A simple model for the charm structure function of nuclei

In this paper, we have investigated the importance of quark charm in nuclear structure functions in the color dipole model at small $x$. The charm structure function per nucleon $F_{2}^{cA}/A$ for light and heavy nuclei in a wide range of transverse separations $\mathrm{r}$ with renormalization and factorization scales are considered. Bounds on the ratio $F^{cA}_{2}/AF^{A}_{2}$ for nuclei are well described with respect to the electron-ion future colliders kinematic range, i.e, EIC and EIcC colliders.


I. Introduction
Electron-nucleus scattering experiments provide vital complementary information to test, assess and validate different nuclear models and data collected from the electron scattering controlled kinematics, large statistics and high precision allow one to constrain nuclear properties and specific interaction processes [1].The study of nuclear structure function and its modification at small Bjorken x, is a very interesting subject in compared to those for free nucleon observed in deep inelastic scattering (DIS) [2].For a nucleus A with Z protons and N = A − Z neutrons, the nuclear parton distribution functions (nPDFs) are defined into the parton distribution functions (PDFs) of a bound proton and neutron (i.e., f p/A i and f n/A i respectively) by the following form where, the bound nucleon PDFs are different from those of a free proton by the nuclear modification factors as The modification of nuclear structure functions at small Bjorken x in comparison with the free nucleon observed in DIS is defined by small-x shadowing followed by antishadowing and EMC-effect, and at the large-x is Fermi motion.At small values of the Bjorken variable x (for x < 0.01), the shadowing effect is seen in nuclear DIS.In the shadowing region, the structure function F 2 per nucleon turns out to be smaller in nuclei than in a free nucleon [3].This effect manifests itself as an inequality F A 2 /(AF N 2 ) < 1, where A is the number of nucleons in a nuclear target.Indeed, unitarity driven nuclear shadowing becomes important at x≪x A = 1/(m N R A ) = 0.15A −1/3 where R A is the radius of the target nucleus and m N is the nucleon mass [4] 1 .Nuclear shadowing, in the laboratory frame, derives from the coherent interaction of qq, qqg,... states.The Fock state expansion of the physical virtual photon reads |γ * >= √ z g Ψ qq |qq > +Φ qqg |qqg >, where Ψ qq and Φ qqg are the light-cone wavefunctions of the qq and qqg states.Here √ z g is the renormalization of the qqg state by the virtual radiative corrections for the qqg state.For the lowest |qq > Fock component of the photon, the interaction of qq dipole of transverse separation r with a nucleon represents with the dipole cross section σ qq (r) [5,6].Indeed, the incoming virtual photon splits into a colorless qq pair long before reaching the nucleus, and this dipole interacts with typical hadronic cross sections which results in absorption.The key feature is the connection of the dipole-target amplitude to the integrated gluon density where, at very low x, the parton saturation models illuminate the behavior of the gluon density.The dipole cross section can be derived from the Balitsky-Kovchegov (BK) equation [9,10], which established a non-linear evolution equation to describe the high energy scattering of a qq dipole on a target in the fixed coupling case based on the concept of saturation.For the study of saturation of nuclei, those have an advantage over protons since they have more gluons to start with.Therefore, non-linear effects in the evolution of the nuclear gluon distribution should set in at much lower energy than for protons.DIS on heavy nuclei is expected to probe the color dipole cross section in a way different from DIS on nucleons.Specifically, the larger the dipole size is the stronger nuclear screening [11][12][13][14][15]. Unitarity constraints for deep-inelastic scattering on nuclei predicted in Ref. [16].Evidently, the nuclear shadowing (screening) depends on high mass diffraction.It is expected that measurements over the extended x and Q2 ranges, which would become possible for future experiments, in particular, for experiments at the Electron-Ion Collider (EIC) [17] and Electron-Ion Collider in China (EIcC) [18], will give more information in order to discriminate between the distinct models of shadowing of the QCD dynamics at small x.These future facilities will probe nuclear structure over a broad range of x and Q 2 and the analysis of the nuclear effects in deep inelastic scattering has been a topic of discussion in the community in recent times 2 .One of the main physics goals of these future QCD laboratories at small x will be to unambiguously unveil the onset of the so-called gluon saturation regime of QCD, which is characterized by a transverse momentum scale, the saturation scale Q s (x), at which non-linearities become of comparable importance to linear evolution.In the last years the analysis of the nuclear effects in deep inelastic scattering (DIS) has been extensively discussed in the literature [19][20][21][22][23][24][25][26][27][28][29][30][31].In this paper there is a good chance to produce interesting new predictions for the charm structure function of nuclei for future experiments in the low x region.These calculations are based on the color dipole picture (CDP) with a characteristic saturation momentum, Q A s .We analyze the charm quark structure functions in nuclei and those ratios in a wide range of r in section II.In this section, the heavy quark structure functions can be combined with the Sudakov form factor. Section III contains our results and conclusions.

