Nuclear shadowing in deep inelastic scattering on nuclei at very small x

Nuclear shadowing corrections to the structure functions of deep inelastic scattering on intermediate-mass nuclei are calculated at very small values of Bjorken x and small values of Q^2 (Q^2<5 Gev^2). The two-component approach developed in previous works of authors for a description of the nucleon structure functions of deep inelastic scattering is used. It is shown that the hard component of the nucleon structure functions that arises, in terms of the colour dipole model, from qq-pairs with a high transverse momentum, is almost not shadowed. It is shown that a change of the slope of the shadowing curve with a decrease of x depends, at small values of x, on the relative contribution of the hard component to the nucleon structure function (this contribution is a function of x and Q^2) and on a size of gluon saturation effects. It is shown that an accounting for saturation effects becomes essential for predictions of shadowing at x<10^-4, depending on a value of Q^2. Results of numerical calculations of nuclear shadowing for several nuclei are compared with available data of the E665 and the NMC collaborations.


I. INTRODUCTION
It is well known that at small values of Bjorken This, by definition, is the effect of nuclear shadowing (see reviews [1][2][3][4][5]). This phenomenon is explained by a destructive interference of amplitudes of single and multiple scatterings of the hadronic fluctuations of the virtual photon on nucleons of the target nucleus (considering the process in the rest frame of the nucleus). So, nuclear shadowing is a coherent effect and results from a coherent scattering of the hadronic fluctuation from at least two nucleons in the target nucleus.
There are two main formalisms which are used for a study of nuclear effects in DIS: Glauber-Gribov formalism [6,7] and Regge-Gribov framework [8][9][10]. In the first case the hadronic components of the virtual photon are rescattered in the target nucleus in a Glauber-like manner, and different models are distinguished by a choice of the mass spectrum of the hadronic fluctuations and a cross section of their interaction with nucleons. The Glauber-Gribov formalism had been exploited in calculations of nuclear shadowing based on the generalized vector dominance (GVD) approach [7,[11][12][13] (the corresponding results of shadowing calculations are reviewed in [2]) and, more recently, in calculations using the colour dipole model [14,15] (for references on these calculations see reviews [3][4][5]).
Regge-Gribov framework uses the connection between nuclear shadowing and a differential cross section for the diffractive dissociation of the projectile. In order to calculate nuclear shadowing effects in this framework one must know the nucleon diffractive structure functions. Calculations of nuclear shadowing effects in Regge-Gribov framework are performed using the different model assumptions because the perturbative QCD is not applicable to the full description of diffractive DIS (especially in the region of small 2 Q ). Three groups of models are most frequently used for calculations of nuclear shadowing: i) aligned jet models (AJMs) [16,17], ii) Regge-motivated models using the concept of partonic pomeron [18,19] and iii) ''leading twist approaches" operating with diffractive parton distributions [20] (the corresponding works studying shadowing are reviewed in [3][4][5]). Assumptions used in i) and ii) are not in conflict with QCD as it may seem. In opposite, it was shown [21] that perturbative QCD models based upon two-gluon exchange can be extrapolated into the nonperturbative region and, performing such an extrapolation, authors of [21] really discovered a dominance of aligned-jet configurations in the diffractive structure function and an arising of a simple picture of the Pomeron structure function.
In the present work we calculate the nuclear shadowing effects for several nuclei, for a broad interval of x and a limited interval of 2 Q , 22 5 Q GeV  . The choice of just this region of 2 Q , for shadowing studies, is determined by two reasons. Firstly, in recent few years the interest was revived to a study of photonuclear interactions of high energy leptons (i.e., lepton-nucleus inelastic interactions dominated by small values of 2 Q ). This interest is connected, in particular, with a planning of new experiments on detection of astrophysical neutrinos of very high energies (see, e.g., [22,23]). The second reason is the following: in our previous papers [24][25][26] we elaborated the two-component model (GVD + perturbative QCD) for a description of DIS just for a region of small and medium 2 Q and, in the present work, we use, for a calculation of the shadowing corrections, the nucleon structure functions of DIS obtained in this two-component model.
It is well known that shadowing effects in interactions of real and quasireal photons with nuclei are rather well described by the vector dominance approach operating with light vector mesons only ( ,,    ) (see, e.g., the detailed review [27]). However, when data with photons of relatively high virtualities ( 22 1 Q GeV  ) appeared, it had been realized that the light mesons explain well the "high twist" shadowing effects while for a description of a weak 2 Q -dependence of shadowing at medium and large 2 Q , discovered by the data, the additional "quasiscaling" or "partonic" mechanism [28] is necessary. The alternative (and more appropriate, in our opinion) approach for a description of shadowing at medium 2 Q is an application of the GVD concept in aligned jet version [16,17,29]. In GVD excited states of light vector mesons ( ', '',...


