Coulomb corrections to inclusive cross sections at the future Electron - Ion Collider

The experimental results of the future electron -- ion ($e A$) collider are expected to constrain the dynamics of the strong interactions at small values of the Bjorken -- $x$ variable and large nuclei. Recently it has been suggested that Coulomb corrections can be important in inclusive and diffractive $eA$ interactions. In this paper we present a detailed investigation of the impact of the Coulomb corrections to some of the observables that will be measured in the future $eA$ collider. In particular, we estimate the magnitude of these corrections for the charm and longitudinal cross sections in inclusive and diffractive interactions. Our results demonstrate that the Coulomb corrections for these observables are negligible, which implies that they can be used to probe the QCD dynamics.


I. INTRODUCTION
The future Electron -Ion Collider (EIC) [1][2][3][4] will allow us to study the hadronic structure in the regime of large partonic densities and high strong field strengths, which are expected to modify the linear evolution equations [5,6] normally used in Quantum Chromodynamics (QCD) at high energies. We expect that the EIC experimental data will be able to establish the presence of gluon saturation effects, the magnitude of the associated non-linear corrections and determine what is the correct theoretical framework for their description (For reviews see, e.g., Ref. [7]). In particular, the enhancement of the non-linear effects with the nuclear mass number has motivated the development of several studies on the implications of these effects in inclusive and exclusive observables which could be measured in Electron -Ion Colliders [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. These studies have demonstrated that the eA collider is the ideal environment to improve our undestanding of the strong interactions at high energies.
Deep inelastic eA scattering is characterized by a large electron energy loss ν (in the target rest frame) and an invariant momentum transfer q 2 ≡ −Q 2 between the incoming and outgoing electron such that x = Q 2 /2m N ν is fixed (m N is the target mass). In terms of Fock states we then view eA scattering as follows: the electron emits a photon (|e >→ |eγ >) with E γ = ν and p 2 t γ ≈ Q 2 . Afterwards the photon splits into a qq pair (|eγ >→ |eqq >), which has electric and color charges, and typically travels a distance l c ≈ 1/m N x, referred to as the coherence length, before interacting in the target. For small x, the photon is converted into a qq pair at a large distance before the scattering. Consequently, the space-time picture of the DIS in the target rest frame can be viewed as the decay of the virtual photon at high energy into a quark-antiquark pair, which subsequently interacts with the target. If the target is a large -A nucleus, it is characterized by a strong color field, generated by the fusion of the color fields of its nucleons. Consequently, in order to describe the dipole -nucleus scattering one must take into account the multiple interactions of the dipole with the nuclear color field [7]. This aspect has been considered in the phenomenological works that have studied the impact of the non -linear corrections to the QCD dynamics on the observables that will be measured in the future eA collider [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. However, these studies have not taken into account the contribution of the Coulomb corrections to the dipole -nucleus interaction. As the qq dipole has electric charge and a large nucleus generates a strong electromagnetic field, we might expect that the nuclear DIS receives a substantial contribution from electromagnetic interactions of the dipole with the nucleus, which is known as Coulomb correction. A first analysis of the impact of these corrections to the DIS off a heavy nucleus has been performed in Ref. [26] (See also Ref. [27]) and has indicated that the Coulomb corrections reach to 20 % of the inclusive cross section and about 40% of the diffractive cross section in the kinematical range of small values of the Bjorken-x variable and low photon virtualities Q 2 . Considering the large magnitude predicted for these corrections, it is important to investigate in more detail how the behavior of different observables will be modified by the Coulomb corrections in the kinematical range that will be probed in the future EIC. Apart from the total inclusive and diffractive cross sections discussed in Ref. [26], the EIC is expected to measure the heavy quark and longitudinal contributions to these cross sections. In this paper we extend the study perfomed in Ref. [26] and analyze the impact of the Coulomb corrections on these quantities. Our goal is to verify if it is possible to disentangle the Coulomb corrections from the QCD interactions in these different observables and, consequently, to probe the QCD dynamics at high energies. This paper is organized as follows. In the next Section we briefly review the description of the photon-nucleus interaction in the dipole approach and establish the relation between the measured inclusive and diffractive cross sections and the color/electric dipole target elastic scattering amplitude. Moreover, the treatment of the strong and electromagnetic interactions in the nuclear field is presented. In Section III we present a detailed study of the Coulomb corrections to the charm and longitudinal contributions to the inclusive and diffractive cross sections. Our focus is on predictions for DIS off a heavy nucleus in the kinematical region of small values of x and Q 2 . Finally, in Section IV we summarize our main results and conclusions.