II. Method
The cross section in the dipole formulation of the photon-nucleon scattering is defined, with respect to the polarization (transverse, T, or longitudinal, L) of the virtual photon, by where Ψ L,T is the corresponding photon wave function in mixed representation and σ dip (x, r) is the dipole crosssection which related to the imaginary part of the (qq)p forward scattering amplitude.The transverse dipole size r and the longitudinal momentum fraction z due to the photon momentum are defined.The variable z, with 0 ≤ z ≤ 1, characterizes the distribution of the momenta between quark and antiquark.The Golec-Biernat and Wusthoff (GBW) model [32] is a model for the so-called dipole cross section, that is used to descrobed the inclusive DIS data for x < 0.01 and all Q 2 .The model reads where Q sat (x) is the saturation scale defined as Q 2 sat (x) = Q 2 0 (x 0 /x) λ for the proton.Three parameters, σ 0 , λ and x 0 , were determined [33] from a fit to the HERA data and have values, σ 0 = 27.32 ± 0.35 mb, λ = 0.248 ± 0.002 and x 0 /10 −4 = 0.42±0.04for the 4-flavor respectively.The fixed parameters are 14 GeV and m c = 1.4 GeV.The quantity x used in expressions above is the modified Bjorken variable, x = x Bj (1 + , with m q being the effective quark mass.This replacing is a simple way to regular the divergence of the cross section.This modification is quite important as heavy quarks contribution are taken into account and implies that the value of the quark mass plays an important role in avoiding the divergence of the cross section.The Bartels, Golec-Biernat and Kowalski (BGK) model [34], is another phenomenological approach to the color dipole cross section and reads The scale µ 2 is connected to the size of the dipole and takes the form µ 2 = C r 2 + µ 2 0 , where the parameters C and µ 0 are determined from a fit to DIS data [33].Here g(x, µ 2 ) is the gluon collinear PDF.In the color transparency domain, r→0, the dipole cross section is related to the gluon density by the following form [35] The gluon distribution in the GBW and BGK models take the form where α s is the running coupling at µ 2 scale [36].The expression for the nuclear gluon distribution xg A (x, µ 2 ) is the same expect for the change Q 2 s →Q 2A s with the replacement the area of the target with the coefficient A 2/3 .Therefore, the gluon distribution for a nuclear target with the mass number A is defined by where In Ref. [25], it was found δ = 0.79±0.02and the nuclear radius is given by the usual parameterization R A = (1.12A1/3 − 0.86A −1/3 ) fm and πR 2 p = 1.55±0.02fm 2 .The charm structure function in nuclei, owing to the dominance of the gluon distribution, in the collinear generalized double asymptotic scaling (DAS) approach [37] is defined by the following form in the small x region as where B 2,g is the collinear Wilson coefficient function in the high energy regime [38] and n denotes the order in running coupling α s .Here, e 2 c is the squared charge of the charm and ξ r = . The default renormalisation and factorization scales are set to be equal µ 2 R = µ 2 r + 4m 2 c and µ 2 F = µ 2 r .In addition, we consider bounds for F cA 2 /F A 2 follows from the standard nuclear dipole picture which gives correlated values in estimate F A L /F A 2 into the higher Fock components of the photon wave function.The nuclear structure function F A 2 can be obtained from the γ * A cross section through the relation where the nuclear cross section is related to the proton cross section by the following form where the γ * p cross section reads [25,28] Here γ E and Γ(0, η) are the Euler constant and the incomplete Γ function respectively, where the fit parameters (i.e., a and b) are a = 1.868 and b = 0.746.The bound value of F cA 2 /F A 2 is obtained by the following form which will be interesting in EIC and EIcC colliders in the future energy range.Especially at the EIC, it is expected to be probed at an essentially low x (up to x∼10 −4 ), thus providing us with new information on the charm quark density in a nuclei.Indeed, both EIC in the small-x region and EIcC at moderate x give us nuclear modification of the structure functions and hadron production in deep inelastic scattering eA collisions and new information on the parton distribution in nuclei [38].

