# Elastic scattering at $\sqrt{s}=6$ GeV up to $\sqrt{s}=13$ TeV
(proton-proton; proton-antiproton; proton-neutron)

O.V. Selyugin<EMAIL_ADDRESS>BLTP, Joint Institute for Nuclear
Research, 141980 Dubna, Moscow region, Russia

###### Abstract

In the framework of the Regge-eikonal model of hadron interaction, based on
the analyticity of the scattering amplitude with taking into account the
hadron structure, the simultaneous analysis is carried out of the 90 sets of
data. These sets include the data obtained at low energies ($\sqrt{s}>3.6$ GeV
and at high energies at FNAL, ISR, $S\bar{p}pS$, TEVATRON and LHC with 4326
experimental points, including the polarization data of analysing power. The
energy and momentum transfer dependence of separate sets of data is analyzed
on the basis of the eikonalized Born amplitude with taking into account two
additional anomalous terms. Different origins of the nonlinear behavior of the
slope of the scattering amplitude are compared. No contribution of hard-
Pomeron in the elastic hadron scattering is shown. However, the importance of
Odderon’s contribution is presented. The form and energy dependence of
different terms of the cross-even and cross-odd parts of the elastic nucleon-
nucleon scattering amplitude is determined in the framework of the High Energy
Generalize Structure (HEGS) model. In the framework of the HEGS model, using
the electromagnetic and gravitomagnetic form factors, the differential cross
sections in the Coulomb Nuclear Interference (CNI) region and at large
momentum transfer are described well in a wide energy region simultaneously.
It is shown that the cross-even part includes the soft pomeron growing like
$\ln^{2}(s)$ and an additional term with a large slope and with and with
energy dependence $\ln^{2}(s)$. The cross-odd part includes the maximal
odderon term and an additional oscillation term with $\ln(s)$. It is shown
that both additional terms are proportional to charge distributions but the
maximal odderon term is proportional to matter distributions. Also, a good
description of proton-neutron differential scattering with 526 experimental
points it is obtained on the basis of the amplitudes taken from the analysis
of $pp$ and $p\bar{p}$ scattering. A good enough description of the
polarization data was also obtained.

###### pacs:

11.80.Cr, 12.40.Nn, 13.85.Dz

##  I Introduction

One of the most important tasks of modern physics is the research into the
basic properties of hadron interactions. Many models predict that soft hadron
interactions will enter a new regime at the LHC: given the huge energy, as the
S-matrix reaches the unitarity limit. The dynamics of strong interactions
finds its most complete representation in elastic scattering. 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 was introduced in the dispersion theory by N.N.
Bogoliubov[1]. Now many questions of hadron interactions are connected with
modern problems of astrophysics such as unitarity and the optical theorem [2],
and problems of baryon-antibaryon symmetry and CP-invariance violation [3] The
main domain of elastic scattering is small angles. Only in this region of
interactions we can measure the basic properties that define the hadron
structure. Their values are connected, on the one hand, with the large-scale
structure of hadrons and, on the other hand, with the first principles that
lead to theorems on the behavior of scattering amplitudes at asymptotic
energies [4, 7].

The research of the structure of the elastic hadron scattering amplitude at
superhigh energies and small momentum transfer - $t$ can give a connection
between the experimental knowledge and the basic asymptotic theorems based on
first principles [5, 6, 7]. It gives information about the hadron interaction
at large distances where the perturbative QCD does not work [8, 9] and a new
theory as, for example, instanton or string theories must be developed.

The structure of hadrons reflected in generalized parton distributions is now
one of the most interesting questions of the physics of strong interactions
(see, for example [12, 15]. It is tightly connected with the spin physics of
the hadron [16]. Some modern accelerator experiments have developed extensive
programs for deep studies of different issues related to this problem. For
example, the Jefferson Laboratory/Electron-Ion Collider (EIC) team to extract
generalized parton distributions (GPDs) [17]. the same problem is posed for
future experiments at SPD of NICA-JINR [18].

Modern studies of elastic scattering of high energy protons lead to several
unexpected results reviewed, e.g., in [9, 10]. Spin amplitudes of the elastic
$NN$ scattering constitute a spin picture of the nucleon. Without knowing of
the spin $NN$-amplitudes, it is impossible to understand the spin observable
of nucleon scattering off nuclei. In the modern picture, the structure of
hadrons is determined by Generalized Distribution functions (GPDs), which
include the corresponding parton distributions (PDFs). The sum rules [20, 21]
allows one to obtain the elastic form factor (electromagnetic and
gravitomagnetic) through the first and second integration moments of GPDs. It
leads to remarkable properties of GPDs, some corresponding to inelastic and
elastic scattering of hadrons. Now some different models examining the
nonperturbative instanton contribution lead to sufficiently large spin effects
at superhigh energies [57], [56]. The research of such spin effects will be a
crucial stone for different models and will help us to understand the
interaction and structure of particles, especially at large distances. There
are large programs of researching spin effects at different accelerators.
Especially, we should like to note the programs at NICA, where the
polarization of both the collider beams will be constructed. So it is very
important to obtain reliable predictions for spin asymmetries at these
energies. In this paper, we extend the model predictions to spin asymmetries
in the NICA energy domain.

The unique experiment carried out by the TOTEM Collaboration at LHC at 13 TeV
gave excellent experimental data on the elastic proton-proton scattering in a
wide region of transfer momenta [29, 30]. It is especially necessary to note
the experimental data obtained at small momentum transfer in the Coulomb-
hadron interference region. The experiment reaches very small $t=8\ 10^{-4}$
GeV2 with small $\Delta t$, which give a large number of experimental points
in a sufficiently small region of momentum transfer. This allows one to carry
out careful analysis of the experimental data to explore some properties of
hadron elastic scattering.

There are two sets of data - at small momentum transfer [29] and at middle and
large momentum transfer [30]. They overlap in some region of the momentum
transfer, which supplies practically the same normalization of both sets of
differential cross sections of elastic proton-proton scattering. Recently, the
first set of data has created a wide discussion of the determination of the
total cross section and the value of $\rho(t=0)$.

There is a very important characteristic of the elastic scattering amplitude
such as the ratio of the real part to imaginary part of the scattering
amplitude - $\rho(s,t)$. It is tightly connected with the integral and
differential dispersion relations. Of course, especially after different
results obtained by the UA4 and UA4/2 Collaborations, physicists understand
that $\rho(s,t=0)$ is not a simple experimental value but heavily dependent on
theoretical assumptions about the momentum depends of the elastic scattering
amplitude. Our analysis of both experimental data obtained by the UA4 and
UA4/2 Collaborations shows a small difference value of $\rho(s,t=0)$ obtained
in both the experiments if the nonlinear behaviour of the elastic scattering
amplitude is taken into account [73]. Hence, this is not an experimental
problem but a theoretical one [58].

For extraction of the sizes of $\sigma_{tot}$ and $\rho(t=0)$ the Coulomb
hadron region of momentum transfer is used (for example [29]). However, the
form of the scattering amplitude assumed for small $t$ and satisfying the
existing experimental data at small momentum transfer, can essentially be
different from experimental data at large $t$. One should take into account
the analysis of the differential cross section at 13 TeV where the diffraction
minimum impacts the form of $d\sigma/dt$ already at $t=-0.35$ GeV2. The
analysis of new effects, discovered on the basis of the experimental data at
13 TeV [58, 75, 76] and associated with the specific properties of the hadron
potential at large distances was carried out with taking account all sets of
experimental data on elastic $pp$-scattering obtained by the TOTEM and ATLAS
Collaborations in a wide momentum transfer region and gave a quantitative
description of all examined experimental data with minimum fitting parameters.

The non-linear behavior of the slope of the differential cross sections at
small momentum transfer, which was announced by the TOTEM Collaboration in the
proton-proton elastic scattering at $8$ TeV, shows that the complex
(complicated) form of strong interactions searched out at low energies remains
at superhigh energies too. This means that the strong interaction is not
simplified at superhigh energies but includes many different parts of hadron
interactions.

Using the existing model of nucleon elastic scattering at high energies
$\sqrt{s}>3.6$ GeV - 14 TeV [38, 39], which involves minimum of free
parameters, we are going to develop its extended version aimed to describe all
available data on cross sections and spin-correlation parameters at lower
energies down to the SPD NICA region. The model will be based on the usage of
known information on GPDs in the nucleon, electro-magnetic and gravitomagnetic
form factors of the nucleon taking into account analyticity and unitarity
requirements and providing compatibility with the high energy limit, where the
pomeron exchange dominates.

##  II Small momentum transfer region

###  II.1 Electromagnetic scattering

The question of the $t$ dependence of the elastic peak is one. One can be
related to optics. At high energy, elastic scattering at small angles has some
similarity with light scattering in the Fresnel region. The amplitude for the
electric field propagating in the $z$ direction is given by the Rayleigh-
Sommerfeld equation

$\displaystyle
E(x,y,z)=-{i\over\lambda}\int\int_{-\infty}^{+\infty}{E(x^{\prime},y^{\prime},0)\frac{e^{ikr}}{r}\cos\theta}dx^{\prime}dy^{\prime}$
(1)

where $r=\sqrt{(x-x^{\prime})^{2}+(y-y^{\prime})^{2}+z^{2}}$,and
$\cos\theta=\frac{z}{r}$. The integral can be performed analytically only for
the simplest geometrical cases, and one of the general forms of expansion over
$r$ near the $z$ axis is

$\displaystyle r=$ $\displaystyle
z\sqrt{1+\frac{\rho^{2}}{z^{2}}}=z\left[1+\frac{\rho^{2}}{2z^{2}}-\frac{1}{8}\left(\frac{\rho^{2}}{z^{2}}\right)^{2}+\cdots\right]$
(2) $\displaystyle=$ $\displaystyle
z+\frac{\rho^{2}}{2z}-\frac{\rho^{4}}{8z^{3}}+\cdots$

with $\rho^{2}=(x-x^{\prime})^{2}+(y-y^{\prime})^{2}$. If we consider only the
first term and the case of a sphere of radius $R$, we obtain the $t$
dependence of the amplitude [33]

$\displaystyle{\cal
A}(t)\propto(1+\cos\theta)\frac{J_{1}(x\sin\theta)}{\sin\theta}$

with $x=kR$. Hence even in the simplest case the elastic peak cannot be
described by a simple exponential. Note that this form of the scattering
amplitude is used in diffractive hadron scattering.