II. THEORETICAL FRAMEWORK
The basic idea of the dipole approach is that the electron-nucleus interaction at high energies can be factorized in terms of the fluctuation of the virtual photon into a qq dipole and the dipole-nucleus scattering. As a consequence, the longitudinal and transverse γ * A cross sections can be expressed as follows [28], where the variable r defines the relative transverse separation of the pair (dipole) and z (1 − z) is the longitudinal momentum fraction of the quark (antiquark) of flavor f . In the case of diffractive eA collisions, the dipole approach predicts that the cross sections can be expressed as follows The photon wave functions Ψ L,T are determined from light cone perturbation theory and are given by and The auxiliary variable ε 2 = z(1 − z) Q 2 + m 2 f depends on the quark mass, m f . The K 0,1 are the Mcdonald functions and the summation is performed over the quark flavors. Here, we consider the light quark masses to be m f = 0.14 GeV (for f = u, d, s) and for the heavy ones we consider m c = 1.4 GeV and m b = 4.5 GeV.
The function N (x, r, b) is the quark dipole-nucleus forward scattering amplitude for a given impact parameter b which encodes all the information about the scattering due to strong and electromagnetic interactions. In order to estimate this quantity we will consider the Glauber multiple scattering approach [29,30], which is justified at high energies when the incident particle passes through the nucleus in a very short time, so that the change of the positions of the nucleons can be neglected. In this model the scattering amplitude obtained by calculating the scattering from instantaneously fixed nucleons is then averaged over the initial and final nucleus wave functions. The main assumption is that the total eikonal phase acquired by the particle when passing through the nucleus at a fixed impact parameter is equal to the sum of the phases from the individual nucleons. During the last decades, the Glauber model has been sucessfully applied to describe proton -nucleus collisions (For a review see e.g. Ref. [31]). In particular, the impact of the Coulomb corrections has been studied by several authors (See e.g. Refs. [30,[32][33][34]), who have demonstrated that it plays an important role in heavy nuclei, influencing the differential cross sections around the minima of the squared momentum transfer (t) distributions. The basic assumption in these studies is that the proton -nucleon phase shift can be described by the sum of purely strong and Coulomb phase shifts [30,32,33]. In Refs. [26,35,36], the Glauber model has been applied to describe the dipole -nucleus interaction and the additivity of the strong and Coulomb phase shifts to estimate the dipole -nucleon was assumed. As a consequence, the dipole -nucleus forward scattering amplitude can be expressed as follows [26,36] where iΓ em and iΓ s are the electromagnetic and strong contribution to the dipole -nucleon elastic scattering amplitude, respectively. At large impact parameters (b > R A ) the strong interaction contribution is expected to be small, i.e. A[iΓ s ] → 0, and hence On the other hand, at small impact parameters (b < R A ), the strong interaction becomes dominant and the dipolenucleus scattering amplitude can be approximated by In what follows we will assume that these asymptotic solutions can be used to estimate the dipole -nucleus scattering amplitude for b ∼ R A , which is the region of interest for the deep inelastic scattering. In order to calculate the inclusive and diffractive cross sections we must specify the models used for the electromagnetic and strong dipolenucleus scattering amplitude. Following Ref. [26] we will assume that where Q s,A is the nuclear saturation scale, which we will assume to be given by is the saturation scale for a proton. This model is a naive generalization for the nucleus of the saturation model proposed in Refs. [37], which captures the main properties of the high energy evolution equations and it is suitable for describing the non -linear physics in the small-x region. In particular, this model implies that N s ∝ r 2 (color transparency) at small pair separations and that the multiple scatterings are ressumed in a Glauber -inspired way. Such model was used in Refs. [9,10] to estimate the impact of the saturation physics in the observables that will be measured in the future eA collider. In our calculations we will consider the parameters obtained from a fit to the HERA data in Ref. [37]: and b<RA For the electromagnetic case we will follow the approach proposed in Refs. [26,36]. In particular, we will assume that the leading term in α for the dipole -nucleon