributions see e.

In order to separate nuclear e! The avor-dependent quark distribuJ tions of the projectile and target are denoted by q and q2 , respectively. Seen from the center-ofD D mass frame the active quarks carry fractions x and x of the beam and target momenta. At x '0. Outside the domain of quarkonium resonances, i. This indicates the absence of strong modi"cations of nuclear sea quark distributions, as compared to those of free nucleons.

Lepton-induced production of heavy quarks The intrinsic heavy-quark c- or b-quark distributions in nucleons or nuclei are expected to be very small. This mechanism is a basic ingredient of the so-called color-singlet model []. In this model the cross section for heavy quark pair production is proportional to the gluon distribution of the target.

A comparison of these cross sections for nucleons and nuclei can then be directly translated into a di! Neutrino scattering from nuclei Deep-inelastic neutrino scattering permits one to separate valence and sea quark distributions. It is therefore a promising tool to investigate modi"cations of the di! The observed nuclear e! The partonic interpretation of structure functions is particularly transparent in the in"nite momentum frame in which the nucleon or nucleus moves with longitudinal momentum PPR.

In this frame the Bjorken variable x has a simple meaning as the fraction of the nucleon momentum carried by a parton when it is struck by the virtual photon. In this description, the scattering cross section is determined by the square of the target ground state wave function for a review and references see e. Instead, it is often preferable to describe the scattering process in the laboratory frame where the target is at rest. Only in that frame the detailed knowledge about nuclear structure in terms of many-body wave functions, meson exchange currents, etc.

Also, the physical e! In this section we elaborate on several aspects relevant to deep-inelastic scattering as viewed in coordinate space. We "rst discuss the coordinate space resolution of the DIS probe. A detailed discussion of nuclear e! In the "nal part we comment on the relationship between lab frame and in"nite momentum frame pictures. Deep-inelastic scattering in coordinate-space We follow here essentially the discussion in Ref. Consider the scattering from a free nucleon with momentum PI" M, 0 and invariant mass M in the laboratory frame.

Assuming that the integrand in 4. Furthermore, Eq. Such a behavior is consistent with approximate Bjorken scaling []. In the G. Consequently, large longitudinal distances are important in the scattering process at small x. This can also be deduced in the framework of time-ordered perturbation theory see Section 4.

The characteristic laboratory frame correlation length l is one-half of that distance. In this section we prepare the facts and return to the underlying dynamics at a later stage. It is useful to express coordinate-space distributions in terms of a suitable dimensionless variable. The dimensionless variable z"y P then plays the role of a coordinate conjugate to Bjorken x.

Flavor indices are suppressed here for simplicity. At leading twist accuracy, the coordinate-space distributions 4. Note that an expansion of the right-hand side of Eqs. The functions Q z , Q z and G z describe the mobility of partons in coordinate-space. Consider, T for example, the valence quark distribution Q z. The matrix element in 4. There the photon converts into a beam of partons which propagates along the light cone and interacts with partons of the target nucleon, probing primarily its sea quark and gluon content. Coordinate-space distributions of free nucleons In this section we discuss the properties of coordinate-space distribution functions of free nucleons.

Examples of the distributions 4. A sum over the u and d quarks is implied in the functions Q and Q []. T Some general features can be observed: the C-even quark distribution Q z rises at small values of z, develops a plateau at z 9 5, and then exhibits a slow rise at very large z. At z:5, the gluon distribution function zG z behaves similarly as Q z. For z 9 5, zG z rises somewhat faster than Q z.

The C-odd or valence quark distribution Q z starts with a "nite value at small z, then begins T to fall at zK3 and vanishes at large z. Recall that in the laboratory frame, the scale zK5 at which a signi"cant change in the behavior of coordinate-space distributions occurs, represents a longitudinal distance comparable to the typical size of a nucleon. At z 5 the coordinate-space distributions are determined by average properties of the corresponding momentum-space distribution functions as expressed by their "rst few moments [,]. For example, the derivative of the C-even quark distribution Q z taken at z"0 equals the fraction of the nucleon light-cone momentum carried by quarks.

