# The energy dependence of the diffraction minimum in the elastic scattering and new LHC data

###### Abstract

The soft diffraction phenomena in the elastic proton-proton scattering are reviewed from the viewpoint of experiments at the LHC (TOTEM and ATLAS collaboration). In the framework of the High Energy Generalized Structure (HEGS) model the form of the diffraction minimum in the nucleon-nucleon elastic scattering in a wide energy region is analyzed. The energy dependencies of the main characteristics of the diffraction dip are obtained. The numerical predictions at LHC energies are presented. The comparison of the model predictions with the new LHC data at TeV is made.

###### keywords:

hadrons, high energies, eikonal, diffraction dip, LHC data^{†}

^{†}journal: Nuclear Physics A

## 1 Introduction

A great amount of experimental and theoretical researches of high energy elastic proton-proton and proton-antiproton scattering in a wide region of the momentum transfer provide reach information on these processes Rev-LHC (), which allows us to narrow the circle of examined models and at the same time to set a number of difficult problems, which are not yet solved, concerning mainly the energy dependence of characteristics of these reactions.

It is just this process that allows the verification of the results obtained from the main principles of quantum field theory: the concept of the scattering amplitude as a unified analytic function of its kinematic variables connecting different reaction channels introduced in the dispersion theory by N.N. Bogoliubov. The recent results obtained at the collider accelerators and in the cosmic experiments show a still continuing growth of the total cross sections, the diffraction peak shrinkage and a slow growth of the relation of the elastic to the total cross sections. A especial question is about the behavior of the phase of the elastic scattering amplitude, which can be presented in the form of the ratio of the real to imaginary part of the scattering amplitude - , which is tightly connected with the dispersion relations. Also, the question about the energy dependence of the spin-flip amplitude has to be noted. In most of the early models, as in the ordinary picture of perturbative quantum chromodynamics (PQCD), the spin effects were suppressed at large energies. However, in some models mog2a (); GKS (); mog2b (); mog2c (); mog2d () the spin-flip amplitudes, which are related with some different nonperturbative processes not decreasing or slowly decreasing with growing energy, were predicted.

The recent results from the LHC pose new questions in the study of the structure of hadronic amplitudes, as its give the important information about the soft hadron processes at super high energies. The new data of the TOTEM and ATLAS Collaborations indeed show that none of the models predicted correctly the elastic cross sections at the LHC.

One of the main problems of the dynamical models is linked to the structure of hadrons which should be presented by the conventional electromagnetic form factors of the hadrons, or via Generalized Parton Distributions (GPDs), under the assumption that hadrons respond to Pomerons in the same way as they do to photon. In practice, many models took into account these assumptions and used some phenomenological forms of the form factors with the extra parameters determined by a fit of the experimental data. For example in DL-PL14 () purely phenomenological exponential form factors are used. Obviously, such exponential form cannot be used at sufficiently large momentum transfer, as does not correspond to the power dependence of the form factors which is require the quark model.

In papers HEGS0 (); HEGS1 (), the dynamical model for a hadron interaction, which takes into account the hadron structure at large distances through the generalized parton distribution functions, was developed to describe quantitatively and simultaneously the proton-proton and proton antiproton elastic scattering at high energies. The model is based on the general quantum field theory principles (analyticity, unitarity and so on) and takes into account the basic information on the structure of a nucleon as a compound system.

The measure of the -dependence of the total cross sections and of - the ratio of the real to imaginary part of the elastic scattering amplitude, is very important as they are connected to each other through the integral dispersion relations. The validity of this relation can be checked at LHC energies The deviation can point out the existence of a fundamental length at TeV energies Khuri1 (); Khuri2 (). However, for such a conclusion we should know with high accuracy the lower energies data as well.

As we do not know exactly, from a theoretical viewpoint, the dependence of the scattering amplitude on and , it is usually assumed that the imaginary and real parts of the spin-non-flip amplitude behave exponentially with the same slope. Similarly, one assumes the imaginary and real parts of the spin-flip amplitudes (without the kinematic factor ) to have an analogous -dependence in the examined domain of momenta transfer. Moreover, one assumes energy independence of the ratio of spin-flip to spin-non-flip parts at small . All this is our theoretical uncertainty.

