A new parton model for the soft interactions at high energies

We propose a new parton model and demonstrate that the model describes the relevant experimental data at high energies. The model is based on Pomeron calculus in 1+1 space-time dimensions, as suggested in Ref. [18] and on simple assumptions regarding the hadron structure, related to the impact parameter dependence of the scattering amplitude. This parton model evolves from QCD, assuming that the unknown non-perturbative corrections lead to fixing size of the interacting dipoles. The advantage of this approach is that it satisfies both t-channel and s-channel unitarity, and can be used for summing all diagrams of Pomeron interactions, including Pomeron loops. We can use this approach for all reactions: dilute-dilute (hadron-hadron), dilute-dense (hadron-nucleus) and dense-dense (nucleus-nucleus)for the scattering of parton systems. Unfortunately, we are still far from being able to tackle this problem in the effective QCD theory at high energy (i.e. in the CGC /saturation approach).


Introduction 1
The model (brief review) 2 New parton model.
However, in perturbative QCD the dependence of the Odderon on energy is crucially affected by the shadowing corrections, which lead to a substantial decrease of the Odderon contribution with increasing energy [15,17,18].
In this paper we wish to study the shadowing corrections to the Odderon contribution using the model that we proposed in Ref. [19,20]. The model is based (i) on Pomeron calculus in 1+1 space-time, suggested in Ref. [21], and (ii) on simple assumptions of hadron structure, related to the impact parameter dependence of the scattering amplitude. This parton model stems from QCD, assuming that the unknown non-perturbative corrections lead to determining the size of the interacting dipoles. The advantage of this approach is that it satisfies both the t-channel and s-channel unitarity, and can be used for summing all diagrams of the Pomeron interaction, including Pomeron loops. In other words, we can use this approach for all possible reactions: dilute-dilute (hadron-hadron), dilute-dense (hadron-nucleus) and dense-dense (nucleus-nucleus), parton systems scattering.
The model gives a fairly good description of four experimental observables: σ tot ,σ el , B el and the single diffraction cross sections, for proton-proton scattering, in a two channel model for the structure of hadrons at high energy. The impact parameter dependance of the scattering amplitudes show that soft interactions at high energies measured at the LHC, have a much richer structure than expected.
The goal of this paper is study the influence of the shadowing corrections on the Odderon contribution in our model.

THE MODEL (BRIEF REVIEW)
Our model includes three essential ingredients: (i) the new parton model for the dipole-dipole scattering amplitude that has been discussed above; (ii) the simplified two channel model that enables us to take into account diffractive production in the low mass region, and (iii) the assumptions for impact parameter dependence of the initial conditions. New parton model.
The model that we employ [19][20][21] is based on three ingredients: 1. The Colour Glass Condensate (GCC) approach (see Ref. [15] for a review), which can be re-written in an equivalent form as the interaction of BFKL Pomerons [22] in a limited range of rapidities ( Y ≤ Y max ): where NPM stands for "new parton model". P andP denote the BFKL Pomeron fields. The fact that it is self dual is evident. This Hamiltonian in the limit of smallP reproduces the Balitsky-Kovchegov Hamiltonian H BK ( see Ref. [21] for details). This condition is important for determining the form of H NPM . γ in Eq. (2) denotes the dipole-dipole scattering amplitude, which in QCD is proportional toᾱ 2 S . 3. The new commutation relations: For small γ and in the regime where P andP are also small, we obtain consistent with the standard BFKL Pomeron calculus (see Ref. [21] for details) . In Ref. [21], it was shown that the scattering matrix for the model is given by where p(η) andp(η) are solutions of the classical equations of motion and have the form: where the parameters β and α should be determined from the boundary conditions: Two channel approximation .
In the two channel approximation we replace the rich structure of the diffractively produced states, by a single state with the wave function ψ D . The observed physical hadronic and diffractive states are written in the form Functions ψ 1 and ψ 2 form a complete set of orthogonal functions {ψ i } which diagonalize the interaction matrix T The unitarity constraints take the form where G in i,k denotes the contribution of all non diffractive inelastic processes, i.e. it is the summed probability for these final states to be produced in the scattering of a state i off a state k. In Eq. (10) √ s = W denotes the energy of the colliding hadrons and b denotes the impact parameter. In our approach we used the solution to Eq. (10) given by Eq. (5) and The general formulae.
Initial conditions: Following Ref. [19] we chose the initial conditions in the form: Both p 0i and masses m i , as well as the Pomeron intercept ∆, are parameters of the model, which are determined by fitting to the relevant data. Note, that S (b, m i ) These equation are the explicit solutions to Eq. (6) and Eq. (7). Amplitudes: In the following equations +a ik log(z) log b ik z a ik + 1 The amplitude is given by