scattering arises from a single photon exchange, with the elastic dipole -nucleon amplitude being approximately real. As demonstrated in Ref. [35], at the Born approximation, the contribution of the imaginary part is proportional to α 2 Z ∼ α in the Zα ∼ 1, while the contribution of the real part is of the order of α 2 Z 2 ∼ 1. Considering the Weizscker -Williams approximation, where the t -channels photons at high energies are assumed to be almost real, the electromagnetic dipole -nucleon elastic scattering amplitude was derived in Ref. [35], being given by Moreover, we will take into account that for the deep inelastic scattering the main contributions for the DIS cross sections comes from values of pair separations of the order r ∼ 1/m f [9,10]. Consequently, we will have that in general r ≪ b, which allows to simplify the expression for the dipole -nucleon scattering amplitude. Such inequality becomes stronger in the case of the heavy quarks. In particular, for b ≫ r the Eq. (11) can be simplified and becomes which implies that and Consequently, we have where m N is the nucleon mass and W is the photon -nucleus center of mass energy. Moreover, b max = s/(4m N m 2 f ) is a long distance cutoff of the b integral, which is directly related to the minimum transverse momentum at which the Weizsacker -Williams approximation still is valid [35]. As in demonstrated in Ref. [35], one have that b max ≫ R A , which justifies the approximations used in the derivation of the Eqs. (11) - (16). Considering the same approximations, for b > R A we find Using that cos 2 [Z iΓ em ] = 1 + O(α 4 r 4 /b 4 ), yields which implies that b>RA As obtained in Ref. [26], the electromagnetic contribution for the inclusive and diffractive cross sections are equal, differently from the strong one. As a consequence we expect that its impact on inclusive and diffractive observables will be different. In the next Section we will estimate how the electromagnetic contribution modifies these observables in the kinematical range that will be probed by the future electron -ion collider.

III. RESULTS AND DISCUSSION
In this Section we present a detailed study of the flavor and polarization decomposition of inclusive and diffractive cross sections including the Coulomb correction. Improving the work done in Ref. [26], we will include in our analysis the contribution of the heavy quarks (charm and bottom). As discussed e.g. in Ref. [9,15], and shown in what follows, the charm contribution is expected to be ≈ 20% of the total cross section at small values of x and low virtualities. Moreover, it is considered an important probe of the nuclear gluon distribution. Similar expectation also motivates  the analysis of the longitudinal cross section (See e.g. [14,38]). Our goal is to verify the impact of the Coulomb corrections in these observables in inclusive and diffractive interactions.
We start our analysis presenting in Fig 1 the ratio between the Coulomb and strong (QCD) contributions to the inclusive (left panel) and diffractive (right panel) cross sections. In Fig. 1 the ratios are shown as a function of photon virtuality Q 2 for a fixed value of Bjorken variable, x = 10 −4 , and three distinct nuclei: Gold (solid line), Silver (dashed line) and Calcium (dotted line). The results are consistent with those obtained in Ref. [26], even including the heavy quark contribution, showing that the Coulomb correction is quite important in the kinematical region of low-Q 2 and small-x and it is enhanced for the heavier nuclei. In the perturbative range that will be probed in the future EIC (Q 2 > ∼ 1 GeV 2 ) and A = 197, the Coulomb contribution is ≈ 10 (21)% for the inclusive (diffractive) cross section. The different impact of the Coulomb interactions on the inclusive and diffractive cross sections can be understood as follows. In contrast to the inclusive cross section, the main contribution to the diffractive cross section comes from larger dipoles (See e.g. Ref. [10]) and the electromagnetic terms are proportional to r 2 [See Eqs. (16) and (19)]. As a consequence, the presence of Coulomb corrections modifies the ratio between the diffractive and the total cross section, which is expected to be measured in the future electron -ion collider. Our predictions for the Q 2 and x dependence of this ratio are presented in Fig. 2. We compare the predictions obtained with the sum of the electromagnetic and strong contributions, denoted total in the figure, with those derived disregarding the Coulomb corrections. We observe that the ratio is enhanced by the electromagnetic contribution, in particular at low Q 2 and small -x. At Q 2 = 1 GeV 2 , this enhancement is ≈ 10%.