Of course, we have plenty of experimental data in the domain of small at low energies . Unfortunately, most of these data come with large errors. The extracted sizes of contradict each other in the different experiments and give a bad in the different models trying to describe the -dependence of (see, for example, the results of the COMPETE Collaboration COMPETE1a (); COMPETE1b (). It is of first importance that a more careful analysis of these experimental data gives in some cases an essentially different extrapolation for . For example, the analysis of the experimental data made in SelyuginYF92 (), which takes into account the uncertainty of the total cross sections (-parameters fit) and the uncertainty of the Luminosity (-parameters fit) gave a , which differs from the original values obtained by the experimental group, by on average. For example, for GeV/c the experimental work gave and for GeV/c . The analysis with free -parameters gave for these values: and , respectively. This kind of picture was confirmed by the independent analysis of the experimental data Kuznetzov1 (); Kuznetzov2 () (GeV/c) of Fajardo Fajardo () and Selyugin SelyuginYF92 (). Both new analysis coincide with each other but differ from the original experimental determination.

The non-trivial procedure of the extraction of the size of from the experimental data on the differential cross sections shows the semi-phenomenological properties of CS-PRL09 (). Its size is dependent on some theoretical assumption SCP-EPJ08 (). For example, a significant discrepancy in the experimental measurement of was found by the UA4 and UA4/2 collaborations at GeV. But a more careful extrapolation SelyuginPL94 () to shows that there is no real contradiction between these measurements and gives for this energy , the same as in the previous phenomenological analysis SelyuginYF92 ().

In this paper, we consider in detail the situation in the region of the diffraction dip where the real part (and possibly spin-flip amplitude) plays the essential role. The proposed model takes into account all known features of the near forward proton-proton and proton-antiproton data, the properties of the spin-non-flip and spin-flip amplitudes, total cross sections, ratios of the real to the imaginary forward amplitudes and Coulomb-nuclear interference phase where the form factors of the nucleons are also taken into account.

## 2 The elastic nucleon scattering in the framework of the HEGS model

The differential cross sections of nucleon-nucleon elastic scattering can be written as the sum of different helicity amplitudes:

(1) |

The HEGS model HEGS0 (); HEGS1 () takes into account all five spiral electromagnetic amplitudes. The electromagnetic amplitude can be calculated in the framework of QED. In the high energy approximation, it can be obtained bgl () for the spin-non-flip amplitudes:

(2) |

where is the electromagnetic fine-structure constant, and for the spin-flip amplitudes:

(3) |

where the form factors are:

(4) |

with relative to the anomalous magnetic moment, and has the conventional dipole form . With the electromagnetic and hadronic interactions included, every amplitude can be described as

(5) |

where , and will be calculated in the second Born approximation in order to allow the evaluation of the Coulomb-hadron interference term . The quantity has been calculated and discussed by many authors (see PRD-Sum () and references therein).

Let us define the hadronic spin-non-flip amplitudes as

(6) |

The model is based on the representation that at high energies a hadron interaction in the non-perturbative regime is determined by the reggeized-gluon exchange. The cross-even part of this amplitude can have two non-perturbative parts, possible standard pomeron () and cross-even part of the 3-non-perturbative gluons (). The interaction of these two objects is proportional to two different form factors of the hadron. This is the main assumption of the model. The second important assumption is that we chose the slope of the second term four times smaller than the slope of the first term, by analogy with the two pomeron cut. Both terms have the same intercept.

The parton picture of hadron structure is represented, in most part, by the parton distribution functions (PDFs). They are determined in deep inelastic processes. The next step in the development of the picture of hadron structure was made by introducing the non-forward structure functions - general parton distributions - GPDs with the spin-independent and the spin-dependent parts. Some of the advantages of GPDs were presented by the sum rules Ji97 (). Using the different momenta of GPDs as a function of we can obtain the different form factors Compton form factors: ; the electromagnetic form factors and the gravimagnetic form factors . In GPD-PRD14 () different PDFs sets were examined and the momentum transfer dependence of the GPDs and was obtained which give the best descriptions of the electromagnetic form factors of the proton and neutron simultaneously. It allows us to calculate two different form factors: the electromagnetic form factors can be represented as the first moments of GPDs and , and the integration of the second moment of GPDs over gives the momentum-transfer representation of the form factor , which is reflects the matter distribution in the hadron (see HEGS0 (); HEGS1 (); GPD-PRD14 ()).

