A model for strong interactions at high energy based on the CGC/saturation approach

We present our first attempt to develop a model for soft interactions at high energy, based on the BFKL Pomeron and the CGC/saturation approach. We construct an eikonal-type model, whose opacity is determined by the exchange of the dressed BFKL Pomeron. The Green's function of the Pomeron is calculated in the framework of the CGC/saturation approach. Using five parameters we achieve a good description of the experimental data at high energies ( $W\,\geq\,0.546\,TeV$). The model results in different behaviour for the single and double diffraction cross sections at high energies. The single diffraction cross section reaches a saturated value (about 10 mb) at high energies, while the double diffraction cross section continues growing slowly


Introduction
The strong interaction at high energies is one of the most difficult and unrewarding problems of high energy physics. The reason for this, is the embryonic stage of our understanding of non-perturbative QCD. Traditionally, we consider the strong interaction at high energy as a typical example of processes that occur at long distances, where the unknown force confining quarks and gluons plays a crucial role, making all our theoretical efforts to treat these processes, fruitless. The description of these processes which we need for practical purposes, is the field of high energy phenomenology, based on Pomeron calculus [1][2][3]. The LHC data [4][5][6][7] showed that models based on this phenomenology failed to provide significant predictions, and were not able to describe the data at high energy [8][9][10][11][12][13]. A glimpse of hope stems from two facts: models that fit the LHC data have been proposed [14][15][16]; and after two decades of experience in high energy phenomenology we have learned, that the more theoretically based the phenomenological input is, the better and more apprehensible, the description of the data we obtain.
In Ref. [17] we reviewed our model which describes successfully all high energy data, including those at the LHC, and which incorporates theoretical ingredients from N=4 SYM [18][19][20] and from perturbative QCD [21][22][23]. In the present paper we wish to improve this approach, by including more theoretical input. First, we introduce a more constructive meaning to our old idea [10,24], that there is only one Pomeron that describes both soft and hard interactions. In perturbative QCD the BFKL Pomeron at high energy takes the following form [21]  G IP (Y, r, R; b) denotes the BFKL Pomeron Green function,ᾱ S the QCD coupling, r and R are the sizes of two interacting dipoles. Y = ln s, where s = W 2 . W denotes the energy of the interaction, and b the impact parameter for the scattering amplitude of two dipoles.
From Eq. (1.1) it is obvious that the BFKL Pomeron is not a pole in angular momentum, but a branch cut, since its Y-dependence has an additional ln s term; it does not reproduce the exponential decrease at large b, which follows from the general properties of analyticity and unitarity [25]; and the exchange of the BFKL Pomeron depends on the sizes of the dipoles, consequently, the BFKL Pomeron does not factorize.
The Pomeron that appears in N=4 SYM [18], corresponds to the BFKL Pomeron in QCD with the following glossary: AdS-CFT correspondence [26]: denotes the QCD-like coupling.
Hereby, we generalize our approach, dealing with the BFKL Pomeron, instead of the simple Regge pole that was used in our previous model.
The second innovation is related to the Pomeron interaction. The LHC data supports the assumption that the dense system of partons (gluons) are produced in the proton-proton interaction at high energy. Such a system of partons appears naturally in the CGC/saturation approach [27][28][29][30][31][32][33], and provides a successful description of the general properties of the average event at the LHC [34], and of the long range rapidity angular correlations [35]. In this paper, we use the CGC/saturation approach to describe the Pomeron interactions, replacing the Pomeron calculus. This strategy allows us not only to treat the Pomeron interactions, but also to include the saturation phenomenon, which was beyond the scope of our previous model. We have previously [36] attempted to include saturation effects using the Balitsky-Kovechegov equation [31]. The approach used in [36] did not include diffraction dissociation processes, and fails to describe the LHC data. We presume that the main shortcoming of our previous attempt was, that the contribution of the Pomeron loops were not taken into account.