###  II.2 Hadron scattering

In reference [32], the TOTEM Collaboration has announced the observation of a
the non-exponential behavior of the differential elastic cross sections at $8$
TeV and small momentum transfer $|t|$.

Of course, the form of the elastic peak depends also on the structure of
particles and the dynamics of the interaction. These two features can be
parameterised by the profile function $\rho(\vec{b})$ (in the space of impact
parameter, $\vec{b}$) or by the elastic form factor (in momentum space). For
example, in [34] different forms of the profile function determined by the
density distributions taking part in the interaction were considered:

$\displaystyle\mathrm{circle\ of\ radius\ }b_{0}=4\mathrm{\ GeV^{-1}}$
$\displaystyle\Rightarrow{\cal A}(t)\propto J_{0}(4\sqrt{|t|});$ (3)
$\displaystyle\mathrm{hollow\ disk\ near\ }b_{0}=2.8\mathrm{\ GeV^{-1}}$
$\displaystyle\Rightarrow{\cal A}(t)\propto e^{2.8t}J_{0}(2.8\sqrt{|t|});$ (4)
$\displaystyle\mathrm{black\ disk\ of\ radius\ 6}\mathrm{\ GeV^{-1}}$
$\displaystyle\Rightarrow{\cal A}(t)\propto J_{1}(6\sqrt{|t|})/(6\sqrt{|t|});$
(5) $\displaystyle\rho(b)\sim e^{-cb}\Rightarrow{\cal A}(t)\propto e^{5t};$
(6) $\displaystyle\rho(b)\sim e^{\mu\sqrt{b_{0}^{2}+b}}\Rightarrow{\cal
A}(t)\propto e^{5(\sqrt{4\mu^{2}-t}-2\mu)}.$ (7)

Figure 1:  Possible $t$ dependences of the scattering amplitude ${\cal A}(t)$,
rescaled to 1 at $t=0$. (solid line - for Eq. (6); long-dashed line - for Eq.
(3); short-dashed line - for Eq. (5); dotted line - for Eq. (4, and dashed-
dotted line - for Eq. (7).

The $t$ dependence of these amplitudes is shown in Fig.1. Clearly, the simple
exponential form (6) is a special separate case. We can see, except for the
last case (eq.7), that if we take a small interval of $t$ (for example,
$0.01<|t|<0.15$ GeV2) the curves give practically the same result. However,
they give the essentially different results on some more values of $t$ or at
$t\rightarrow 0$. Hence, to obtain the true form of the scattering amplitude,
it is necessary to take experimental data in a wider interval of $t$ and
tending to the limit $t\rightarrow 0$.

This simplest approximation of the differential cross section by one exponent
with a constant slope can lead to artificial new effects. For example, in the
experiment at the Protvino accelerator on the elastic proton-proton scattering
there were found ”oscillations” at small momentum transfer. However, in [35]
it was shown that such ”oscillations” can be appear when the differential
cross section is described by two exponentials with some different slopes, by
the one exponential with the constant slope.

The investigation of the energy and $t$ dependence of the slope of the elastic
peak can lead to new information about the structure of interacting particles.
Early measurements at ISR revealed four slopes of differential cross section
in different regions of momentum transfer, which result from the complicated
structure of nucleons.

###  II.3 Non-linear slope

The complicated $t$ dependence of the slope can have many origins. First of
all, it comes from the unitarization procedure of the Born term of the elastic
scattering amplitude. In many purely phenomenological analyses is represented
by the $Ct^{2}$ term in the slope that mimics the unitarization procedure. The
hadron structure is reflected also not only at large momentum transfer but
also at small $t$. This especially it is concerns the meson clouds whose
interaction in many models is added to the central part of the hadrons (for
example, in Pumplin model [36] and the Dubna Dynamical (DD) model [37])) which
leads to the $\sqrt{t_{0}+t}$ dependence of the scattering amplitude. Such
complicated hadron structure can be reflected in the presence of the two form
factors \- electromagnetic and mater form factor of the hadrons (for example,
the HEGS model [38, 39]).

Another term of the slope, which is commonly represented as

$\displaystyle\sqrt{4\mu^{2}-t}-2\mu,$ (8)

used in many phenomenological descriptions of the elastic differential cross
sections to explain the ”break” in the differential cross sections. Note that
an additional term in the slope like $\sqrt{t_{0}-t}$ was early obtained whose
first approximation can be related to absorbtion corrections that can produce
a set of canceling Regge cuts [42, 43] and leads to the Schwarz type
trajectories [44] $\alpha(t)=1+\gamma t^{1/2}$. A more complicated form was
obtained in [45] $\alpha_{\pm}(t)=1\pm\gamma t^{1/2}+2\rho(1/2\
\gamma^{2}t)^{3/2}(-ln(t))^{1/2}$. The appearence of a complex trajectory
greatly complicates the picture and requires additional research. Hence in
[46], based on the works [54, 55], it was proposed to use the simplest form
$\alpha(t)=1.041-0.15\sqrt{t_{0}-t}$. However, as we show such behavior has a
really pure phenomenological basis and can be replaced by either a simpler a
form (for example eq.(11) or more complicated form used in the HEGS-model
[39].

Many models based on the famous works [47, 48] researched the non-linear
behavior of the scattering amplitude.

Based on the works in [49] it was obtained

$\displaystyle\alpha_{P}(q^{2})=1-C_{p}q^{2}-(\sigma_{\pi\pi}/32\pi^{2})h_{1}(q^{2}).$
(9)

where

$\displaystyle h_{1}(q^{2})=\frac{q^{2}}{\pi}$ (10)
$\displaystyle[\frac{8\mu^{2}}{q^{2}}-(\frac{4\mu^{2}+q^{2}}{q^{2}})^{3/2}ln\frac{\sqrt{4\mu^{2}+q^{2}}+q}{\sqrt{4\mu^{2}+q^{2}}-q}+ln\frac{m^{2}}{\mu^{2}}],$

with $q^{2}=-t$ . Note that we have removed the misprint in this equation, as
made in [51]. They obtained the limits of the representation in the brackets
at $q^{2}>>4\mu^{2}$; hence, the slope grows in order $q^{2}$ with small
logarithmic suppression. At small $t$ ($q^{2}\ll 4\mu^{2}$ they predicted that
the representation in the brackets goes to $ln(m^{2}/\mu^{2})-8/3$. Note that
the authors aimed to explain the deviation of the slope from the constant at
non-small momentum transfer (in the domain $-t=0.4$ GeV2. However, in this
domain the impact of the diffraction minimum is already felt. Hence this
domain of $t$ is usually described by an additional term like $ct^{2}$, which
were proposed early (for example [53]).

Figure 2:  The $t$ dependence of the different parts of the terms of eq.(13)
(solid line - the sum all three terms situated into brackets; the dots-dashed
line the first term is multiplied by ($-1$); the long-dashed line - the second
term, the short-dashed line - the sum of the two first terms).  Figure 3:  The
$t$ dependence of the additional terms to $\alpha^{\prime}t$ (solid line -
eq.(13); long-dashed line - the eq.(8); short-dashed line - the eq.(8) without
extraction $2\mu$ and multiplied on $0.1$.

Practically at the same time a similar equation was obtained in [50]

$\displaystyle
D_{R}^{NN}(t)=n[A-(\frac{(4\mu^{2}+q^{2})^{3/2}}{q})ln\frac{\sqrt{4\mu^{2}+q^{2}}+q}{\sqrt{4\mu^{2}+q^{2}}-q}]^{-1},$
(11)

They proposed to approximate this representation by

$\displaystyle D_{R}^{NN}(t)\sim[\frac{1}{5\mu^{2}-t}+C_{const}]$ (12)

For numerical calculation and comparison with experimental data, they took
$C_{const}=24.3$ GeV-2.

In [51], using the calculations [49] and removing the misprint, the authors
proposed for $h_{1}$, with taking into account of the meson form factor with
$\Lambda_{\pi}^{2}=m_{\rho}^{2}$ GeV2 ), the following equation:

$\displaystyle
h_{1}(q^{2})=\frac{4}{\tau}f^{2}_{\pi}(t)[2\tau-(1+\tau)^{3/2}ln\frac{\sqrt{1+\tau}+1}{\sqrt{1+\tau}-1}+ln\frac{m^{2}}{\mu^{2}}],$
(13)

where $\tau=4\mu^{2}/q^{2}$ and

$\displaystyle f_{\pi}(t)=\frac{\Lambda_{\pi}^{2}}{\Lambda_{\pi}^{2}-t}$ (14)

Really for the slope, which is multiplied by $q^{2}$, there are two divergence
terms with different signs plus a small constant term. The divergence terms
cancels each other and the rest have a slow $t$ dependence (see Fig.2). The
cancelation of the two terms as $t\rightarrow 0$ is not full and the rest have
also some indefiniteness and strongly depend on $t$. In Fig.2, the $t$
dependence of the different parts of the term of eq.(13) in the brackets are
shown. The cancelation of the two diverse terms leads to the negative
contribution to the standard constant slope. However, the sum of all three
terms give a positive additional contribution with small $t$ dependence.