The r 2 dependence of the electromagnetic contribution has a direct impact on the Coulomb corrections to the heavy quark cross sections. As these cross sections are dominated by small dipoles, we expect that the effect of the Coulomb corrections will be smaller here than in the case of light quark production. This expectation is confirmed by the results shown in Fig. 3, where we present the flavor decomposition of the ratio between the electromagnetic and strong contributions. We observe that for Q 2 ≈ 1 GeV 2 the Coulomb corrections can be disregarded in the charm and bottom production in the inclusive case and are of order of 2 % in diffractive interactions. At larger values of Q 2 , the heavy quark contribution to the total cross sections increases as well as the Coulomb corrections. We have verified that the impact of the Coulomb corrections to the charm and bottom production is almost x independent in the range 1 ≤ Q 2 ≤ 10 GeV 2 . Another interesting aspect is that the inclusion of the heavy quarks decreases the magnitude of the Coulomb corrections for the total cross sections in comparison to those obtained considering only light quarks, denoted by u + d + s in the figure.
Let us now analyze more carefully the impact of the Coulomb corrections on the charm cross section, which is expected to be measured in the future electron -ion collider. In Fig. 4 we present our predictions for the Q 2 (left panel) and x (right panel) dependence of the ratio between the charm and the total cross sections for inclusive and diffractive interactions. The predictions obtained disregarding the Coulomb corrections are denoted as strong in the figure. The charm contribution increases with Q 2 and at smaller values of x. Moreover, it is larger for inclusive processes. Our results indicate that the inclusion of the Coulomb corrections implies a mild decreasing of the ratios. In particular for inclusive processes, the Q 2 and x dependence of the ratios are only slightly modified by Coulomb corrections in the perturbative Q 2 range that will probed in the future electron -ion collider. This result implies that the study of the charm production is a good probe of the high energy regime of the QCD dynamics.
Let us now consider the impact of the Coulomb corrections on the longitudinal and transverse cross sections in inclusive and diffractive interactions. In Fig. 5 (upper panels) we present our results for the ratio between the electromagnetic and strong longitudinal cross sections. On the left (right) panel we show the inclusive (diffractive) cross sections. In the inclusive case, the ratio rapidly decreases with Q 2 and becomes smaller than 5 % for Q 2 > ∼ 1.5 GeV 2 . In the diffractive case, the Coulomb correction is a factor 2 larger in the same kinematical range. As expected from our previous analysis, the charm contribution is small in the 1 ≤ Q 2 ≤ 10 GeV 2 range. The results presented in the lower panels of Fig. 5 are analogous to those of the upper panels but refer to the transverse cross sections. They indicate that the Coulomb corrections to both longitudinal and transverse cross sections are comparable. Moreover, compared to the strong cross sections, they are small everywhere, except in the very low Q 2 (Q 2 ≃ 0.1 GeV 2 ) region, where dipoles of larger size dominate the cross sections. In the figures we also see that for the charm cross sections, the Coulomb corrections are very small. The fact that the longitudinal cross section is not sensitive to the Coulomb corrections is illustrated in a different way in Fig. 6 where we present the Q 2 and x dependence of the ratio σ L /σ tot in inclusive and diffractive interactions. These results demonstrate that the inclusion of the Coulomb corrections has no impact on the ratio. The same is true for the transverse cross sections. Therefore, the study of the longitudinal and transverse cross sections can also be useful to understand the QCD dynamics at small -x.
A final comment is in order. In the analysis presented above we have assumed, following Refs. [26,36], that the modelling of the strong dipole -nucleus scattering amplitude is given by Eq. (8). However, as discussed in Refs. [15,19], a more realistic model for N s (x, r, b) can be derived using the Glauber -Mueller (GM) approach [39]. In particular, in Ref. [15] two of the authors have demonstrated that the GM approach is able to describe the current data for the nuclear structure function. In order to verify the dependence of our predictions on the model used to describe N s we also have estimated the different observables discussed in this Section using the GM approach and have obtained that the contribution of the Coulomb corrections still are smaller than those presented here.

IV. SUMMARY
In this paper we have studied in detail the Coulomb contribution to some of the observables that will be measured in the future eA collider. In particular, we study the impact of these corrections on the total, charm and longitudinal cross sections of inclusive and diffractive interactions. Our analysis is motivated by the fact that these observables are expected to probe the QCD dynamics and constrain the description of the non -linear effects. Our results indicate that the Coulomb corrections to the total cross sections are important at low -Q 2 and small values of x and are larger for diffractive interactions. In particular, the ratio between the diffractive and total cross sections are enhanced by ≈ 10%. In contrast, our results indicate the impact of the Coulomb corrections on the transverse and longitudinal cross sections is small and it is negligible on the charm cross sections. Therefore, these observables (especially the latter) can be considered clean probes of the QCD dynamics.