) are included in the mass spectrum of the hadronic fluctuations of the virtual photon [11,13]. In such an approach the photoabsorption cross section * N   contains the GVD (i.e., non-perturbative) part and the perturbative term taking into account a contribution of those hadronic fluctuations of the virtual photon whose interactions with the target nucleon are described by perturbative QCD. Just this approach is used for calculations of the DIS structure functions in [24][25][26].
The plan of the paper is as follows. In the second Section the main assumptions underlying our approach are discussed and some key formulas are briefly derived. In the third Section the shadowing correction due to soft interactions of the hadronic fluctuations of the virtual photon with nucleons of the target nucleus is studied. In the fourth Section the contribution to shadowing from the hard interactions of non-aligned qq -pairs, produced by the virtual photon, with nucleons is considered. In the fifth Section results of the numerical calculations of the shadowing coefficients for several nuclei and for a broad interval of Bjorken x are shown. Discussions of the results and conclusions are given in the sixth Section.

II. OUTLINE OF THE MODEL
Consider, at first, a simplest case when the hadronic fluctuations are described by the separate vector mesons.
GVD approach (see, e.g., [30] for the historical review) starts from the spectral representation for the transverse photon absorption cross section  We assume, further, in accord with QCD (in its large c N limit) and Regge theory (see, e.g., [31]), that the mass squared of the family member obeys "equal spacing rule" with respect to the index n , With such a spectrum the parton-hadron duality condition, leads to the following relation for the photon-vector meson couplings of the family members: .
Note, that we do not need the analogous relation for the electromagnetic decay widths of the vector mesons (see the recent work [32] where the question of the validity of eq. (9) is discussed).
Substituting eq. (2) in eq. (1) we obtain the relation and, using the Glauber-Gribov formalism, calculate the cross section on a nucleus. In leading order (i.e., for the coherent scattering on two nucleons) one obtains the expression where   n CM is the nuclear factor depending on nucleon densities inside of the nucleus and on the coherent length (see, e.g., [33]), The same expression for A T  can be obtained using the Regge-Gribov framework. Here, one needs the differential cross section for the diffractive production which, in the GVD approach (assuming the diagonal approximation, as in eq. (1), and the approximation of a zero width, as in eq. (4)), is given by the formula [34] In these equations, X M is the invariant mass of the Xsystem produced in the diffractive process.
As mentioned in the Introduction, we use in the present work the aligned jet version of GVD. In this version, all members of GVD sums, in particular, in eqs. (10) and (11) (18) in GVD expressions for , TL  .
In calculations of the nucleon structure functions of DIS, in the region of small and medium 2 Q , the number of those vector mesons which saturate the GVD sums is around 8-9, if their masses are given by eq.(7) with 2 a  . We assume that those vector mesons which almost do not contribute to the total photoabsorption cross section (but contribute noticeably to the diffractive cross section) form a high-mass continuum. For a description of the diffractive cross section in the region of large invariant masses X M it is natural to use Regge parameterization. Concretely, the triple-pomeron limit should be good enough. The corresponding parameterization is rather simple, Here, So, now, for a case of the transverse virtual photons one has the sum: where soft T  is given by eq.(10) (with replacements introduced in eq. (18)). From here, and everywhere below in the text, we change the notation designating by T  just the sum of the soft and hard parts. If we assume, on a moment, that hadronic configurations which constitute the hard component interact completely incoherently with nucleons of the target nucleus (due to the colour transparency phenomenon), then we have, in an approximation of the pure  -dominance, the simple expression [1] for the shadowing effect: . , In a general case, however, one must take into account the shadowing effect from the hard component and, also, include into a consideration the contribution from longitudinal virtual photons. By definition, the shadowing coefficient is given by the general formula It is convenient, for more clarity, to introduce two shadowing coefficients, soft and hard ones, If there is no shadowing in the hard part then 1 hard   . The total shadowing coefficient is given by the sum: 22 22 .