In Fig. 3, we show the $t$ dependence of a different form of the slopes with
the full kinematic coefficient. One can see very different $t$ dependence ,
especially in the form $\sqrt{4\mu^{2}+q^{2}}$ which was proposed in the work
[46] without the extraction of $2\mu$.

Some models of hadron interactions at high energy suppose that the slope has a
slow $t$ dependence. For example, the Dubna dynamical model (DDm) [37], which
takes into account the contribution to hadron interactions from the meson
cloud of the nucleon and uses the standard eikonal form of the unitarization,
leads to the scattering amplitude in the form

$T(s,t)=-is\sum_{n=1}^{\infty}\frac{\mu}{(n^{2}\mu^{2}-t)^{3/2}}(1-b\sqrt{n^{2}\mu^{2}-t})e^{-b\sqrt{n^{2}\mu^{2}-t}}.$
(15)

The analysis of the high energy data on the proton-antiproton scattering in
the framework of this model shows the obvious non-exponential behavior of the
differential cross sections. For $\sqrt{s}=540$ GeV this shows the change in
the slope from $16.8$ GeV-2 at $t=-0.001$ GeV2 up to $14.9$ GeV-2 at $t=-0.12$
GeV2. For the same $t$ the size of $\rho(s,t)$ changes from $0.141$ up to
$0.089$. For Tevatron energy $\sqrt{s}=1800$ GeV it shows the change of the
slope from $18.1$ GeV-2 at $t=-0.001$ GeV2 up to $15.9$ GeV-2 at $t=-0.12$
GeV2 and again the size of $\rho(s,t)$ changes from $0.182$ up to $0.143$.
Hence the model shows the continuously decreasing slope and $\rho$ at small
$t$. Practically the same results were obtained in [36] in the framework of
the model that also uses the eikonal unitarization.

The analysis [73] of the high precision data obtained at $Sp\bar{P}S$ at
$\sqrt{s}=541$ GeV in the UA4/2 experiment shows the existence in the slope of
the term is proportional $\sqrt{t}$. This term can be related with the nearest
$\pi$ meson threshold or, as was shown in [73], such behavior of the
differential cross sections can reflect the presence of the contribution of
the spin-flip amplitude. As was noted in [79]), the analytic $S-$matrix
theory, perturbative quantum chromodynamics and the data require Regge
trajectories to be nonlinear complex functions [80, 81]. The Pomeron
trajectory has threshold singularities, the lowest one being due to the two-
pion exchange required by the $t-$channel unitarity [49]. This threshold
singularity appears in different forms in various models (see [79]). In the
recent high energy general structure model (HEGS) [39], a small additional
term is introduced into the slope which reflects some possible small non-
linear properties of the intercept. As a result, the slope is taken in the
form

$\displaystyle B(s,t)\
=-\alpha_{1}\ln(\hat{s})t(1-d_{1}t/\ln(\hat{s}e^{d_{2}\alpha_{1}t\ln(\hat{s})}).$
(16)

This form leads to the standard form of the slope as $t\rightarrow 0$ and
$t\rightarrow\infty$ Note that our additional term at large energies has a
similar form as the additional term to the slope coming from $\pi-loop$
examined in [49] and recently in [51].

##  III Experimental database

In our researches we use widest possible region of experimental data. The
energy region begins from $\sqrt{s}=3.5-3.8$ GeV for proton-antiproton
scattering. At these energies new data were obtained in the high precision
experiment on elastic $p\bar{p}$ scattering at small angles. It has four sets
at different energies in the momentum transfer region $|t|=0.000986--0.02$
GeV. The experiment is very important as they obtained the value of
$\rho(s,t=0)$ with a remarkably small error and with the size near zero. It is
essentially different from the value of $\rho(s,t=0)$ obtained in the
framework of the dispersion relation analysis by P. Kroll, which was carried
out on the basis of old experimental data with large errors.

Low-energy proton-antiproton data from $\sqrt{s}=11.54$ GeV up to the final
ISR energy $\sqrt{s}=62$ GeV are represented in eleven sets. Then we include
seven sets of experimental data obtained at the $Sp\bar{p}S$ collider at
energies around $\sqrt{s}=540-630$ GeV. The data Tevatron at
$\sqrt{s}=1800-1960$ GeV are represented in four sets. The latter data are
presented the maximal energy obtained for the proton-antiproton scattering at
accelerators. In whole for the $p\bar{p}$ elastic scattering we have 33 sets
of the different experiments.

For the proton-proton elastic scattering we take into account the sixty five
sets of the different experiments from the low energy $\sqrt{s}=6.1$ GeV up to
maximal the LHC energy $\sqrt{s}=13$ TeV. Especially note the high precisions
experimental data obtained at small momentum transfer in the FNAL
collaborations at $\sqrt{s}=9.8,9.9,10.6,12.3$ GeV and at
$\sqrt{s}=19.4,22.2,23.9,27.4$ GeV which started from a very small momentum
transfer $t=-0.00049$ GeV2. They is can be compared with the data obtained by
the UA4/2 Collaboration at $Sp\bar{p}S$ collider, which is start from
$t=-0.000875$ GeV2 and with the data obtained at the LHC by the TOTEM
Collaboration at $\sqrt{s}=13$ TeV which reached the $t=-0.00029$ GeV2. Of
course, the last case achieved smallest possible angles of scattering. We take
into account the experimental data at a high value of momentum transfer up to
momentum transfer $-t=10-14$ GeV2. The corresponding experimental data were
obtained at energies $\sqrt{s}=19.4,27.4,52.8$ GeV.

Summarizing all the sets of experimental data on elastic scattering at not
large angles, we took into account $115$ sets of different experiments which
included $4326$ experimental points on proton-proton and proton-antiproton
elastic scattering.

We included the data for the spin correlation parameter $A_{N}(s,t)$ of the
polarized proton-proton elastic scattering. This set of data includes 235
experimental data for relatively small energies $\sqrt{s}=3.63$ GeV and up to
$\sqrt{s}=23.4$ GeV. During our fitting procedure sufficiently good
descriptions were obtained.

For the first time, we also included in our research the elastic proton-
neutron experimental data. The corresponding data were taken into account
beginning from the $\sqrt{s}=4.5$ GeV up to maximal energy proton-neutron
collisions $\sqrt{s}=27.19$ GeV obtained at accelerators. It should be noted
note that such energy represented an average between minimum and maximum
energies. For example, maximum energy obtained in the FNAL presented average
between $P_{L}=340$ GeV/c and $P_{L}=400$ GeV/c. It is very important that in
such experiments a very small momentum transfer was reached, for example
$-t_{min}=0.23\ 10^{-4}$ GeV2 at $\sqrt{s}=23.193$ GeV. On the whole we took
into account $24$ sets of experimental data of different experiments which
supply $526$ of the experimental data. Hence on the whole, we took into our
analysis $5027$ experimental data on the elastic nucleon-nucleon scattering.

##  IV Main amplitudes in the high energy generalized structure (HEGS) model

###  IV.1 Asymptotic part of the scattering amplitude

The model is based on the idea that at high energies a hadron interaction in
the non-perturbative regime is determined by the reggenized-gluon exchange.
The cross-even part of this amplitude can have two non-perturbative parts,
possible standard pomeron - $(P_{2np})$ and cross-even part of 3-non-
perturbative gluons ($P_{3np}$). 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 choose the
slope of the second term four times smaller than the slope of the first term,
by analogy with the two pomeron cuts. Both terms have the same intercept.

###  IV.2 Form factors

Since nucleons are not point particles, their structure must be taken into
account. Usually, especially in the 60s - 70s, the researchers proposed that
the hadron form factor is proportional to the charge distribution into the
hadron, which can be obtained from electron-nucleon scattering.

This is primarily due to the electromagnetic structure of the nucleon which
can be obtained from the electron-hadron elastic scattering. In the Born
approximation, the Feynman amplitude for the elastic electron-proton
scattering is

$\displaystyle M_{ep\rightarrow
ep}=\frac{1}{q^{2}}[e\bar{u}(k_{2})\gamma^{\mu}u(k_{1}][e\bar{U}(p_{2}\Gamma_{\mu}(p_{1},p_{2})U(p_{1}],$
(17)

where $u$ and $U$ are the electron and nucleon Dirac spinors,

$\displaystyle\Gamma^{\mu}=F_{1}(t)\gamma^{\mu}\ +\
F_{2}(t)\frac{i\sigma^{\mu\nu}q_{\nu}}{2m},$ (18)

where $m$ is the nucleon mass, $\kappa$ is the anomalous part of the magnetic
moment and $t=-q^{2}=-(p-p^{\prime})^{2}$ is the square of the momentum
transfer of the nucleon. The functions $F_{1}(t)$ and $F_{2}(t)$ are named the
Dirac and Pauli form factors, which depend upon the nucleon structure.

