# A new parton model for the soft interactions at high energies

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## Abstract

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 Kovner et al. (JHEP 1608:031, 2016), 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 the 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).

## 1 Introduction

The equivalence between the CGC approach and the BFKL Pomeron calculus is very important, since (1) it shows that the CGC approach satisfies t-channel unitarity, which is a prerequisite for any effective theory at high energies^{1} and (2) it allows us to consider elastic and diffraction processes, and processes of multiparticle generation, on the same footing. The last statement follows from the AGK cutting rules [18, 19], which has been proven for inclusive production [20], but not for correlations [21].

*P*and \(\bar{P}\) denote the BFKL Pomeron fields.

*x*,

*y*) on the dipole (

*u*,

*v*)

In Ref. [30] it is shown that the Hamiltonian of Eq. (2) cannot be the correct one, since it violates s-channel unitarity. In other words, we can use \(\mathcal{H}_\mathrm{B}\) only for summing the large Pomeron loops in the MPSI approximation, but cannot use it for a general description of the high energy interaction in QCD, nor even for the description of the DIS, which is given by Balitsky–Kovchegov equation [31, 32, 33], and was considered to be the most reliable equation in the framework of the CGC approach. It should be stressed, that in the CGC approach, we do not have a general Hamiltonian that describes the interaction of two dense, or of two dilute systems of partons, although some work in this direction has been done [34]. Unfortunately, no progress in formulating the BFKL Pomeron calculus for these systems has been achieved in Ref. [30].

*Y*is reduced to

*N*we obtain the non-linear equation:

In this model we encountered the same problems with *s*-channel unitarity as in QCD at high energy, however we have identified the Pomeron Hamiltonian which cures all these problems [30]. In the next section we will briefly review our finding. In Sect. 3 we propose a model based on this Hamiltonian, while in Sect. 4 we compare the predictions of this model with the relevant experimental data. We summarize our results in the Conclusions.

## 2 A new parton model

Below we give a short review of sections 4 and 5 of Ref. [30], where \(1+1\) Reggeon Field Theory (RFT) is discussed. We emphasis section 5, which contains the results that we are going to use in building our model.

### 2.1 Violation of the s-channel unitarity in 1 + 1 RFT

*m*dipoles on a target consisting of \(\bar{n}\) dipoles, is given as by

*P*and \(\bar{P}\) to satisfy the dilute limit algebra with the commutator Eq. (3), such that

*s*-channel unitarity, we consider the infinitesimal evolution (\(\delta Y\)) of the projectile and target wave functions with the Hamiltonian \(\mathcal{H}_\mathrm{BK}\):

*n*-dipole state has the meaning of, the probability to find this number of dipoles in the wave function. The projectile evolution, given by Eq. (11) is unitary: all the probabilities in the evolved state remain positive and smaller than unity, and the sum of the probabilities add up to unity.

On the other hand, the target evolution is non-unitary. Indeed, we face two difficulties with the unitarity: (1) the probability to find the initial state \(|\bar{n}\rangle \) after a short interval of evolution exceeds unity; and (2) the probability to find a state \(|\bar{n} - 1\rangle \) is negative. The coefficients still sum to unity as for the projectile, but clearly the target evolution violates unitarity.

*s*-channel unitarity as in Eqs. (11) and (12) we obtain for the evolution of the target wave function:

### 2.2 Commutators

*P*and \(\bar{P}\) are also small, we obtain

In the calculation of an amplitude of the type of Eq. (17), once all the factors of \(1-\bar{P}\) are commuted through to the left, then in the remaining matrix element \(\bar{P}\) operates on the \(\delta \)-function and thus vanishes. The remaining factors of \((1-P)\) also become unity, since a factor of \(\Phi \) is equivalent to a derivative acting on the \(\delta \)-function, and vanishes when integrated over \(\bar{P}\).

It should be stressed, that the modification of the Pomeron algebra is not a matter of choice, but is necessary to obtain the amplitude of Eq. (17), which is unitary for arbitrary numbers of colliding dipoles. However, the question of the unitarity of the evolution is a completely separate one. We will examine the s-channel unitarity for the Braun Hamiltonian, since we plan to use this Hamiltonian for the description of hadron–hadron scattering at high energy.

*m*and \(\bar{n}\). It is shown in Ref. [30] that the terms that include the four Pomeron interaction, do not help.

### 2.3 The Hamiltonian of the new parton model

It is interesting that Eq. (28) displays saturation behavior very similar to that expected from QCD evolution, namely at large \(\bar{n}\), the change in the wave function is independent of the number of dipoles \(\bar{n}\). In the BK approach in QCD, the wave function never saturates (see Eq. (11)), and saturation of the scattering amplitudes is due to multiple scattering effects.

### 2.4 Equations of motion and the scattering amplitude

*Y*in all our formulae is actually equal to \(\Delta Y\). The exponential “BFKL-like” growth continues until the Pomeron reaches the value \(P(Y)=1\).

*P*(

*Y*). On the other hand, the scattering amplitude in NPM depends on \(\bar{P}\). Nevertheless the two models should be approximately the same in the regime where the BK evolution applies. The results of the estimates in Ref. [30] shows that in the region close to saturation the differences between BK and NPM are quite significant. We will continue the comparison of the two approaches in the following sections, where we construct a realistic model based on the NPM approach.

## 3 The model

To build a model we need to solve two problems: (1) to express parameters \(\alpha \) and \(\beta \) in Eq. (33) in terms of \(p_0\) and \(\bar{p}_0\), and to take the integral over \(\eta \) in Eq. (40); (2) to introduce the non-perturbative structure of hadrons.