The same is true for the gluon distribution zG z the momentum fractions carried by quarks and by gluons are in fact approximately equal, a well-known experimental fact. At z'10 the coordinate-space distributions are determined by the small-x behavior of the corresponding momentum-space distributions. T The fact that Q z and zG z extend over large distances has a natural interpretation in the laboratory frame.

For similar reasons, the valence quark distribution Q z has a pronounced tail which extends to distances beyond the nucleon radius. A detailed and instructive discussion of this frequently ignored T feature can be found in Ref. Finally we illustrate the relevance of large distances in deep-inelastic scattering at small x.

Contributions from di! Coordinate-space distributions of nuclei The implications for scattering from nuclear targets, especially for coherence phenomena, are now obvious. If one compares, in the laboratory frame, the longitudinal correlation length l from Eq. Possible modi"cations of the coordinate distribution functions 4. Modi"cations of the coordinate distribution functions are now expected to come from the coherent scattering on at least two nucleons in the target. This suggests that the nuclear modi"cations seen in coordinate-space distributions will be quite di! Furthermore, the ratios of valence quark and gluon G.

The intrinsic structure of individual nucleons is evidently not much a! For a detailed discussion see Ref. It is interesting to observe that in coordinate-space, shadowing sets in at approximately the same value of l for all sorts of partons. In momentum space, shadowing is found to start at di! Finally note that the shadowing e! The results shown in Fig. At small Bjorken-x, the subsequent QCD evolution of this pair rapidly induces a cascade of gluons.

This o! In summary, a coordinate-space representation which selects contributions from di! The two basic time orderings are shown in Figs. For small Bjorken-x the pair production process b dominates the scattering amplitude, as already mentioned. This can also be easily seen in time-ordered perturbation theory as follows see e. For large energy transfers? O When analyzing the spectral representation of the scattering amplitude one observes that the bulk contribution to process b results from those hadronic components in the photon wave function G.

Deep-inelastic scattering at small x;1 in the laboratory frame proceeds via hadronic uctuation present in the photon wave function. The ratio in Eq. Hence pair production, Fig. On the other hand, at x'0. For x 0. For larger values of the Bjorken variable, x'0. At the same time the process in Fig.

Now the incoherent scattering from the hadronic constituents of the nucleus dominates. Nuclear deep-inelastic scattering in the inxnite momentum frame Let us "nally view the deep-inelastic scattering process in the so-called in"nite momentum frame where the target momentum is large.

In this frame the standard parton model applies in which a snapshot of the target at the short time scale of the interaction reveals an ensemble of almost non-interacting partons, i. Therefore, at x;0. One can anticipate that, at x;0. In the lab frame, this is where the quark and gluon uctuations of the photon interact simultaneously with the parton content of several nucleons.

Shadowing in unpolarized deep-inelastic scattering As outlined in Section 3. For small values of the Bjorken variable x 0. The analogous behavior is observed for real photons at large energies l'3 GeV. Multiple scattering becomes important as soon as the lab frame coherence length for the hadronic uctuations of the photon propagator exceeds the average distance between two nucleons in the nuclear target. At extremely small x i. It is of great interest to investigate the transition of the observed shadowing phenomena into this new domain, accessible by collider experiments, but so far unexplored for nuclear systems.

In this section we "rst concentrate on the relationship between di! Then we investigate perturbative and non-perturbative QCD aspects of shadowing. After that we summarize existing models which successfully describe data. Finally, we outline implications of shadowing for nuclear parton distributions. Diwractive production and nuclear shadowing In the shadowing region, di!

This suggests that the di! For this e! Shadowing results from the coherent scattering of a hadronic uctuation from at least two nucleons in the target. Since the longitudinal propagation length j of a di!