Parton cascade of one dipole in the saturation region
The parton cascade which originates from the decay of a gluon to two gluons in QCD, can be described equally well in two ways. The first describes the change of probability to have n gluons at rapidity Y , due to the decay of one gluon to two, using the QCD expression for this decay. The equation in this approach is a linear functional equation for the generating functional (see Refs. [30,32,37]). The alternate way is to sum the Pomeron fan diagrams (see Fig. 1) in the framework of the BFKL Pomeron calculus [38].
In this paper we use the solution of the functional equation which was proposed and discussed in Ref. [39]. For completeness of presentation, we repeat the main ideas of the solution, and explain the physical meaning of the phenomenological parameters that we have introduced in our model.
The first simplification arises when we consider the interaction of the parton cascade with a large target (say with a heavy nucleus). In this case the functional equation reduces to the non-linear Balitsky-Kovchegov equation. The solution of this equation has three distinct kinematic regions.
where Q s denotes the saturation scale [27][28][29]. The non-linear corrections are small and the solution is the BFKL Pomeron; 2. r 2 Q 2 s (Y, b) ∼ 1 (vicinity of the saturation scale). The scattering amplitude has the following form [40,41] A where φ 0 is a constant and γ cr can be found from 3. r 2 Q 2 s (Y, b) ≫ 1 (deeply inside the saturation domain). The amplitude approaches unity [42]: viz.
In spite of our understanding of all qualitative features of the solution, we do not have an analytical solution equation [33], which we need to reconstruct the parton cascade. The parton cascade can be described as the amplitude for the production of dipoles of size r i at impact parameters b i . This amplitude can be written as The solution to the non-linear equation is of the following general form Unfortunately, we cannot find the coefficient C n , for the general non-linear equation. For the case of the simplified BFKL kernel (see Refs. [39,42]) the solution can be found, and we can suggest a simple formula that provides a very accurate solution of Eq. (2.5) (see Ref. [39]).
This formula allows us to find C n (φ 0 , r), and to reconstruct the amplitude of Eq. (2.4).

Summing Pomeron loops (MPSI approximation)
It was shown in Ref. [43], that in the BFKL Pomeron calculus for the parton cascade (see Fig. 1), the integration over rapidities of the triple Pomeron vertices, suggests that the value of the typical rapidity is of the order of Y − Y i ∼ 1/∆ BFKL . Consequently, only large Pomeron loops with rapidity of order Y , contribute at high energies [44]. To sum such loops we use the MPSI approximation developed in Ref. [44]. The essence of this approximation is to use the t-channel unitarity constraint, which is satisfied by the one BFKL Pomeron exchange. Indeed, at any value of Y ′ , the BFKL Pomeron has the following property from t-channel unitarity [27,45] (see Fig. 2 The MPSI approximation is illustrated in Fig. 2-a, where the first non-trivial loop diagram is presented. This approximation enables us to evaluate the Pomeron loops, using the fan diagram structure of the parton cascade. The general MPSI equation for the sum of enhanced Pomeron diagrams, has the form which leads to a new Pomeron Green function (dressed Pomeron).
In the last equation we used Eq. (2.6) and z is given by Figure 2: MPSI approximation: the simplest diagram( Fig. 2-a) and one Pomeron contribution ( Fig. 2 Since for the proton-proton scattering r = R z > 0, we are dealing with parton cascades in the saturation domain.
In Ref. [46] the MPSI approximation as well as the equivalence of the CGC/saturation approach and the BFKL Pomeron calculus, was proven for a wide range of rapidities: (2.11) For larger Y the MPSI approximation does not give the exact answer, since we have not introduced the vertex of the four Pomeron interaction, which violates the simple structure of the parton cascade shown in Fig. 1. The errors that stem from neglecting the four Pomeron interaction, have been evaluated in Ref. [17].

Model: main formulae and parameters
In this section we describe our model. Its main ingredient is the sum of the Pomeron loops, that leads to a new dressed Pomeron Green function.