The parameters and -dependence of the GPDs are determined by the standard parton distribution functions, so by the experimental data on the deep inelastic scattering, and by the experimental data for the electromagnetic form factors (see GPD-ST-PRD09 ()). The calculations of the form factors were carried out in GPD-PRD14 ().

Hence, the Born term of the elastic hadron amplitude can now be written as

where the last term represents the Odderon contribution and the upper sign is related to and the lower sign to . and have the standard Regge form: ; ; , and . The slope of the scattering amplitude has the standard logarithmic dependence on the energy with GeV and with some small additional term HEGS1 () which reflect the small non-linear behavior of at small momentum transfer Sel-Df16 (). The final elastic hadron scattering amplitude is obtained after unitarization of the Born term. So, at first, we have to calculate the eikonal phase

(8) |

where , and then obtain the final hadron scattering amplitude

(9) |

At large our model calculations are extended up to GeV. We added a small contribution of the energy independent part of the spin flip amplitude in the form similar to the proposed in Kuraev-SF () and analyzed in W-Kur ().

(10) |

The model is very simple from the viewpoint of the number of fitting parameters and functions. There are no any artificial functions or any cuts which bound the separate parts of the amplitude by some region of momentum transfer.

experimental points were included in our analysis in the energy region GeV TeV and in the region of momentum transfer GeV. The experimental data of the proton-proton and proton-antiproton elastic scattering are included in 92 separate sets of 32 experiments data-Sp (); Land-Bron () including recent data of the TOTEM Collaboration at TeV TOTEM-8nexp (). The whole Coulomb-hadron interference region, where the experimental errors are remarkably small, was included in our examination of the experimental data. As the result, it was obtained with the parameters and the low energy parameters .

Such a simple form of the scattering amplitude in the huge region of energy requires careful determination of the slope of the scattering amplitude. In the present model, a small additional term is introduced into the slope, which reflects some possible small nonlinear properties of the intercept and leads to the standard form of the slope as and HEGS1 ().

## 3 The differential cross sections at LHC energies

Let us see the predictions of the HEGS model for the LHC energies. The result of the calculations of the differential cross sections of the elastic proton-proton scattering at TeV and TeV are presented in Table 1 and Table 2. In the model, the data on the total cross section are not included in the fitting procedure as its value is extracted from the corresponding differential cross sections by one or another procedure. Hence, the sizes of are obtained in the model through the optic theorem by the calculation of the imaginary part of the hadronic amplitude at zero value of the momentum transfer. The corresponding values of the obtained total cross sections show a good coincidence with the experimental data HEGS1 (); Sel-Df16 () in the wide energy region.

The arithmetic mean on the value of the total cross sections obtained by the different methods by the TOTEM Collaboration at TeV is mb and at TeV - mb. The ATLAS Collaboration for these energies give the value of the and mb. Obviously, there is a large difference between the data of the TOTEM and ATLAS Collaborations which grows at TeV.

Let us see the experimental data of the differential cross sections in the small momentum transfer region at LHC energies. Now there are five sets of experimental data on the elastic scattering at LHC energies and small momentum transfer: it is the data of the TOTEM Collaborations at TeV T7a (); T7b (), at TeV T8a (), and the data of the ATLAS Collaborations at TeV ATL7 () and at TeV ATL8 (). Recently, there have appeared preliminary non-normalized data at TeV Diff16 () of the TOTEM Collaboration.

The comparison of the predictions of the HEGS model with these data Sel-Df16 () shows that the main problem of these data is concentrated in the normalization of the differential cross sections. The data of the ATLAS Collaboration practically exactly coincide with the model calculations, and the additional normalization is really near unity. However, the TOTEM data require additional normalization respectively, at TeV and TeV. The data at TeV have no normalization. However, their form coincides with the form of the model predictions sufficiently well.