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The leading contribution to nuclear shadowing comes from double scattering. Its mechanism is best illustrated for a deuterium target on which we focus next. The cH-deuteron cross section can be written as the sum of single and double scattering parts as illustrated in Fig. For large energies, l'3 GeV, or small values of the Bjorken variable, x 0. At the high energies involved it is a good approxi Fig.

We neglect the spin and isospin dependence for unpolarized scattering [71]. Of course, these degrees of freedom play a crucial role in polarized scattering as we will discuss in Section 7. The corresponding cross sections are obtained from the imaginary part of the forward scattering amplitude indicated by the dashed line. The sum is taken over all di! The limits of integration de"ne the kinematically permitted range of di! Note that the minimal momentum transfer required to produce a hadronic state di! It becomes even more transparent for x;0. In this limit the magnitude of shadowing is determined just by the ratio of di!

To verify this let us parametrize the t-dependence of the di! In the 6 di! Clearly, the soft deuteron form factor selects momenta such that the double scattering correction in 5. Integrated deuteron form factor F from Eq. From Eq. Combining Eqs. One "nds that shadowing at x;0. The e! Shadowing for heavy nuclei The di! It is an empirical fact that nuclear shadowing increases with the nuclear mass number A of the target see Section 3. For A'2 the hadronic state which is produced in the interaction of the photon with one of the nucleons in the target may scatter coherently from more than two nucleons.

However, double scattering still dominates since the probability that the propagating hadron interacts with several nucleons along its path decreases with the number of scatterers. The cos[ z! Note that nuclear short-range correlations are relevant only if the coherence length of the di! In this case the shadowing e!

With increasing photon energies or decreasing x down to x;0. Then Fig.

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Double scattering contribution to deep-inelastic scattering from nuclei. The more detailed results are discussed in Section 5. We restrict ourselves to the double scattering correction 5. For x;0. For the nuclear densities in Eq. Using again typical values for the ratio of di! A more detailed investigation of the connection between HERA data on di!

Inelastic transitions between di! They cannot be treated in a model-independent way. Estimates of such higher-order di! For rising energies the relative importance of inelastic transitions is expected to grow []. Given the important role of di! The coherence lengths j of hadronic states with small masses become comparable with nuclear dimensions for x 0. As j increases with decreasing x, the shadowing e!

At x;0. At asymptotically large energies Regge phenomenology suggests eK0. R "log c! Once a major fraction of di! The quantity log 1! R as a function of log x for data taken on lead [78]. Note this asymptotic behavior sets in when the coherence lengths j of low mass hadronic uctuations of the photon exceed by far the 6 dimension of the nucleus. Shadowing for real photons Data on the di! They are useful to gain insight into the relative importance of o, u and mesons, as compared to heavier states, for nuclear shadowing with real photons. Nuclear shadowing at photon energies l up to about GeV is largely determined by the coherent multiple scattering of those di!

With rising energies 4 additional contributions to shadowing from di! The shadowing correction 5. The observed energy dependence of shadowing in Fig. The experimental data are taken from the E collaboration [92]. The energy values of the data have to be understood as average values which correspond to di! The kinematic conditions of such experiments imply that the data for x 0. The corresponding energy transfers are typically 50 GeV l GeV. The conclusions just drawn for real photons apply here too: nuclear shadowing as measured by E and NMC receives major contributions from the di!

In the intermediate range 0. The following section gives a schematic view of the scales involved, as outlined in Ref. Sizes, scales and shadowing Consider DIS at small x in the lab frame. In this frame of reference the important feature is the nuclear interaction of hadronic uctuations of the virtual photon see Section 4. Since the photon and its hadronic con"gurations carry high energy, the transverse separations and longitudinal momenta of their quark and gluon constituents are approximately conserved during the scattering process.