Dressed Pomeron
The resulting Green function of the Pomeron is given by Eq. (2.9). Using Eq. (2.7) and Eq. (2.9) we obtain the following expression (see Refs. [17,39] for more details ): can be found from Eq. (2.9) and has the form: where we used two inputs: r = R and The parameter λ =ᾱ S χ (γ cr ) /(1− γ cr ) in leading order of perturbative QCD. From phenomenology λ turns out to have the value λ = 0.2÷0.3 [48][49][50]. S (b) is a pure phenomenological profile function which we choose to be of the form The parameter m represents the inverse size of the dipole m ∼ 1/r = 1/R. Unfortunately, we have no theoretical estimate for this mass. It maybe large, reflecting the masses of glueballs and the small size of the typical dipole in a hadron [51]. Note, that S (b) has a correct, exponential decrease at large b. This is an advantage of our approach, as it enables us to introduce a non-pertutbative scale in a physical motivated way, for the observable that characterizes the principle property of the parton cascade. Therefore, in the framework of our approach we do not face the theoretical problem of large b behaviour, which is the main unsolved problem in the CGC/saturation approach [52].

Interaction of dressed Pomerons
The interaction of a dressed Pomeron with a hadron is a non-perturbative problem, which cannot be solved at the moment. From the microscopic point of view this interaction depends on the size of a typical dipole in a hadron, on the probability of finding such a dipole, and on the interaction coupling. Since this interaction originates at long distances, we cannot calculate it even in the CGC/saturation approach. Introducing two phenomenological constants: g and m 1 , we describe the vertex of the hadron-Pomeron interaction as follows To account for the interaction of the dressed Pomerons with hadrons, we use the strategy that has been suggested in Ref. [53], and which is based on the fact that we anticipate the value of g in Eq. (3.4) will be large. In this case we can evaluate the scattering amplitude in the following kinematic region of rapidities: The difference with our previous model reviewed in Ref. [17] lies in the value of G 3IP which was a phenomenological parameter, while now we are able to estimate it from the CGC/saturation approach.
Finally, the opacity Ω has the form In Eq. (3.6) we assumed that m ≫ m 1 . The factor 1.29 stem from estimates of the triple Pomeron vertex in the CGC/saturation approach.

Elastic amplitude
The elastic amplitude is (3.8)

Single diffraction
The cross section for single diffraction can be written as whereN SD (Y ) = d 2 b ′ N SD (Y, b ′ ) and N SD (Y, b) has been calculated in Ref. [39]. It has the form: . This definition of T is valid only in the region where T < 1. A more general formula is given in Ref. [39]. Eq. (3.10) which sums the diagrams of Fig. 4-a, where the double wavy lines crossed by the dotted one, denote the dressed Pomeron structure in terms of the produced particle.
In Reggeon calculus it is referred to as the cut Pomeron. We would like to emphasis that in our approach this contribution is the solution to the equation for single diffractive production of Ref. [54], which is given in Ref. [39]. The profile function for the single diffraction production is taken from Eq.3.25 of Ref. [10].

Double diffraction
The double diffraction cross section has the form where T = T (Y, b) of Eq. (3.2) with the same comments as for Eq. (3.10). Eq. (3.11) sums the diagrams shown in Fig. 4 The profile S DD (b) is given by

Phenomenological parameters
In this section we summarize our phenomenological parameters and provide theoretical estimates for them. Altogether, we have five parameters: g, φ 0 , λ, m and m 1 .
• λ in the CGC/saturation approach, can be calculated in the leading order of perturbative QCD. It characterizes the energy dependence of the saturation scale in proton-proton collisions. Theoretical estimates give λ =ᾱ S χ (γ cr ) /(1 − γ cr ) ≈ 4.88ᾱ S in leading order of perturbative QCD, where γ cr = 0.37. However, the estimates with a running QCD coupling, as well as CGC/saturation phenomenology, lead to λ = 0.2 ÷ 0.3. (1−γcr ) . The exact value of φ 0 cannot be determined without specifying the Pomeron-hadron interaction in more detail than we have. However, φ 0 ∝ᾱ 2 S so we expect φ 0 to be small.
• m 1 and m are pure phenomenological parameters, in our formulae we assumed that m ≫ m 1 . We make this assumption to simplify the formula.
• g is a pure phenomenological parameter which we assumed to be larger thanG 3IP .