## 4 The dip region

Now let us examine the form of the differential cross section in the region of the momentum transfer where the diffractive properties of the elastic scattering appear most strongly - it is the region of the diffraction dip. The form and the energy dependence of the diffraction minimum are very sensitive to the different parts of the scattering amplitude. The change of the sign of the imaginary part of the scattering amplitude determines the position of the minimum and its movement with changing the energy. The real part of the scattering amplitude determines the size of the dip. Hence, it depends heavily on the odderon contribution. The spin-flip amplitude gives the contribution to the differential cross sections additively. So the measurement of the form and energy dependence of the diffraction minimum with high precision is an important task for future experiments. In Fig.1, the description of the diffraction minimum in our model is shown for ISR energies. The HEGS model reproduces sufficiently well the energy dependence and the form of the diffraction dip. In this energy region the diffraction minimum reaches the sharpest dip at GeV. Note that at this energy the value of also changes its sign in the proton-proton scattering. The cross sections in the model are obtained by the crossing without changing the model parameters. And for the proton-antiproton scattering the same situation with correlations between the sizes of and takes place at low energy (approximately at GeV).

The HEGS model reproduces at very small and large and provides a qualitative description of the dip region at GeV, for GeV and GeV for TeV (Fig.2). Note that it gives a good description for the proton-proton and proton-antiproton elastic scattering or GeV and for GeV. The diffraction minimum at TeV and TeV is reproduced sufficiently well too (Fig.3).

The dependence of the position of the diffraction minimum on is determined in most part by the growth of the total cross sections and the slope of the imaginary part of the scattering amplitude. In the framework of the geometrical scaling approximation DiasDeDeus () it was proposed in Alberi-81 () that the basic values of the elastic scattering are proportional to the effective radius of the interaction , , , . In this case, we have . Now most models use the hypothesis of the factorization eikonal Henzi-76 () where the eikonal phase . The experimental data in most part support the last hypothesis and the geometrical scaling valid only in some region of the energy on the average.

Figure 2 shows such a dependence obtained in the HEGS model in the huge energy interval. The energy dependence of the position of the diffraction minimum can be reproduced by the simple approximation, with a right value as ,

(11) |

where and are the free parameters and (as in the HEGS model), with ; (the short-dashed line in Fig.4a for the simplest case with two parameters), and the case and with , if it is taken into account as a free parameter (hard line in Fig.4a). In Fig.4a the errors of the data obtained from the model calculations are taken as from the size of . A good approximation can be obtained else by

(12) |

In this case and and (long-dashed line in Fig.4). Despite that this equation gives a better approximation of the energy dependence of , it has a bad asymptotic value as .

,GeV | , GeV | , TeV | , GeV | |||
---|---|---|---|---|---|---|

In Nemes13 (), the assumption about the scaling properties of and was introduced as where mb GeV. In comparing the form of eq.(11) with the approximation of the total cross section (for example PDG PDG-tot ()), such scaling properties are confirmed at first sight. However, the detailed examination shows some difference. The scaling properties are performed on the average, see Fig.4b (hard line) and the product of (eq.(11), ) and PDG(PDG-tot ()) (Fig.4b - short-dashed line). Really, the energy dependence of such a product has a complicated form. The calculations in the framework of the HEGS model and with and energy independence of the product of (eq.(11) with ) and PDG(PDG-tot ()) (Fig.4b long-dashed line) show practically the same dependence.