For example, the contribution of a hadronic uctuation to double scattering, which dominates shadowing, is proportional to its weight in the photon wave function multiplied by the square of its interaction cross section. These properties and their consequences for the cross sections in Eqs.

E The energy dependence of nuclear shadowing for x These hadronic con"gurations are expected to interact like ordinary hadrons.

Note, those observations can be applied to di! Its energy dependence is expected to behave similarly as in hadron collisions. For limitations to this simple picture see Sections 5. Nuclear shadowing and parton conxgurations of the photon The results of the previous sections are elucidated by making contact with the underlying basic QCD and the parton structure of the virtual photon. The photon wave function can be decomposed in a Fock space expansion, 5.

Let us now have a closer look at this minimal Fock Q component. The quark has a four-momentum kI" k , k with k" k , k. Since the OO, modi"ed Bessel functions in Eqs. The reasoning goes as follows. All non-perturbative Q e! The situation is di! Double scattering gives a negative correction proportional to the squared cross section of the hadronic uctuation. Only those hadronic con"gurations with large interaction cross sections contribute signi"cantly to shadowing.

The exponential in 5. For small-sized uctuations, interesting e! Contributions from the vector mesons o, u and turn out to be particularly important. As before we restrict ourselves to lab-frame descriptions. We do not aim for completeness but rather emphasize common features of various models and their implications for the underlying mechanism of nuclear shadowing.

Vector mesons and aligned jets As discussed in Section 5. Models which combine aspects of vector meson dominance and the aligned-jet picture [] are described in Refs. Their starting point is the hadronic spectrum of the virtual photon exchanged in the deep-inelastic scattering process. The sum in Eq. The high-energy virtual photon with 54 G. The full line has been obtained in Ref. The dashed line indicates the contribution of the vector mesons o, u and. The data are from the NMC []. The ansatz neglects contributions to the forward virtual photon scattering amplitude in which the mass k can change during the interaction.

The color singlet nature of hadronic uctuations of the virtual photon implies that their interaction cross section is proportional to their transverse size. Their cross sections should be comparable to typical hadronic cross sections. Their cross sections should therefore be small. With these ingredients, Eq. A comparison with data from NMC is shown in Fig. Results from Ref. In the spectral ansatz 5. Data are taken from [83]. Shadowing in Xe. Details of the calculation are given in Ref.

The dashed curve shows the contribution of vector mesons o, u and , while the solid curve includes pomeron exchange. The data are from the E collaboration [77]. A comparison of results from Ref. Vector meson dominance and pomeron exchange As indicated in Eqs. Their contributions can be described within the framework of vector meson dominance see e. Neglecting transitions between di! Most descriptions concentrate on the dominant contribution from pomeron exchange. Investigations of shadowing e! For a free nucleon this leads to the following picture: prior to its interaction with the target the virtual photon uctuates into a hadronic state with invariant mass k.

Here the integrals over initial and "nal hadronic uctuations and their propagators are made explicit. L LL L The next step in simpli"cation is to consider only diagonal m"n and nearest o! An extension of this approach to nuclear targets involves multiple scattering of hadronic uctuations from several nucleons. The multiple scattering process is described by a coupled channel optical model [,] which accounts for the shadowing criteria in Eq.

Vector mesons and quark scattering We add a few remarks and references about approaches dealing with DIS in terms of quark dynamics. The starting point in Ref. O, 58 G. In Ref. One "nds that vector mesons carry more than half of the shadowing e! Green function methods The previously mentioned models have outlined in di! Several questions are faced in this context. All those aspects can be uni"ed within a coordinate-space Green function approach. We follow here the presentations in Refs. The longitudinal z- direction is de"ned by the photon three-momentum, as usual.

The color dipole cross section p b has the characteristic color screening behavior, i. It satis"es a wave equation [] which can be made plausible by the following considerations. Note the di! OO ii No absorption: Take the limit pP0 in the wave equation 5. Inserting Eq. In this case the wave equation 5. Instructive results are discussed in Ref. Meson exchange and shadowing Up to now we have concentrated on di! The coherent interaction of the photon with several nucleons in the target nucleus can also involve non-di! These are commonly described by the exchange of mesons and sub-leading reggeons.