III. Numerical Results
In the present paper we consider the charm structure function of the deep inelastic scattering of nuclei, which is directly related with the gluon distribution of nuclei in the CDP approach at low x.In this model, the ratios xg A (x, µ 2 )/Axg(x, µ 2 ) and F cA 2 (x, µ 2 r )/AF c 2 (x, µ 2 r ) are independent of the variables and depend on the mass number A by the following form which gives a plateau behavior in the region x≤0.01, which is similar with the results F A 2 /AF 2 in Refs.[25,28].In Fig. 1, we plot this ratio for a wide range of A and observe that this ratio is independent of x and r.We observe that the ratio of R A rapid drop for A < 50 followed by a slow rise for larger A. The minimum values of the ratio are found around A≈56 where these nuclei are the most tightly bound [39].The increase of binding energy to A≈56 decreases the momentum carried by parton distributions in comparison with other nuclei.
Our numerical results for charm structure functions of nuclei per nucleon, F cA 2 /A are shown in Fig. 2  uncertainties of the renormalization and factorization scales in a wide range of r for x = 0.0013 and 0.0130.These results for F cA 2 /A, in Fig. 2, increase as r decreases for light and heavy nuclei.We observe that the results of F cA 2 /A for light and heavy nuclei, at very low r, will increase at the EIC according to the kinematic coverage of the deep inelastic scattering process.The uncertainties due to the renormalization and factorization scales increase as r increase.Owing to the x − Q 2 EIC kinematics3 , the new information on the charm structure function in nuclei can be achieved with x = 0.0130 for r 0.06 GeV −1 and with x = 0.0013 for r 0.2 GeV −1 .As a result, we predict that at low r, the charm structure function will increases at the EIC than the EIcC at high inelasticity according to Table I.In Figs. 3 and 4, we plot the ratio F cA 2 /AF A 2 for nuclei A = 12 and A = 208, respectively, as a function of r with  and have the largest uncertainties at r > 10 −1 GeV −1 .The maximum value of F cA 2 /AF A 2 , in accordance to the EIC kinematic range, is ≃ 1.3 and 5.5 for nuclei A = 12 and A = 208 respectively.Indeed, the importance of the nuclear structure function ratios will depend on the values of F cA 2 /AF A 2 , where these bounds can further restrict the kinematical range of the applicability of the dipone picture in the future electron-ion colliders (i.e., EIC and EIcC).
Summarizing, a simple model for the charm structure functions in nuclei, in the region of small x, has been presented.We analyzed F cA 2 using the gluon density from the GBW and BGK models, inspired by the DAS approach, within the color dipole model to the future electron-ion colliders kinematic range at EIC and EIcC, in a wide range of transverse separations r.Our results indicate that the study of the charm structure functions in the eA process at EIC is ideal for considering the heavy quark effects present in the nuclear structure functions, which, in turn, is a crucial ingredient to estimate the bounds of the processes which will be studied in future accelerators.We have considered the charm structure function F cA 2 /A per nucleon in light and heavy nuclei, then have obtained bounds on F cA 2 /AF A 2 at moderate and large r with the renormalization and factorization scales.We demonstrated the importance of the contributions of F cA 2 /A and F cA 2 /AF A 2 at small r in the EIC and EIcC colliders.The uncertainties of these results are due to the standard variations in the renormalization and factorization scales which increase as r increases.The focus of this paper was to provide an analytical charm structure function per nucleon in nuclei for studying high energy lepton-nucleus phenomena at future colliders such as EIC and the EIcC.

FIG. 2 : 2 r(
FIG. 2: Results of the charm structure function per nucleon F cA 2 /A for light and heavy nuclei in a wide range of the transverse separation r[GeV −1 ] with x = 0.0013 and x = 0.0130.The uncertainties are due to µ 2 = µ 2 r + 4m 2 c (dashed lines) and µ 2 = µ 2 r

TABLE I :
The transverse separation range of r in the future facilities(i.e., EIcC and EIC) with the inelasticity y≤1 for x = 0.0013 and 0.0130.