However, it is not obvious that strong interactions have to be proportional to
the electromagnetic properties of hadrons. Taking into account this fact in
one of the famous models, Bourrely-Soffer-Wu used some modification of the
form factor with free parameters which was obtained from the description of
the differential cross section of the hadron scattering. For example, it takes
the form

$\displaystyle
G(t)=\frac{1}{1-t/m^{2}_{1}}\frac{1}{1-t/m^{2}_{2}}\frac{a^{2}+t}{a^{2}-t}.$
(19)

However, this allows some freedom in the $t$-dependence of the scattering
amplitude. It is necessary to take into account the parton distribution (PDFs)
in the hadrons. But PDFs depend on the Bjorken variable $x$. In the 80s some
relations between PDfs and hadron form factor there were proposed.

Note that the function like GPDs(x,t, $\xi=0$) was used already in the old
”Valon” model proposed by Sanevich and Valin in 1986 [23]. In the model, the
hadron elastic form factor was obtained by the integration function
$L(x)G(x,t)$ where $L(x)$ corresponds to the parton function and $G(x)$
corresponds to the additional function which depends from momentum transfer
and $x$. In modern language $L(x)G(x,t)$ exactly corresponds to the GPDs. The
scattering amplitude will be

$\displaystyle M_{AB}(s,t)=K_{A}(q^{2})K_{B}(q^{2})V(s,q^{2});$ (20)

where $V(s,q^{2})$ is a potential of strong interaction and $K_{AB}(q^{2})$
are the corresponding form factors.

$\displaystyle
K_{p}(q^{2})=\frac{1}{3}\int_{0}^{1}dx[2L_{p}^{u}(x)T_{p}^{u}(\vec{k});$ (21)

where $\vec{k}=(1-x)\vec{q}$, $k^{2}=(1-x)^{2}q^{2}$ and

$T_{p}^{u}(\vec{k})=e^{6.1k^{2}};T_{p}^{d}(\vec{k})=e^{3k^{2}}$

. This form factor can be obtained taking into account PDF of particle
interactions which is multiplied by some function depending on momentum
transfer $t$ and Boirken variable $x$.

Many different forms of the $t$-dependence of GPDs were proposed. In the quark
di-quark model the form of GPDs consists of three parts - PDFs, function
distribution and Regge-like function. In other works (see e.g. [59]), the
description of the $t$-dependence of GPDs was developed in a more complicated
picture using the polynomial forms with respect to $x$.

Commonly, the form of $GPDs(x,\xi,t)$ is determined through the exclusive deep
inelastic processes of type $\gamma^{*}p\rightarrow Vp,$ where $V$ stands for
a photon or a vector meson. However, such processes have a narrow region of
momentum transfer and in most models the $t$-dependence of GPDs is taken in
factorization form with the Gaussian form of the $t$-dependence. Really, this
form of $GPDs(x,\xi,t)$ can not be used to build space structure of the
hadrons, as for that one needs to integrate over $t$ in a maximally wide
region.

The conjunction between the momentum transfer and the impact parameter allows
one to obtain a space parton distribution that has a probability conditions
[14]. The connections between the deep-inelastic scattering, from which we can
obtain the $x$-dependence of parton distributions, and the elastic electron-
nucleon scattering, where the form factors of the nucleons are obtained, can
be derived by using the sum rules [19, 20, 21, 22]. The form factors, which
are obtained in different reactions, can be calculated as the Mellin moments
of GPDs. Using the electromagnetic (calculated as the zero Mellin moment of
GPDs) and gravitomagnetic form factors (calculated as the first moment of
GPDs) in the hadron scattering amplitude, one can obtain a quantitative
description of hadron elastic scattering in a wide region of energy and
transfer momenta.

Using the electromagnetic (calculated as the zero Mellin moment of GPDs) and
gravitomagnetic form factors (calculated as the first moment of GPDs) in the
hadron scattering amplitude one can obtain a quantitative description of
hadron elastic scattering in a wide region of energy and transfer momenta.

The proton and neutron Dirac form factors are defined as

$\displaystyle F_{1}^{p}(t)=e_{u}F_{1}^{u}(t)+e_{d}F_{1}^{d}(t),$ (22)
$\displaystyle F_{1}^{n}(t)=e_{u}F_{1}^{d}(t)+e_{d}F_{1}^{u}(t),$

where $e_{u}=2/3$ and $e_{d}=-1/3$ are the relevant quark electric charges. As
a result, the $t$-dependence of the $GPDs(x,\xi=0,t)$ can be determined from
the analysis of the nucleon form factors for which experimental data exist in
a wide region of momentum transfer. It is a unique situation as it unites
elastic and inelastic processes.

In the limit $t\rightarrow 0$, the functions $H^{q}(x,t)$ reduce to usual
quark densities in the proton:

${\cal\ H}^{u}(x,t=0)=u_{v}(x),\ \ \ {\cal H}^{d}(x,t=0)=d_{v}(x)$

with the integrals

$\int_{0}^{1}u_{v}(x)dx=2,\ \ \ \int_{0}^{1}d_{v}(x)dx=1$

normalized to the number of $u$ and $d$ valence quarks in the proton. The
energy-momentum tensor $T_{\mu\nu}$ [24, 20, 21, 13] contains three
gravitation form factors (GFF) $A^{Q,G}(t)$, $B^{Q,G}(t)$, and $C^{Q.G}(t)$.
We will scrutinize the first one which corresponds to the matter distribution
in a nucleon. This form factor contains quark and gluon contributions

$A^{Q,G}(t)=A_{q}^{Q,G}(t)+A_{g}^{Q,G}(t).$

To obtain the true form of the proton and neutron form factors, it is
important to have the true form of the momentum transfer dependence of GPDs.
Let us choose the $t$-dependence of GPDs in a simple form ${\cal{H}}^{q}(x,t)\
=q(x)\ exp[a_{+}\ f(x)\ t],$ with $f(x)=(1-x)^{2}/x^{\beta}$ [99]. The
isotopic invariance can be used to relate the proton and neutron GPDs; hence
we have the same parameters for the proton and neutron GPDs.

The complex analysis of the corresponding description of the electromagnetic
form factors of the proton and neutron by different PDF sets (24 cases) was
carried out in [25]. These PDFs include the leading order (LO), next leading
order (NLO) and next-next leading order (NNLO) determination of the parton
distribution functions. They used different forms of the $x$ dependence of
PDFs. We slightly complicated the form of GPDs in comparison with the equation
used in [99], but it is the simplest one as compared to other works

$\displaystyle{\cal{H}}^{u}(x,t)\ =q(x)^{u}\ e^{2a_{H}f(x)_{u}\ t};$ (23)
$\displaystyle{\cal{H_{d}}}^{d}(x,t)\ =q(x)^{d}\ e^{2a_{H}f_{d}(x)\ t};$
$\displaystyle{\cal{E}}^{u}(x,t)\ =q(x)^{u}(1-x)^{\gamma_{u}}\ e^{2a_{E}\
f(x)_{u}\ t};$ (24) $\displaystyle{\cal{E_{d}}}^{d}(x,t)\
=q(x)^{d}(1-x)^{\gamma_{d}}\ e^{2a_{E}f_{d}(x)\ t},$

where
$f_{u}(x)=\frac{(1-x)^{2+\epsilon_{u}}}{(x_{0}+x)^{m}},f_{d}(x)=(1+\epsilon_{0})(\frac{(1-x)^{1+\epsilon_{d}}}{(x_{0}+x)^{m}}).$

The hadron form factors will be obtained by integration over $x$ in the whole
range of $x$ \- $(0-1)$. Hence the obtained form factors will be dependent on
the $x$-dependence of the forms of PDF at the ends of the integration region.
The Collaborations determined the PDF sets from the inelastic processes only
in some region of $x$, which is only approximated to $x=0$ and $x=1$. Some
PDFs have the polynomial form of $x$ with different power. Some other have the
exponential dependence of $x$. As a result, the behavior of PDFs, when
$x\rightarrow 0$ or $x\rightarrow 1$, can impact the form of the calculated
form factors.

In that work, 24 different PDF were analyzed. On the basis of our GPDs with,
for example, the PDFs ABM12 [82], we calculated the hadron form factors by the
numerical integration and then by fitting these integral results by the
standard dipole form with some additional parameters

$F_{1}(t)=(4m_{p}-\mu t)/(4m_{p}-t)\ \tilde{G}_{d}(t),$

with

$\tilde{G}_{d}(t)=1/(1+q/a_{1}+q^{2}/a_{2}^{2}+q^{3}/a_{3}^{3})^{2}$

which is slightly different from the standard dipole form on two additional
terms with small sizes of coefficients. The matter form factor

$\displaystyle A(t)=\int^{1}_{0}x\
dx[q_{u}(x)e^{2\alpha_{H}f(x)_{u}/t}+q_{d}(x)e^{2\alpha_{H}f_{d}(x)/t}]$ (25)

is fitted by the simple dipole form $A(t)=\Lambda^{4}/(\Lambda^{2}-t)^{2}$
with $\Lambda^{2}=1.6$ GeV2. These form factors will be used in our model of
proton-proton and proton-antiproton elastic scattering and further in one of
the vertices of pion-nucleon elastic scattering.

To check the momentum dependence of the spin-dependent part of GPDs
$E_{u,d}(x,\xi=0,t)$, we can calculate the magnetic transition form factor,
which is determined by the difference of $E_{u}(x,\xi=0,t)$ and
$E_{d}(x,\xi=0,t)$. For the magnetic $N\rightarrow\Delta$ transition form
factor $G^{*}_{M}(t)$, in the large $N_{c}$ limit, the relevant
$GPD_{N\Delta}$ can be expressed in terms of the isovector GPD yielding the
sum rules [13]

The experimental data exist up to $-t=8$ GeV2 and our results show a
sufficiently good coincidence with experimental data. It is confirmed that the
form of the momentum transfer dependence of $E(x,\xi,t)$ determined in our
model is right.