Other interesting characteristics of the diffraction minimum are the height of the dip, which is reflected in the difference between the sizes of the minimum and the second maximum of the differential cross sections, and the width of the diffraction dip at half its height. These values are determined on the one hand, by the slope of the imaginary part in this domain and, on the other hand, by the contributions and and dependence of the real part of the spin-non-flip scattering amplitude and by the contribution of the spin-flip scattering amplitude. The depth of the dip can be represented as relations between the maximum and minimum of the dip - . The energy dependence of these values is represented in Table 3. We can see that grows faster at low energies and reaches the maximum in the domain around GeV where the diffraction dip has a maximum value. It reflects that the real part of the spin-non-flip amplitude in the elastic proton-proton scattering changes its sign at . As was noted above, it is a remarkable fact that the size of the real part in the dip region hardly correlates with the size of the real part at . This takes place also for proton-antiproton scattering. The real part in that case changes its sign at approximately around GeV. And the diffraction dip in scattering also has its maximal value at that energy.

At higher energy, we can see from Table 3 that reaches its minimal value at GeV. It means that the real part of the scattering amplitude relative to the imaginary part has a maximal value in this energy region. At higher energy is increasing weakly with respect to the slowly decreasing value of . The inverse value of in some approximation shows the ratio of the real to imaginary part of the scattering amplitude in this domain of . It can be approximated

(13) |

with , , , GeV, with ; and , , , GeV, with . The energy dependence is shown in Fig.5a with and .

Another characteristic of the form of the dip of the differential cross sections is the value of , GeV - the width of the dip at half its height. Again, this value grows at low energies and after GeV slowly decreases. However, the width decreases faster than . When it changes two times (in the domain TeV), the changes on only. It can be represented in the form

(14) |

with , , , GeV. The results of the model calculation and the approximation by eq.(14) are shown in Fig.5b. This value essentially changes in the region where changes its sign. At large energies it has small changes.

From a more profound analysis the slope of the real part in the dip region and some possible contribution of the spin-flip amplitude can be obtained . However, it requires high precision data in the dip region and a more complicated special work.

## 5 Conclusions

The form and energy dependence of the diffraction minimum of the differential cross sections of the elastic hadron-hadron scattering gave the valuable information about the structure of the hadron scattering amplitude and hence about the dynamics of strong interactions. The diffraction minimum corresponds to the change of the sign of the imaginary part of the spin-non-flip hadronic scattering amplitude and is created under a strong impact of the unitarization procedure. Its dip depends on the contributions of the real part of the spin-non-flip amplitude and the whole contribution of the spin-flip scattering amplitude. The HEGS model reproduces well the form and the energy dependence of the diffraction dip of the proton-proton and proton antiproton elastic scattering. The predictions of the model in most part reproduce the form of the differential cross section at TeV. It means that the energy dependence of the scattering amplitude determined in the HEGS model and unitarization procedure in the form of the standard eikonal representation satisfies the experimental data in the huge energy region (from GeV up to TeV). It should be noted that the real part of the scattering amplitude, on which the form and energy dependence of the diffraction dip heavily depend, is determined in the framework of the HEGS model only by the complex , and hence it is tightly connected with the imaginary part of the scattering amplitude and satisfies the analyticity and the dispersion relations. It allows one to determined the energy dependence of the main characteristic of the diffraction dip - the position of the minimum , its width at half of the difference between the maximum and minimum of the dip, and the ratio of the maximum to minimum of the diffraction dip and its inverse value . These characteristics are important for the analysis of the structure of the elastic scattering amplitude in the domain of the diffraction dip. There is a remarkable fact that the size of the real part in the dip region hardly correlates with the size of the real part at . We find that the scaling properties are performed on the average. Really, the energy dependence of the product has a complicated form. The calculations in the framework of the HEGS model and with and energy independence of the product of (eq.(11) with ) and PDG(PDG-tot ()) (Fig.4b long-dashed line) show practically the same dependence. It is shown that the width of the dip of the differential cross sections decreases faster than . Of course, the HEGS model is oversimplified, and to reproduce quantitatively the different thin structures of the scattering amplitude, a wider analysis is needed. This concerns the fixed intercept taken from the deep inelastic processes and the fixed Regge slope , as well as the form of the spin-flip amplitude. Such analysis requires the use of a wider circle of experimental data, including the polarization data and the normalized new data on the elastic scattering at TeV.

Acknowledgement: The author would like to thank J. -R. Cudell for helpful discussions, gratefully acknowledges the financial support from FRNS and would like to thank the University of Liège where part of this work was done.

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