Cross sections and the values of the parameters
We determine the five parameters that our model depends on, by fitting to experimental data for the following set of the observable: total, inelastic and elastic cross sections, for single and double diffractive production cross sections, and for the slope of the forward elastic differential cross section. We fit to the high energy data with W ≥ 0.546 TeV. The quality of the fit can be seen from Fig. 5 and the values of parameters are presented in Table 1.  Our first observation is that the values of all parameters are in agreement with our expectations given in section 3.4. Second, the overall fit has χ 2 /d.o.f. ≈ 1 and therefore, we have a good description of the data.
We discuss some regularities in our fit, which could be useful for further and deeper understanding of the microscopic physics. • We obtain a very good description both of values and of energy dependence for total, inelastic and elastic cross sections in a wide energy range: W = 0.546 ÷ 57 T eV . • The values of B el are rather close to the experimental ones, but a glance at Fig. 5-d shows that the energy behavior in our model is milder that the experimental one.

•
Our single diffractive production cross section has values within the experimental error, but our model prefers a lower cross section at W = 7T eV than the one given by ALICE [4]. The model predictions shown in Table 2 support the idea that the single diffraction is saturated at high energy † .

•
The striking feature of our model and, perhaps of the data, is that the double diffractive cross section increases with energy (see Table 2).  Table 2: Cross sections at high energies predicted by the model. † As far as we know, K. Goulianos was the first to predict this feature of single diffraction production at high energies [56], based on a different point of view of high energy interactions, than ours.

Partial amplitudes
. We believe that the information contained in the impact parameter dependence of partial amplitude, is instructive for understanding the nature of strong interactions at high energy. Our present model which is a single channel model has only one elastic amplitude, A el (s, b) which is shown in Fig. 6-a.
One can consider our proton as a gray disk (A el (s, b = 0) < 1), even at energies as high as W = 57 T eV . This behaviour mostly stems from the b-dependence of the Green function of the dressed Pomeron (see Fig. 6-b).
The typical b increases with energy. Note, that such an increase has been seen in the behaviour of B el versus energy (see Fig. 5-d). This behaviour is due to strong saturation in dipole-dipole scattering, as the slope of the Pomeron trajectory for the BFKL Pomeron, is equal to zero.
In Fig. 7 dσ SD (s, b) /db 2 (Eq. (3.9)) and dσ DD (s, b) /db 2 (Eq. (3.11)) are plotted. One can see that the single and double diffraction production have quite different distributions in b. dσ DD (s, b) /db 2 , as is expected, has a peripheral form having a minimum at b = 0, the maximum and the width of the bdistribution of dσ DD (s, b) /db 2 grows considerably with energy. On the other hand, the peripheral nature of the single diffractive production starts to appear only at high energies. The typical distribution has two maxima at b = 0 and b ≈ 1 f m, both decrease with energy, while the width of the distribution slowly increases with energy. Such unexpected behaviour stems mostly from the rather transparent dipole-dipole interaction which appears due to the values of the fitted parameters in our model.