Now let us calculate the moments of the GPDs with inverse power of $x$. It
gives us the Compton form factors. The results of our calculations of the
Compton form factors coincide well with the existing experimental data.
$R_{V}(t)$ and $R_{T}(t)$ have a similar momentum transfer dependence but
differ essentially in size. On the contrary, the axial form factor $R_{A}$ has
an essentially different $t$ dependence.

A good description of the variable form factors and elastic scattering of
hadrons gives a large support of our determination of the momentum transfer
dependence of GPDs. Based on this determination of GPDs, one can calculate the
gravitomagnetic radius of the nucleon using the integral representation of the
form factor and make the numerical differentiation over $t$ as $t\rightarrow
0$. This method allows us to obtain a concrete form of the form factor by
fitting the result of the integration of the GPDs over $x$. As a result, the
gravitomagnetic radius is determined as

$\displaystyle<{r_{A}}^{2}>=-\frac{6}{A(0)}\frac{dA(t)}{dt}|_{t=0};$ (26)

We used the same procedure as for our calculations of the matter radius. As a
result, the Dirac radius is determined from the zero Mellin moment of GPDs

$\displaystyle<{r_{D}}^{2}>=-\frac{6}{F(0)}\frac{dF(t)}{dt}|_{t=0};$ (27)

where $F(t)=\int_{0}^{1}(e_{u}\ q_{u}(x)\ +\ e_{d}\ q_{d}(x))\ e^{-\alpha\
tf(x)}dx$.

One can obtain the gravitational form factors of quarks which are related to
the second moments of GPDs. For $\xi=0$, one has

$\displaystyle\int^{1}_{0}dx\ x{\cal{H}}_{q}(x,t)=A_{q}(t);\,\ \int^{1}_{0}dx\
x{\cal{E}}_{q}(x,t)=B_{q}(t).$ (28)

The parameters of the phenomenological form of GPDs can be obtained from the
analysis of the experimental data for the proton and neutron electromagnetic
form factors simultaneously. Our determination of the momentum transfer
dependence of GPDs of hadrons allows us to obtain good quantitative
descriptions of different form factors, including the Compton,
electromagnetic, transition and gravitomagnetic form factor simultaneously.

##  V Model approximation

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

$\displaystyle\frac{d\sigma}{dt}=\frac{2\pi}{s^{2}}(|\Phi_{1}|^{2}+|\Phi_{2}|^{2}+|\Phi_{3}|^{2}+|\Phi_{4}|^{2}+4|\Phi_{5}|^{2}.$
(29)

and the spin correlation parameter $A_{N}(s,t)$ is

$\displaystyle A_{N}\frac{s^{2}}{4\pi}\frac{d\sigma}{dt}=$ (30)
$\displaystyle-[Im(\Phi_{1}(s,t)+\Phi_{2}(s,t)+\Phi_{3}(s,t)-\Phi_{4})(s,t)\Phi^{*}_{5}(s,t)]$

The HEGS model [38, 39] 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 for the spin-non-
flip amplitudes:

$\displaystyle\Phi^{em}_{1}(t)=\alpha
f_{1}^{2}(t)\frac{s-2m^{2}}{t};\Phi^{em}_{3}(t)=\Phi^{em}_{1};$ (31)

and for the spin-flip amplitudes, with the electromagnetic and hadronic
interactions included, every amplitude $\Phi_{i}(s,t)$ can be described as

$\displaystyle\Phi_{i}(s,t)=\Phi^{em}_{i}\exp{(i\alpha\varphi(s,t))}+\Phi^{h}_{i}(s,t),$
(32)

where $\varphi(s,t)=\varphi_{C}(t)-\varphi_{Ch}(s,t)$, and $\varphi_{C}(t)$
will be calculated in the second Born approximation in order to allow the
evaluation of the Coulomb-hadron interference term $\varphi_{Ch}(s,t)$. The
quantity $\varphi(s,t)$ has been calculated at large momentum transfer
including the region of the diffraction minimum [77, 78].

###  V.1 Electromagnetic amplitudes and phase factor

The electromagnetic amplitude can be calculated in the framework of QED in the
one-photon approximation,

$\displaystyle\Phi^{em}_{1}(t)=\alpha
f_{1}^{2}\frac{s-2m^{2}}{t},\Phi^{em}_{3}(t)=\Phi^{em}_{1}(t),$
$\displaystyle\Phi^{em}_{2}(t)=\alpha\frac{f_{2}^{2}(t)}{4m^{2}},$
$\displaystyle\Phi^{em}_{4}(t)=-\phi^{em}_{2}(t),\Phi^{em}_{5}(t)=\alpha\frac{s}{2m\sqrt{|t|}}f_{1}^{2}.$
(33)

For numerical calculations let us take
$-t=x(1)$ \- momentum transfer ( $x(1)$ is taken from experimental data) and
$q^{2}=-t$ GeV2.

( $fc1=1$ for $pp$ and $fc1=-1.$ \- for $p\bar{p}$ reactions),

$\Phi_{1}^{em}=-fc1\ \alpha_{em}(f1t^{2}/q^{2})e^{i\alpha_{em}\varphi_{CN}},$
$\Phi_{2}^{em}=fc1\alpha_{em}f2t^{2}/(pm24)e^{i\alpha_{em}*\varphi_{CN}},$
$\Phi_{3}^{em}=-fc1\alpha_{em}f1t^{2}/q^{2}e^{i\alpha_{em}*\varphi_{CN}},$
$\Phi_{4}^{em}=-\Phi_{2em},$
$\Phi_{5}^{em}=-fc1\alpha_{em}f1tf2t/(2m_{p}q)e^{i\alpha_{em}\varphi_{CN}}.$

The Coulomb-hadron interference phase is calculated with the dipole
electromagnetic form factor [78]. For calculation of the Coulomb-hadron phase
we take the energy dependence of the slope in form $b_{sl}=6+0.75\ln(s)$.

We take $\Lambda^{2}=0.71$; and $\gamma=.577215665$.
$\phi_{a}=q^{2}(2\Lambda^{2}+q^{2})/\Lambda^{4}\ln{(\Lambda^{2}+q^{2})^{2}/(\Lambda^{2}q^{2})};$
$\phi_{b}=(\Lambda^{2}+q^{2})^{2}/(\Lambda^{4}(4.\Lambda^{2}+q^{2})^{2}*q\sqrt{(}4\Lambda^{2}+q^{2}));$
$(4\Lambda^{4}(\Lambda^{2}+7q^{2})+q^{4}(10\Lambda^{2}+q^{2}))\ln(4\Lambda^{2}/(\sqrt{(}4\Lambda^{2}+q^{2})+q)^{2});$
$\phi_{c}=(2\Lambda^{4}-17\Lambda^{2}q^{2}-q^{4})/(4\Lambda^{2}+q^{2})^{2};$

$\phi_{1-3}=\phi_{a}+\phi_{b}+\phi_{c}$ ;
$\phi_{CN}=-\ln{b_{sl}q^{2}/2.}+\gamma+\ln(1.+8.d0/(b_{sl}al2))-\phi_{1-3})$

.

Figure 4:  The comparison of the HEGS model calculation of $d\sigma/dt$ of
$pp$ scattering at $\sqrt{s}=13$ TeV, $p\bar{p}$ scattering at $\sqrt{s}=53$
GeV and $pn$ scattering at $\sqrt{s}=23$ GeV

.

Figure 5:  The HEGS model calculation of $d\sigma/dt$ of $pp$ scattering at
$\sqrt{s}=13$ TeV.

###  V.2 The hadron scattering amplitude

.

Figure 6:  The comparison of the HEGS model calculation of $d\sigma/dt$ of
$pp$ scattering at $\sqrt{s}=30.6$ GeV (solid line and full ) and $\bar{p}p$
at $\sqrt{s}=30.4$ GeV (dashed line and open triangles). In both cases the
additional normalization $n=1$). scattering.  Figure 7: The dependence of the
imaginary part of the hadron scattering amplitude on $s$ and $t$ calculated in
the model at the energy $\sqrt{s}=0.51,7,13$ TeV

Figure 8:  [top] $sigma_{tot}(s)$: dashed line - $p\bar{p}$ scattering, solid
line -$pp$ scattering; [middle] $\rho(s,t=0)$: dashed line - $p\bar{p}$
scattering, solid line -$pp$ scattering; [down] the real part of the
scattering amplitude: dashed line - $p\bar{p}$ scattering, solid line -$pp$
scattering.

Let us define the hadron spin-non-flip amplitude as

$\displaystyle F^{h}_{\rm nf}(s,t)$ $\displaystyle=$
$\displaystyle\left[\Phi_{1h}(s,t)+\Phi_{3h}(s,t)\right]/2;$ (34)

At small $t$,the CNI region, there are two contributions coming from the
electromagnetic and the strong interaction.

On the one hand, the interference of such contribution gives the possibility
to determine the size of the real part of the scattering amplitude. On the
other hand, to determine the form of the imaginary part of the hadronic
amplitude, it is necessary to extract the Coulomb contribution and the
interference term. As the electromagnetic amplitude and its main contribution
at $t\rightarrow 0$ are well known, the measure of experimental data at very
small $t$ gives the possibility to improve the normalization of the
differential cross sections.

The existence of the different sets of the experimental data also allows to
improve the separate normalization of the experimental data. To compare the
different sets it is need to have some gauge.