Odderon exchange
As has been mentioned, we view the Odderon as a reggeon with negative signature and with the intercept α Odd (t = 0)=1. Generally speaking its contribution to the scattering amplitude has the following form: where η − is a signature factor η = tan 1 2 π α Odd (t) − i , g i Odd is the vertex for the interaction of the Odderon with state i, and α Odd denotes the trajectory. The Odderon appears naturally in perturbative QCD. As one can see from Fig. 1 the QCD Odderon describes the exchange of three gluons and all the interactions between them. The QCD Odderon has the trajectory with the intercept equal to 1 and which does not depend on t [16,17]. Hence, the Odderon only contributes to the real part of the scattering amplitude. For an estimate we will use the following form of the Odderon contribution: where sign plus corresponds to proton-antiproton scattering, while sign minus describes the proton-proton collisions. The value of σ 0 was evaluated in Ref. [24] (see also Ref. [25]) in the framework of perturbative QCD. It turns out that σ 0 = 20.6ᾱ 3 S mb. In perturbative QCD, we expect that B is smaller than for the elastic scattering. We choose B = 5.6 − 6 GeV −2 for our estimates [5,20]. In Eq. (21) we assume that g i Odd (b) in Eq. (20) does not depend on i.

Shadowing corrections
In the two channel model the elastic amplitude is equal to (see Fig. 2 Eq. (22) is the series whose general term is proportional to Ω n ik /n!. In the case of Odderon exchange we need to replace one of Ω ik by O ik (s, b). Hence Ω n ik /n! should be replaced by O ik (s, b) n Ω n−1 ik /n! = O ik (s, b)Ω n−1 ik /(n − 1)!. Finally, we have (see also Ref. [13])

Numerical estimates
In this section we make estimates using our model for Ω ik , with parameters that are given by Table I. In Fig. 3 we plot the b dependence of the Odderon contribution. One can see that the shadowing corrections lead to a considerable ρ=Re/Im, Set I ρ=Re/Im, Set II  Table I are so small that cannot be seen in the figure.
suppression of the Odderon contribution at small b in comparison with Eq. (21) (see red line in Fig. 3). This suppression is much smaller than in our approach, based on CGC [13]. The reason for this is that in our model the value of A el (s, b) tuns out to be smaller than 1 even at very high energies. Due to this O (W, b = 0) = 0 even at W ≈ 100T eV . In Fig. 4 we plot the contribution of the Odderon to the ratio of ρ = Re/Im parts of the scattering amplitude as function of energy. One sees the influence of the shadowing corrections, which induce the energy dependence of this ratio on energy. Eq. (21) shows that the Odderon does not depend on energy without these corrections. This induced energy dependence turns out to be rather large causing a decrease of ρ in the energy range: W = 0.5 ÷ 13 TeV. However, this effect is much smaller than in our previous estimates [13] and the value of ρ does not contradict the experimental data[2? -4].
The shadowing correction has a remarkable effect on the t-dependence of the scattering amplitude (see Fig. 5 ). We see that the shadowing corrections lead to a narrower distribution over t, than the input given by Eq. (21), which is shown in Fig. 5

CONCLUSIONS
In this paper we discussed the Odderon contribution in our model [20] that provides a fairly good description of σ tot ,σ el ,σ diff and B el , especially as related to the energy dependence of these observables. We showed that the shadowing corrections are large and induced considerable dependence on energy for the Odderon contribution, which in perturbative QCD is energy independent . However, this energy dependence does not contradict the experimental data for ρ = Re/Im, if we assume that the Odderon gives a contribution of about 1mb at W=7 TeV. This fact is in striking contrast to our estimates for the CGC based model [13]. The reason for this difference is that the elastic scattering amplitude in the two channel model does not reach the unitarity limit (A el (W, b = 0) = 1 even at very high energies.   Table I) while Fig. 5-b shows the Odderon contribution for set II of parameters.

Set I
We believe that our estimates will be useful for further discussion of the Odderon contribution, especially the t-dependence of the Odderon, which turns out to be quite different from the bare Odderon given by Eq. (21).