Modi"cations to nuclear structure functions at small x through meson exchange have been investigated in Refs. In this work signi"cant e! Here, as in di! Contributions from the exchange of other mesons, e. For the double scattering contribution through pion exchange one "nds in analogy with Eq. The form factor in Eq. The momentum-space wave function of the deuteron with polarization m is denoted by tK.

Note that the energy of the exchanged pion is determined by k "M! It is common to factorize the semi-inclusive di!

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Here, however, only the region x'0. An extraction of the pion structure function at small x from semi-exclusive reactions at HERA has been discussed recently in [,]. The resulting pionic correction dppH to double scattering turns out to be positive, i. The relative weight of dppH decreases with decreasing x. Interpretation of nuclear shadowing in the inxnite momentum frame In this section we brie y discuss how nuclear shadowing develops in the in"nite momentum frame where the parton model for deep-inelastic scattering can be applied. We found in Section 4.

One then expects that the interaction of partons belonging to di! Shadowing at x 0. At the same time parton fusion leads to an enhancement of partons at x'0. Procedures for modeling nuclear parton distributions at small x have been proposed in Refs. Recombination e! It should be mentioned that the calculation of the recombination e! Nuclear parton distributions at small x Any quantitative QCD analysis of high-energy processes involving nuclei requires a detailed knowledge of nuclear parton distributions.

In this section we outline the empirical information on their di! Let us "rst focus on the nuclear gluon distribution. In leading order perturbation theory and in the limit x;0. The intimate relation between sea quarks and gluons through DGLAP evolution then also suggests shadowing for gluons. The enhancement of nuclear gluon distributions around xK0. The empirical information on this sum rule applied to quarks has been presented in Section 3. It implies that the momentum carried by gluons is, within errors, equal in nucleons and nuclei, i.

Assuming the latter to be located in the region 0. Note that the close relation between shadowing and di! It suggests signi"cantly larger shadowing for gluons than for quarks. Nuclear e! T Typical results from Ref. Momentum-space ratios from Ref. The kinematic range where enhancement takes place is related to processes which involve typical longitudinal distances of 1 fm in the laboratory.

This is the region where components of the nuclear wave function with overlapping parton distributions should be relevant. In such a picture the enhancement of gluons should increase with the density of the nuclear target. More detailed information on nuclear parton distributions is certainly needed.

A more quantitative separation of nuclear e! However, an extraction of nuclear parton distributions in hadron production processes from nuclei, e. Nuclear structure functions at large Bjorken-x Deep-inelastic scattering from nuclei probes the nuclear parton distributions. On the other hand, conventional nuclear physics works well with the concept that nuclei are composed of interacting hadronic constituents, primarily nucleons and pions. In this kinematic region, incoherent scattering from hadronic constituents of the target nucleus dominates.

Such processes explore the quark distributions of nucleons bound in the nucleus. To gain "rst insights suppose that the nucleus is described by nucleons moving in a mean "eld. The quark substructure of bound nucleons may di! First, there is a purely kinematical e! This e! To illustrate this recall that for a free nucleon the light-cone momentum fraction of partons cannot exceed x"1.

A nucleon bound in a nucleus carries a non-vanishing momentum which adds to the momenta of individual partons in that nucleon. As a consequence light-cone momentum fractions up to x"A are possible in principle, although the extreme situation in which a single parton carries all of the nuclear momentum will of course be very highly improbable. On the other hand, intrinsic properties of bound nucleons, e. This may lead to additional, dynamical modi"cations of their partonic structure. Impulse approximation Nuclei are, in many respects, dilute systems.