Let us take the calculations of the differential cross sections carried out in
the framework of the new High energy generalized structure (HEGS) model [38,
39] as such a gauge. The model has only a few free parameters and it
quantitatively describes experimental data in a wide domain of the momentum
transfer, including the data in the CNI region, in a very wide energy region
(from $\sqrt{(}s)=6$ GeV up to LHC energies simultaneously with the same
numbering of the free parameters. The HEGS model assumes a Born term for the
scattering amplitude which gets unitarization via the standard eikonal
representation to obtain the full elastic scattering amplitude. The scattering
amplitude has exact $s\leftrightarrow u$ crossing symmetry as it is written in
terms of the complexified Mandelstam variable $\hat{s}=se^{-i\pi/2}$ that
determines its real part. The scattering amplitude also satisfies the integral
dispersion relation at large $s$. It can be thought of as the simplest unified
analytic function of its kinematic variables connecting different reaction
channels without additional terms for separate regions of momentum transfer or
energy. Note that the model reproduces the diffraction minimum of the
differential cross section in a wide energy region [69]. HEGS model describes
the experimental data at low momentum transfer, including the Coulomb-hadron
interference region, and hence includes all five electromagnetic spin
amplitudes and the Coulomb-hadron interference phase.

Let us determine the Born terms of the elastic nucleon-nucleon scattering
amplitude using both (electromagnetic and gravitomagnetic) form factors

$\displaystyle F_{Pom2}^{Born}(s,t)=$ $\displaystyle h_{Pom2}\
G^{2}_{em}(t)F_{a}(s,t);$ (35) $\displaystyle F_{Pom3}^{Born}(s,t)=$
$\displaystyle h_{Pom3}\ A^{2}_{gr}(t)F_{b}(s,t);$ $\displaystyle
F_{Odd3}^{Born}(s,t)=$ $\displaystyle h_{Odd3}\ A^{2}_{gr}(t)F_{c}(s,t);$

where $F_{a}(s,t)$ and $F_{b,c}(s,t)$ have the standard Regge form:

$\displaystyle F_{a}(s,t)\ =\hat{s}^{\epsilon_{1}}\ e^{B(\hat{s})\ t};\ \ \
F_{b,c}(s,t)\ =\hat{s}^{\epsilon_{1}}\ e^{B(\hat{s})/4\ t},$ (36)

with $\hat{s}=s\ e^{-i\pi/2}/s_{0}$ ; $s_{0}=4m^{2}_{p}\ {\rm GeV^{2}}$, and
$h_{Odd3}=ih_{3}t/(1-r_{0}^{2}t)$. The intercept of all main parts of the
scattering amplitudes $1+\Delta_{1}=1.11$ was chosen as arithmetic means from
different works including inelastic scattering. Hence, at the asymptotic
energy we have the universality of the energy behavior of the elastic hadron
scattering amplitudes. The main part of the slope of the scattering amplitude
has the standard logarithmic dependence on the energy $B(s)=\alpha^{\prime}\
ln(\hat{s})$ with $\alpha^{\prime}=0.24$ GeV-2. It is taken with some
correction (eq. 16).

Both the hadron electromagnetic and gravitomagnetic form factors were used in
the framework of the high energy generalized structure (HEGS) model of elastic
nucleon-nucleon scattering. This allowed us to build the model with a minimum
number of fitting parameters [38, 39, 40]. The Born term of the elastic hadron
amplitude can now be written as

$\displaystyle F_{h}^{Born}(s,t)=$ $\displaystyle h_{1}\ G^{2}(t)\ F_{a}(s,t)\
(1+r_{1}/\hat{s}^{0.5})$ $\displaystyle+h_{2}\ A^{2}(t)\ F_{b}(s,t)\ $
$\displaystyle\pm h_{odd}\ A^{2}(t)F_{b}(s,t)\
(h_{as}+r_{2}/\hat{s}^{0.75})(-t)/(1-r_{d}t)),$

where both (electromagnetic and gravitomagnetic) form factors are used. The
constant parameters are determined by the fitting procedure.

So, to extend the model to low energies, it is necessary to take into account
the contributions of the second Reggions. To avoid a substantially increasing
number of fitting parameters we introduce the effective terms which represent
the contributions of the sums of different Reggions. For the terms with the
energy dependence order $1/\sqrt{\hat{s}}$ we take for the $pp$ scattering the
cross even part in the form

$\displaystyle F_{R1}=ih_{R1}/\sqrt{\hat{s}}e^{b_{R1}t\ln{\hat{s}}};$ (38)

and for the $p\bar{p}$ the cross odd term in the form

$\displaystyle F_{R1}=h_{R1}/\sqrt{\hat{s}}e^{b_{R1}t\ln{\hat{s}}}$ (39)

where the value $b_{R1}$ is fixed by unity. For the cross even part fast
decreasing with growing energy, we take term in the form

$\displaystyle F_{cr.ev.}=h_{R2}/\hat{s}e^{b_{R2}t\ln{\hat{s}}};$ (40)

As a result, the low energy terms require only three additional fitting
parameters.

The model is very simple from the viewpoint of the number of terms of the
scattering amplitude and fitting parameters. There are no any artificial
functions or any cuts which bound the separate parts of the amplitude by some
region of momentum transfer. In the framework of the model, the description of
experimental data was obtained simultaneously at the large momentum transfer
and in the Coulomb-hadron region in the energy region from $\sqrt{s}=6$ GeV up
to LHC energies. The model gives a very good quantitative description of the
recent experimental data at $\sqrt{s}=13$ TeV [75].

The final elastic hadron scattering amplitude is obtained after unitarization
of the Born term. So, at first, we have to calculate the eikonal phase

$\chi(s,b)\ =-\frac{1}{2\pi}\ \int\ d^{2}q\ e^{i\vec{b}\cdot\vec{q}}\ F^{\rm
Born}_{h}(s,q^{2})$

and then to obtain the final hadron scattering amplitude

$F_{h}(s,t)=is\ \int\ b\ J_{0}(bq)(1-\exp[\chi(s,b)])db.$

The essential property of the model is that the real part of the scattering
amplitude is obtained automatically through the complex $\hat{s}$ only. The
scattering amplitude has exact $s\leftrightarrow u$ crossing symmetry.

An example of such description is shown in Fig. 6 for proton-proton at
$\sqrt{s}=30.6$ GeV and for proton-antiproton scattering at $\sqrt{s}=30.4$
GeV. The difference comes from the interference term of the Coulomb and the
real part of the hadron amplitudes. The $\chi^{2}/N=1.1$ for $pp$ and
$\chi^{2}/N=1.03$ for $p\bar{p}$ scattering with additional normalization
$k_{pp}=1.005$ and $k_{p\bar{p}}=0.974$, respectively. The figure presents the
data without additional normalization. Note that this $\chi^{2}$ was obtained
as separate part of the full fit of the $4327$ experimental data in a wide
energy and momentum transfer region [39].

Let us compare the predictions of the HEGS model for the differential elastic
cross section at small $t$ with the LHC data. In the fitting procedure only
the statistical errors are taken into account. The systematic errors are
reflected through an additional normalization coefficient which is the same
for all the data of a given set. The different normalization coefficients have
practically random distributions at small $t$.

Figure 9:  The real part of the scattering amplitude coming from complex
$\hat{s}$ in the Regge representation (short dashed line - complex $\hat{s}$
only in the exponential eq.(43); solid line full representation eq.(44)

The data of the TOTEM at $\sqrt{s}=7$ TeV are consistent and their mean value
is equal to $98.5$ mb. The ATLAS Collaboration, using their differential cross
section data in a region of $t$ where the Coulomb-hadron interference is
negligible, obtained the value $\sigma_{tot}=95.35\pm 2.0$ mb. The difference
between the two results, $\sigma_{tot}(s)$(T.) - $\sigma_{tot}(s)$(A.) =
$3.15$ mb, is about 1 $\sigma$. At $\sqrt{s}=8$ TeV, the measured value of
$\sigma_{tot}$ grows, especially in the case of the TOTEM Collaboration and
the difference between the results of the two collaborations grows to
$\Delta(\sigma_{tot}(s)$(T.) - $\sigma_{tot}(s)$(A.) $=5.6$ mb, i.e. 1.9
$\sigma$. This is reminiscent of the old situation with the measurement of the
total cross sections at the Tevatron at $\sqrt{s}=1.8$ TeV via the luminosity-
independent method by different collaborations.

Of course, we cannot say that the normalization of the ATLAS data is better
than that of the TOTEM data simply because it coincides with the HEGS
predictions. But this exercise may point to the main reason for the different
values of the total cross sections obtained by the two collaborations. This
does not exclude some further problems with the analysis of the experimental
data, e.g. those related to the analysis of the TOTEM data at $\sqrt{s}=7$ TeV
[68].

The position of the diffraction minimum $t_{min}(s,t)$ moves to low momentum
transfer continuously [41]. It is interesting that the velocity of changing
the position of the diffraction minimum changes very slowly. For example, from
ISR energy $\sqrt{s}=53$ GeV up to SPS energy $\sqrt{s}=540$ GeV such position
changes with a speed of $0.11$ GeV2 per $100$ GeV. Between $\sqrt{s}=540$ GeV
and $\sqrt{s}=7$ TeV such speed is two times less and equals $0.006$, at last
between $7$ and $13$ TeV the position of minimum changes with a speed of
$0.002$ per $100$ GeV. Approximately, scaling of this process can be
represented as $t_{min}\ln{s/s_{0}}=const$. After the second bump the slope of
the differential cross sections increases with energy. It corresponds to the
grows of the slope of the diffraction peak.