### 3.1 Explicit solutions

### 3.2 Non-perturbative structure of hadrons

- 1.
The BFKL Pomeron exchange occurs at fixed impact parameters. In other words the Green function of the Pomeron \(\propto \,\delta ^{(2)}\left( \varvec{b}\right) \). The triple Pomeron vertex (see Fig. 1) does not change the impact parameters. We have discussed this property in the introduction.

- 2.
For nucleus–nucleus collisions the main contributions stems from the ’net’ diagrams of Fig. 1a [27, 28, 29]. In these diagrams the dependence on the impact parameters are concentrated in the vertices of the Pomeron interaction with the nuclei, this dependence has a clear meaning: i.e. the number of nucleons in a nucleus at fixed impact parameter.

- 3.For DIS we have the following formula for the total cross section:where \(Y \,=\,\ln \left( 1/x_{Bj}\right) \) and \(x_{Bj}\) is the Bjorken$$\begin{aligned} N\left( Q, Y; b\right)= & {} \int \frac{d^2 r}{4\,\pi } \int ^1_0 d z \, \Psi _{\gamma ^*}\nonumber \\&\times \left( Q, r, z\right) \,N\left( r, Y; b\right) \,\Psi ^*_V\left( r,z\right) \end{aligned}$$(47)
*x*.*z*is the fraction of energy carried by quark.*Q*denotes the photon virtuality. \( N\left( r, Y; b\right) \) is the scattering amplitude for a dipole of size*r*at impact parameter*b*which is the solution to the Balitsky–Kovchegov equation [31, 32, 33]. \(\Psi _{\gamma ^*}\left( Q, r, z\right) \) is the wave function of the dipole in the virtual photon.Equation (47) splits the calculation of the scattering amplitude into two steps: (1) calculation of the wave functions, and (2) estimates of the dipole scattering amplitude.

- 4.The initial condition for the amplitude \(N\left( r, Y; b\right) \) can be written as follows [43, 44, 45, 46, 47, 48]where \(N^\mathrm{BA}\left( r,Q_T\right) \) is the amplitude for dipole scattering in the Borm approximation . \(T\left( b \right) \) denotes the number of the nucleons at fixed impact parameter$$\begin{aligned}&N\left( r, Y=Y_0 ,b\right) \nonumber \\&\quad =\,i \Bigg ( 1 \,\,-\,\,\exp \Big ( - N^\mathrm{BA}\left( r, Q_T=0\right) \,T_A\left( b \right) \Big )\Bigg )\nonumber \\ \end{aligned}$$(48)
*b*.

*r*that we have in the nucleus at impact parameter

*b*. Equation (48) takes the form

*p*(

*y*) and \(\bar{p}(y)\), considering \(\gamma S_A(b) \,\ll \,1\):

*m*in the two interacting nucleons. It should be stressed that in Eq. (53) we assumed that the typical size of the dipole that is described by the new parton model is 2 /

*m*. In principle, we can introduce this size as a new parameter whose value we will need to determine from comparison with the experimental data.

*b*.

## 4 Comparison with experimental data

### 4.1 Fitting \(\sigma _\mathrm{tot}, \sigma _\mathrm{el}\) and \(B_\mathrm{el}\)

*b*-dependence we suggested a specific form for

*b*-dependence (see Eq. (50)) which is characterized by the dimensional factor

*m*. All three parameters were determined by fitting to the experimental data. We choose to describe three observables: total and elastic cross section and the elastic slope. They have the following expressions through the partial amplitudes:

Fitted parameters

\(\Delta \) | \(p_0\) | m (GeV) | \(\chi ^2\)/d.o.f. |
---|---|---|---|

0.6488 ± 0.030 | 0.489 ± 0.030 | 0.867 ± 0.005 | 1.3 |

From Fig. 2 we see that we fail to describe the experimental data for \( W < 1\) TeV . However, we would like to stress that we used a very naive model for the hadron structure. Our previous experience [3, 4] shows that we need to take into account the processes of diffraction production, which have been neglected in this model. Our main goal is to demonstrate that the suggested model is able to describe the experimental data at high energies.

### 4.2 \(A_{el}(Y, b)\)

Such behaviour of the scattering amplitude \(A_\mathrm{el}(X)\) reflects the fact that in our approach there is no fixed point (1,0) (or/and (0,1)), as occurs in the Braun Hamiltonian, and the scattering of hadrons or nucleus cannot be reduced to BK evolution at high energies.

## 5 Conclusions

In this paper we showed that the experimental data at high energies, can be described in the framework of the new parton model. The model is based on the Pomeron calculus in 1 + 1 space-time, suggested in Ref. [10], and on simple assumptions on the 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 fixing the 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 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. Unfortunately, we are still far from tackling this problem in the framework of QCD effective theory at high energy (CGC/saturation approach).

We achieved quite good descriptions of the three experimental observables: \(\sigma _\mathrm{tot}\),\(\sigma _\mathrm{el}\) and \(B_\mathrm{el}\), especially regarding the energy dependence of these observables. We consider this paper as the first attempt to show that the new parton model can be relevant to the discussion of the experimental data. In spite of the embryonic state of the theory of the quark-gluon confinement, we hope that our model can be viewed as the first step in the right direction regarding the theoretical description of the dilute–dilute parton system scattering at high energy at which, we believe, that such systems become dense.

We are aware that our model is very naive in the description of the hadron structure. We are planning to include diffraction production in our formalism, and to develop a theoretical approach in the framework of the new parton model, so as to be able to treat processes of multiparticle generation.

## Footnotes

## Notes

### Acknowledgements

We thank our colleagues at Tel Aviv University and UTFSM for encouraging discussions. This research was supported by the BSF Grant 2012124, by Proyecto Basal FB 0821 (Chile) , Fondecyt (Chile) Grants 1170319 and 1180118 and by CONICYT Grant PIA ACT1406.

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