Final state interactions of the scattered hadron with the residual nuclear system are neglected at high energy. One should note, however, that the validity of this approximation, illustrated in Fig. Given the small average momenta of the weakly bound nucleons, their quark sub-structure is described by structure functions similar to those of free nucleons [,]. This becomes immediately obvious from the following simple kinematic consideration. In the laboratory frame deep-inelastic scattering from a nucleon bound in a nucleus involves the removal energy,!

The energy of the interacting nucleon is then: p "M e! The squared four momentum of the active nucleon is obviously not restricted by its free invariant mass. It is determined by the nuclear wave function which describes the momentum distributions of bound target nucleons as well as the mass spectrum of the residual nuclear system.

This quark belongs to a nucleon with momentum p which is removed from the target nucleus. It is determined by the momentum-space amplitude W p "1 A! Note that the spectral function is normalized to A, the total number of nucleons in the nucleus. This leads to the proper normalization of the nucleon distribution function in Eq. The nuclear structure functions are then obtained by a convolution over the squared fourmomentum of the interacting nucleons and their light-cone momentum fraction.

This implies that the nucleon light-cone distribution 6. Expanding the bound nucleon structure function in Eq. Then 1e2 coincides with the separation energy. Corrections to 0 Eq. Let us brie y discuss the physical meaning of the di! The second term on the right-hand side of Eq. As such it is determined by nuclear binding. In the range 0. Finally, the fourth term in 6. Note that this contribution enters at the same order as binding and Fermi-motion corrections. Nevertheless, such e! An important and not yet completely solved problem with respect to the binding and Fermimotion corrections in Eq.

In a simple nuclear shell G. This in turn causes an increase of the average separation energy 1e2 [,]. This sum rule is exact if only two-body forces are present in the nuclear Hamiltonian. We refer in this context to a calculation [] of the spectral function of nuclear matter based on a variational method. This calculation shows that there is a signi"cant probability to "nd nucleons with high momentum and large separation energies.

An integration of the spectral function of Ref. In order to estimate these quantities for heavy nuclei one usually assumes [] that the high-momentum components of the nucleon momentum distribution are about the same as in nuclear matter. Together with Eq. We observe that a qualitative understanding of the EMC e! One should note, of course, that the presentation of nuclear e! The impulse approximation picture of nuclear deep-inelastic scattering can also be maintained in a relativistically covariant way [].

Here, however, a simple factorization of nuclear structure functions into nuclear and nucleon parts as in Eq. A relativistic calculation of nuclear structure functions requires relativistic nuclear wave functions as well as Fig. Nevertheless, relativistic e! In the region x'1, where nuclear structure functions are very small however, larger deviations are expected. In this context a word of caution is in order. A description of nuclear structure functions based on nucleons alone is necessarily incomplete since it violates the momentum sum rule [1].

Non-nucleonic degrees of freedom are brie y discussed in Section 6. Beyond the impulse approximation The quality of the impulse approximation has frequently been questioned see e. Here we give a brief summary of possible shortcomings in terms of models for nuclear deep-inelastic scattering which go beyond this approximation. Quark exchange in nuclei The impulse approximation includes only incoherent scattering processes from hadronic constituents of the target nucleus. Deutsch et al. Nishiguchi, K. Nagai, H. Sano , Long-range nematic order and anomalous fluctuations in suspensions of swimming filamentous bacteria , Physical Review E , vol.

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Schimansky-geier, and P. Romanczuk , Active Brownian particles with velocity-alignment and active fluctuations , New Journal of Physics , vol. Peshkov, I. Aranson, E. Bertin, H. Romanczuk and L. Schimansky-geier , Mean-field theory of collective motion due to velocity alignment , Ecological Complexity , vol. Toner , Reanalysis of the hydrodynamic theory of fluid, polar-ordered flocks , Physical Review E , vol.

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Ginelli, S. Mishra, A. Peshkov et al. Solon and J. Caussin, A. Solon, A. Peshkov, H.