Figure 10: The dependence of the real part of the hadron scattering amplitude
on $s$ and $t$ calculated in the model at the energy $\sqrt{s}=0.51,7,13$ TeV

The behavior of the imaginary part of the scattering amplitude over momentum
transfer is presented in Fig. 7 for the energies $\sqrt{s}=0.51,7,13$ TeV.
Again, we can see the point of crossover in the region of $|t|=0.2$ GeV2.
Despite the essential grows of the size of the imaginary part of the
scattering amplitude at very small momentum transfer, its slope slightly
changes with $t$ in the region of the Coulomb nuclear interference. The size
of slope is practically proportional to the size of the total cross section in
that region. However at larger $t$, for example at $|t|=0.1$ GeV2, it grows
essentially faster.

It should be noted that the size of the slope of the differential cross
sections is determined in that region of $t$ by the CNI interference term
which is proportional to $\alpha/t$ . It allows us to analyze [71] the first
points of the unique experiment carried out by the ATLAS Collaboration [31].
The point of $t$, where the imaginary part changes its sign, determines the
position of the diffraction minimum. But it slightly moves at some large $t$
by the contribution of the real part of the elastic hadron scattering
amplitude.

Figure 11: The size of the $\rho(s,t)$ \- ratio of the real to imaginary part
of the hadron scattering amplitude is calculated in the model at the energy
$\sqrt{s}=0.51,7,13$ TeV depending on $s$ and $t$.

It is interesting that the form of the real part of the hadron elastic
scattering amplitude it is similar to its imaginary part. Of course, they are
not proportional to each other as their connection has to satisfy the
dispersion relations [58] which require, for example, the changing size of the
real part at sufficiently small momentum transfer. Really, in Fig. 11, we see
that the real part change its sign essentially earlier than the position of
the diffraction minimum. At LHC energies this happens in the area of momentum
transfer approximately equal to $0.2$ GeV2. But before that, we can see the
crossing point at $|t|\sim 0.06$ GeV2. Likely, the behavior of the imaginary
part the slope of the real part at very small momentum transfer is also
practically proportional to the size of the total cross section at different
energies and grows at larger momentum transfers. It can be see that the real
part at LHC energies has the negative maximum at approximately $|t|=0.3$ GeV2
situated near the diffraction minimum. Hence, it essentially impacts the form
and size of the diffraction minimum in the differential cross sections. In
Fig. 11, the ratio $\rho(s,t)$ of the real to imaginary part of the hadron
elastic scattering amplitude is presented for different energies. Such a
complicated structure of $\rho(s,t)$ are determined by the changes of the sign
of the real and imaginary parts of the scattering amplitude. At small momentum
transfer, the size of $\rho(s,t)$ is small as the real part changes its sign.
Contrary, when the imaginary part changes its sign, the size of $\rho(s,t)$
grows very faster. The energy dependence of $\rho(s,t)$ is due to the movement
of the position of the diffraction minimum, hence with the energy dependence
of the imaginary part of the scattering amplitude.

Especially the size and energy dependence of the real part of the hadronic
amplitude impact on the differential cross sections at CNI region. The model
descriptions of differential cross section at small momentum transfer and at
low energies are presented in Figures 5, 15-17. In these figures are presented
the experimental data with high precision and we show only statistical errors,
which were taking into account in out fitting procedure. Obviously, that the
model reproduce the experimental data very good at the wide energy region.

Figure 12:  The slopes of the Born term of the scattering amplitude [top] and
the uniterized scattering amplitude [bottom] ( long dashed line -the slope of
the imaginary part, the short-dashed line - the slope of the real part and the
solid line - the slope of the differential cross sections.

###  V.3 The real part of the elastic scattering amplitude

One of the origins of the non-linear behavior of the differential cross
sections may arise the different $t$ dependence of the imaginary and real
parts of the scattering amplitude. In most part, in the different approaches
it is supposed that this $t$ dependence is the same for both parts of the
scattering amplitude. It should be noted the importance of determining the
size of the real part of the scattering amplitude was emphasized in many work
of Andre Martin. If at the LHC the value of $\rho(s,t)$ is measured at high
precision, it will give the possibility to check up the validity of the
dispersion relations [79].

In the analysis of the experimental data [65] two cases were considered. One
is so-called ”central” case in which the ratio of the real to imaginary parts
of the scattering amplitude is independent of momentum transfer or slightly
decreases. The other, so-called ”peripheral” case takes into account the
assumption that $\rho(s,t)$ grows with momentum transfer. Really, the last
case contradicts the dispersion relations; hence, it has non-physical
motivation.

Really, as the scattering amplitude has to be an analytic function of its
kinematic variable, let us take the energy dependence of the scattering
amplitude through the complex $\hat{s}=se^{-i\pi/2}$, and it must satisfy the
dispersion relations.

For simplicity, very often they use, the so-called, local or derivative
dispersion relations (see for example [45]) to determine the real part of the
scattering amplitude. For example, the COMPETE Collaboration used

$\displaystyle ReF_{+}(E,0)=$ (41)
$\displaystyle(\frac{E}{m_{p}})^{\alpha}tan[\frac{\pi}{2}(\alpha-1+E\frac{d}{dE}]ImF_{+}(E,0)/(\frac{E}{m_{p}})^{\alpha}.$

A different form of the derivative dispersion relation was taken as [90]

$\displaystyle ReF_{+}(E,0)=$ (42)
$\displaystyle(\frac{\pi}{ln(s/s_{0}})\frac{d}{d\tau}[\tau
ImF_{+}(s,t)/Im_{+}(s,t=0)]ImF_{+}(s,t=0),$

where $\tau=t(ln(s/s_{0})^{2}$ and as $s\rightarrow\infty$. To satisfie these
relations, the scattering amplitude has to be a unified analytic function of
its kinematic variables connecting different reaction channels.

In most cases, the real part is taken as proportional to the imaginary part of
the scattering amplitude. So the slopes of both parts equals each other. There
is also some unusual assumption about the growth of the real part of the
scattering amplitude at small momentum transfer relative to its imaginary part
(so-called - the peripheral case [65]). Obviously, both assumptions do not
satisfy the dispersion relations, especially the last one.

.  .

Figure 13: The ratio of the cross-odd part of the scattering amplitude of
elastic proton-proton scattering to its cross even part (a) at 13 TeV; (b) at
9.26 GeV

.  .

Figure 14: The contributions of the Odderon part to the differential cross
sections in the region of the dip a)[top] $-t=0.45$ GeV2 ; b) [down] $-t=1.45$
GeV2.

Let us examine the origins of the complex $t$ dependence of the real part.
Take the scattering amplitude in the form

$\displaystyle F(s,t)\ =\ hs^{\Delta}e^{BtLn(\hat{s})}.$ (43)

where the complex $\hat{s}$ is used only in the exponential. In this case, the
real part will be negative (in Fig. 11, it is shown by the short dashed line).
and essentially different from the behavior of the imaginary part (long-dashed
line). Now let us use $\hat{s}$ is used in all parts of the scattering
amplitude

$\displaystyle F(s,t)\ =\ h\hat{s}^{\Delta}e^{Bt\ln(\hat{s})}.$ (44)

The dispersion relations lead to the fact that the slope of the real part of
the scattering amplitude must be larger than the slope of imaginary part. For
example, if the imaginary part of the spin-nonflip hadron elastic scattering
amplitude takes a simple exponential form $ImF_{+}\sim he^{Bt}$, then from
eq.(V.3) we have that the real part of the $F_{+}$ will be
$ReF_{+}=(1.+Bt)e^{Bt}$. Hence it has zero in the region of momentum transfer
around $-0.1--0.15$ GeV2

At last, but not least, it should be noted that the unitarization procedure
has a strong influence on the $t$ dependence of the real part of the
scattering amplitude. For example, take the standard eikonal form of
unitarization. If the Born term is taken in the ordinary exponential form
(eq.44), then the imaginary part has a constant slope (see Fig. 12 [top]) and
the real part has first zero at small momentum transfer. After eikonalization
both slopes of the imaginary and real parts of the scattering amplitude have a
strong $t$ dependence (Fig. 12 [bottom]).

##  VI New effects in diffraction elastic scattering at small angles

In the fitting procedure of the experimental data, only statistical errors
were taken into account. As the systematic errors are mostly determined by
indefiniteness of luminosity, they are taken into account as an additional
normalization coefficient. This method essentially decreases the space of a
possible form of the scattering amplitude. This allowed us to find the
manifistation of some small effects at 13 TeV experimental data for the first
time [19-21]. Our further researches with taking into account a wider range of
experimental data confirm such new effects. We determined the new anomalous
term with a large slope as

$\displaystyle f_{an}(t)=$ $\displaystyle ih_{an}\ln{(\hat{s}/s_{0})}/k$
$\displaystyle\exp[-\alpha_{an}(|t|+(2t)^{2}/t_{n})\ln{(\hat{s}/s_{0})}]\
F_{em}^{2}(t);$

where $h_{an}$ is the constant determining the size of the anomalous term with
a large slope - $\alpha_{an}$; $F_{em}(t)$ is the electromagnetic form factor,
which was determined from the GPDs [25], and $\ k=\ln(13000^{2}\ {\rm
GeV}^{2}/s_{0})\ $ is introduced for normalization of $h_{an}$ at $13$ TeV,
$t_{n}=1$ GeV2-normalization factor (see Appendix B for definitions of
$\hat{s}$ and $s_{0}$). Such a form adds only two additional fitting
parameters, and this term is supposed to grow with energy of order
$\ln{(\hat{s}/s_{0})}$. The term has a large imaginary part and a small real
part determined by the complex $\hat{s}$.