III. SHADOWING OF THE SOFT COMPONENT
According to the previous Section, the soft component of the nuclear structure function (and the corresponding photoabsorption cross section) contains the vector meson part and the high-mass continuum part, Consider, at first, the vector meson part. The shadowing correction is given by the quantity The contribution to this correction from the transverse photons can be extracted from eq. (11): '...
For taking into account the higher rescattering terms we introduce in the right part of eq. (11) an eikonal factor F (inside of the integral over the impact parameter in eq. (12)). This factor is ,, Here, the factor  is the ratio [26] for details). The contribution of the high-mass continuum into the soft shadowing correction is given by the formula followed from eq. (16): For a comparison with experimental data it is more convenient to use the following variables: To our knowledge, the most recent parameterizations which are valid for a broad interval of values of the variables are based on the well-known BEKW model [21]. Specifically, (neglecting the small contribution from the longitudinal virtual photons) one has [37,38] Here  

IV. SHADOWING OF THE HARD COMPONENT
The qq -pairs, produced by the virtual photons, with high transverse momenta have relatively small transverse sizes and their interaction with the nucleon can be described, in a language of Regge theory, by an exchange of the perturbative ("hard") pomeron [39]. In terms of the colour dipole model, the photoabsorption cross sections on the nucleon are given by the integrals and it is assumed that the s -dependence of this cross section is more strong than in a case of the soft pomeron.
The corresponding shadowing correction is calculated with a help of the general formula of eq. (16). In this formula, the nuclear factor    (45) Finally, the shadowing correction for the hard component is given by the expression (summing over photon polarizations) 1 , , and , hard TL  are calculated using eq. (42).

V. RESULTS OF CALCULATIONS OF SHADOWING COEFFICIENTS
According to derivations of the previous Sections the total shadowing coefficient is calculated using the expression In our previous work [43] we supposed (in calculations of shadowing) that the soft hadronic fluctuations of the virtual photon consist of the one separate vector meson ( 0  ) and the continuum with the border mass 1.5 GeV. In the present work we used the GVD approach and, respectively, took into account excited states of the meson family (eight mesons, in addition to 0  ) with masses determined by eq. (7) with a =2. The border mass of the continuum is equal to 3.3 GeV.
Calculating shadowing corrections from separated vector mesons we should, for a comparison with data in the region of large values of Bjorken x , slightly modify eq. (33) and its analogue in a case of the longitudinal photon, inserting into their integrands the "damping factor" (see, e.g., [44,45]). This factor is necessary because in the region of For a calculation of the shadowing correction due to the high mass continuum we use eqs. (35)(36)(37)(38)(39) To take into account phenomenologically the effects of gluon saturation we modified the exponential term in the formula (55) [26]. Namely, we assumed that Q . a) solid curve: our main result, dashed curve: the same, but without taking into account gluon saturation effects; b) the same as a) but for 22 5 Q GeV  .
On Fig.6 the A-dependence of  is shown for two values of x , for a characteristic value of 2 Q . This figure shows the size of the predicted change of shadowing in a region of very small x where there is no data.
On Fig.7 we show the comparison of results of our calculations of the shadowing coefficients with experimental data from E665 [46] and NMC [47] collaborations (the collection of experimental points is borrowed from the paper [48]). Different Finally, Fig. 8 shows a comparison of our predictions for nuclear shadowing for 208 Pb, at 2 Q =3 GeV 2 , with the results of calculations using models of other authors. One can see that the disagreement between different predictions fastly grows with a decrease of x . This disagreement is rather large even at x = (10 -4 -10 -5 ) where the gluon saturation effect which seems to be the main source of uncertainty is still relatively small.

VI. CONCLUSIONS
In the present paper, for a description of the fluctuations of the virtual photon, i.e., intermediate states that interact strongly with the target nucleon, the hadronic (rather than quark-gluon) representation is used. Even when we consider the hard interactions of non-aligned qq -pairs we use for a calculation of the corresponding cross section the phenomenological concepts of hadron dominance [49,50] and hard pomeron [39] rather than perturbative QCD directly (calculations of the hard part of * N   using the framework of perturbative QCD are carried out, e.g., in [52,53]). An use of the hadronic basis has, though, a large advantage: it allows to neglect non-diagonal transitions (such as corresponding to a production of qq -pairs and qqg systems by the virtual photons. We assumed, in eq. (39), that our high-mass continuum arises from ( qqg + highmass qq ) -part of   At smallest values of x a modification of the formula for the hard cross section, hard qqN  , seems to be necessary, due to an influence of gluon saturation effects. The modification used in the present paper was suggested in [26]. The predictions of [26] for the nucleon structure function 2N F at smallest x and 2 Q obtained with an use of this modification are close to those of GBW model [53]. As shown in the paper (see Fig. 5) an accounting for the gluon saturation effects is very essential for predictions of shadowing at x  10 -4 -10 -5 .