Comparison with other models on the market
In brief, this model is a one channel eikonal -type model, with a dressed Pomeron whose form has been adapted from the CGC/saturation approach. It differs from our previous model (see review [17]) which is a two component model having three different partial amplitudes (see Fig. 8-a). The striking feature of the two component model is that two amplitudes become black at low energies. The resulting elastic amplitude in the two component model is shown in Fig. 8-b, and is similar to that of our present model. This is not surprising, as both models provide a good description of the data. However, the fact that in our present approach none of ingredients: elastic amplitude and Green's function of dressed Pomeron, reach the black limit, looks surprising, especially so, since the recent model proposed by KMR (see Ref. [57] and Fig. 9), supports the fact that two of the partial amplitudes are black.
At first sight, the situation when two quite different models describe the available experimental data equally well, looks discouraging. We wish to emphasis, our two models: the present one and the two component model, lead to completely different predictions for single diffraction: in the first the cross section is saturated, while in the second it grows with energy.
Over the past few years a number of models have been constructed [13,57,58] based on Reggeon Field Theory whose results for energies below that of the LHC are similar i.e. they adhere to the general trend of the experimental data, in that their results for σ tot , σ el , B el , σ SD and σ DD increase with increasing energy.
The same applies to the Monte Carlo programs MBR [59] (which is an "event generator" based on an enhanced PYTHIA8 simulation) and QGSJET-II [16].
Following the appearance of the preliminary results for single and double diffraction cross sections by the TOTEM Collaboration [60] and the CMS Collaboration [61] which suggests that the growth of σ SD and σ DD maybe leveling off (or even decreasing) at W = 7 TeV. KMR [15] have modified their model by including energy dependent couplings, so as to be in accord with the TOTEM results. We would like to stress that the published results of the ALICE collaboration [62] have the single and double diffractive cross sections still increasing at LHC energies.

Conclusions
In this paper we present our attempt to develop a consistent approach based on the BFKL Pomeron and the CGC/saturation approach for soft interactions at high energy. We construct an eikonal-type model whose opacity is determined by the exchange of the dressed BFKL Pomeron. The Green's function of the Pomeron is calculated in the framework of the CGC/saturation approach. Having only five parameters we obtain a good description of the experimental data at high energies ( W ≥ 0.546 T eV ). One of these five parameters λ, determines the energy dependence of the saturation scale, its value λ = 0.323 is a bit higher than the values that has been found from the description of the DIS and heavy ion scattering data, but it is close to them.
Using the value of λ from the fit we can estimate the value of the intercept of the BFKL Pomeron since λ = 4.88ᾱ S while ∆ BFKL = 2.8ᾱ S ≈ 0.2. From Eq. (2.11) we see that we can trust the MPSI approximation for Y ≤ 36, and therefore, the MPSI approximation provides the exact answer for the entire kinematic region of energies.
In our model we find different behaviour for the single and double diffraction cross sections at high energies. The single diffraction reaches a saturated value (about 10 mb) at high energies, while the double diffraction cross section grows steadily. The reason for this, is the different energy and impact parameter dependences, of the diagrams describing σ sd (Fig.4-a) and σ dd (Fig.4-b).
It turns out that in the model, all ingredients are far away from being a black disc, in contradiction to our previous model. This illustrates how important it is to find a theoretical approach for soft processes.
We consider this paper as an attempt to expand the CGC/saturation approach to describe soft processes at high energy. We plan to include more details of the CGC/saturation theory in our model and, in particular, to account for the running QCD coupling. This paper provides a good illustration of how the LHC data has stimulated our thinking.

Acknowledgements
We thank our colleagues at Tel Aviv university and UTFSM for encouraging discussions. Our special thanks go to Carlos Contreras, Alex Kovner and Misha Lublinsky for elucidating discussions on the subject of this  (Fig. 5-a), inelastic ( Fig. 5-b), elastic cross sections (Fig. 5-c), as well as the elastic slope (B el , Fig. 5-d) and single diffraction (Fig. 5-e) and double diffraction (Fig. 5-f) cross sections. The solid lines show our present fit. The data has been taken from Ref. [55] for energies less than the LHC energy. At the LHC energy for total and elastic cross section we use TOTEM data [7] and for single and double diffraction cross sections are taken from Ref. [4].  : Partial amplitudes (T ik (b) ≡ A ik (b)) of the Durham group's model [57]. The figure is taken from Ref. [57]