It is proportional to electromagnetics form factors, and the analysis of the
experimental data above $6$ GeV gives the sizes of the constant
$h_{an}=0.997\pm 0.006$. It is less than obtained in the fit of only high
energy data, but now it has a more complicated energy dependence and a very
small error.

Figure 15: The comparison of the HEGS model calculation of $d\sigma/dt$ of
$pp$ scattering at $\sqrt{s}=27.4$ TeV [left]; and $p\bar{p}$ scattering at
$\sqrt{s}=30$ GeV [right].

Figure 16: The comparison of the HEGS model calculation of $d\sigma/dt$ of
$pp$ scattering at $\sqrt{s}=19.4$ TeV [left]; and at $\sqrt{s}=12.3$ GeV
[right].

Figure 17: The comparison of the HEGS model calculation of $d\sigma/dt$ of
$pp$ scattering at $\sqrt{s}=9.9$ TeV [left]; and at $\sqrt{s}=6.1$ GeV
[right].

Our method helps us to find a small oscillation effect in the differential
cross section at small momentum transfer. Such oscillation can be determined
by an additional oscillation term in the scattering amplitude. Our fitting
procedure takes the oscillatory function

$\displaystyle f_{osc}(t)=$ $\displaystyle ih_{osc}(1\pm
i)(\ln{(\hat{s}/s_{0})}/k+h_{s}/\hat{s})\ J_{1}(\tau)/\tau\ A^{2}(t),$
$\displaystyle\tau=\pi\ (\phi_{0}-t)/t_{0};$

here $J_{1}(\tau)$ is the Bessel function of the first order;
$t_{0}=1/[a_{p}/(\ln{(\hat{s}/s_{0})}/k)]$, where $a_{p}=17.15$ GeV-2 is the
fitting parameter that leads to AKM scaling on $\ln{(\hat{s}/s_{0})}$; $A(t)$
is the gravitomagnetic form factor, which was determined from the GPDs [25],
and $h_{osc}$ is the constant that determines the amplitude of the oscillatory
term with the period determined by $\tau$. This form has only a few additional
fitting parameters and allows one to represent a wide range of possible
oscillation functions. The phase $\phi_{0}$ is obtained near zero and has a
different sign for $pp$ and $p\bar{p}$ scattering. Inclusion in the fitting
procedure of the data of $p\bar{p}$ elastic scattering shows that part of the
oscillation function changes its sign for the crossing reactions. As a result,
the plus sign is related with $pp$ and minus with $p\bar{p}$ elastic
scattering. Hence, this part is the crossing-odd amplitude, which has the same
simple form for $pp$ and $p\bar{p}$ scattering only with a different sign.

The wider energy region used in this analysis allows one to reveal the
logarithmic energy dependence of the oscillation term. Let us compare the
constant (size) of the oscillation function of three independent analyses
(only $13$ TeV, all LHC data, all data above $500$ GeV). Now from our fitting
procedure we obtain $h_{osc}=0.227\pm 0.007\ {\rm GeV}^{-2}$ The size of
$h_{osc}$ is obtained as is smaller than obtained in the only high energy
data; however, the error is decreased essentially. Perhaps, this reflects a
more complicated form of energy dependence obtained now. Note, despite the
logarithmic growth of the oscillation term, its relative contribution
decreases as the main scattering amplitude grows as $\ln^{2}(s)$.

Figure 18:  The HEGS model calculation of $d\sigma/dt$ of $pn$ scattering
(top) at $\sqrt{s}=26$ GeV; (down) $\sqrt{s}=19.4$ GeV; [left] with TOTEM data
[66] (requires the additional normalization of the data $k=0.91$); [right]
with TOTEM data [96, 97] (with the additional normalization of the data
$k=0.915$).

Figure 19:  The HEGS model calculation of $d\sigma/dt$ of $pn$ scattering
(top) at $\sqrt{s}=13.7$ GeV; (down) $\sqrt{s}=4.8$ GeV; [left] with TOTEM
data [66] (requires the additional normalization of the data $k=0.91$);
[right] with TOTEM data [96, 97] (with the additional normalization of the
data $k=0.915$).

##  VII Proton-proton and proton-antiproton elastic scattering and Odderon
contributions

At high energies and in the region of small momentum transfer the difference
between the $pp$ and $p]bar{p}$ differential cross sections comes in most part
from the CNI term, as the real part of the Coulomb amplitude has a different
sign in these reactions. In the standard fitting procedure, one neglects the
$\alpha^{2}$ term the equation takes the form:

$\displaystyle d\sigma/dt=$
$\displaystyle\pi[(F_{C}(t))^{2}\\!+\\!(\rho(s,t)^{2}+1)(ImF_{N}(s,t))^{2})$
(47) $\displaystyle+$ $\displaystyle
2(\rho(s,t)+\alpha\varphi(t))F_{C}(t)ImF_{N}(s,t)],$

where $F_{C}(t)=\mp\ 2\alpha G^{2}(t)/|t|$ is the Coulomb amplitude (the upper
sign is for $pp$, the lower sign is for $p\bar{p}$) and $G^{2}(t)$ is the
proton electromagnetic form factor squared; $ReF_{N}(s,t)$ and $ImF_{N}(s,t)$
are the real and imaginary parts of the hadron amplitude;
$\rho(s,t)=ReF_{N}(s,t)/ImF_{N}(s,t)$. The formula (VII) is used for the fit
of experimental data in getting hadron amplitudes and the Coulomb-hadron phase
in order to obtain the value of $\rho(s,t)$.

It is supposed that the real part of the $pp$ and $p\bar{p}$ scattering have
the same sign, at least at small $t$. Such a difference, obtained in the HEGS
model is presented in Fig. 6, where the comparison of the HEGS model
calculation of $d\sigma/dt$ of $pp$ scattering at $\sqrt{s}=30.6$ GeV (solid
line and full squares) and $\bar{p}p$ at $\sqrt{s}=30.4$ GeV (dashed line and
open triangles) is shown. In both cases the additional normalization $n=1$.

Of course, the Odderon amplitude changes the size and $t$ dependence of the
real par of the full amplitude. In the recent paper [91] the different cases
(with and without Odderon contributions) were analysed. It was noted that the
effect of incorporating the Odderon becomes notably significant when analysing
specific subsets of data. It is remarkable, that the authors note ”we will get
too large $\rho^{\bar{P}p}$ at $\sqrt{s}\sim 541$ GeV in disagreement with the
data UA4/2. As a result, they restore an old problem of the value of
$\rho(s,t)$ at $\sqrt{s}\sim 540$ GeV. However, as was noted in the
introduction, this problem strongly depends on the form of the non-linear
slope and can be solved in [73]. Now, in the present model, the Odderon
amplitude essentially decreases as $t\rightarrow 0$. The value of
$\rho(\sqrt{s}=541\rm{GeV},t=0)=0.122$, which coincides with the result of the
UA4/2 Collaboration.

In Fig. 13, the ratio of the cross-odd part of the scattering amplitude of
elastic proton-proton scattering to its cross even part at 13 TeV and at 9.26
GeV is presented. It is seen that this ratio is small at 13 TeV, except for
the position of the diffraction minimum. At low energy, this ratio is small at
small t but increases essentially in the region of large momentum transfer.

However, the Odderon contribution is very important at the diffraction
minimum. The real part of the scattering amplitude, which is positive at $t=0$
and $\sqrt{s}>30$ GeV, changes its sign at larger $t$ thus corresponding to
the dispersion relations, but, in any case it fulfills the diffraction dip in
the differential cross sections. In Fig. 14, it is shown the sizes of the
differential cross sections in the region of the diffraction minimum with and
without the Odderon contributions at $\sqrt{s}$ around $53$ GeV and at $10$
TeV.

##  VIII Proton-neutron elastic scattering

We take in our analysis $24$ sets of the proton-neutron experimental data from
$\sqrt{s}>4.5$ GeV up to maximum energy $\sqrt{s}>27.5$ GeV where they
correspond to experimental data with a total number of experimental points
$N=526$. As in the case of $pp$ scattering, we include in our fitting
procedure only statistical errors with taken into account the systematic
errors as additional normalization of a separate set. We take the amplitudes
of the model obtained in the case of $pp$ and $p\bar{p}$ scattering and fix
the parameters of the main terms and two anomalous terms, except for the terms
of the second Reggions. The electromagnetic amplitudes are removed in this
case. As a result, in the $pn$ case, only parameters of second effective
Reggion are obtained from the fitting procedure. The
$\sum_{i,j}\chi^{2}_{i,j}=567$. The corresponding obtained description of the
differential cross section are present in Figs. 18, 19. It is clear that the
HEGS model very well works in the case of proton-neutron elastic scattering.
As in the case of $pn$ scattering, there are many experimental data at small
momentum transfer, where the contributions of our two anomalous terms are
important, the successful description of the differential cross section
confirms the existence of these terms.

Figure 20: The analyzing power $A_{N}$ of pp - scattering calculated: a) at
$\sqrt{s}=3.62\ $GeV (small $t$) (the experimental data [102]), and b) at
$\sqrt{s}=3.62\ $GeV (larger $t$).

Figure 21: The analyzing power $A_{N}$ of pp - scattering calculated: a) at
$\sqrt{s}=9.2\ $GeV, (the experimental data [102]), and b) at $\sqrt{s}=13.7\
$GeV (points - the experimental data [102]).

Figure 22: The analyzing power $A_{N}$ of pp - scattering calculated: a) at
$\sqrt{s}=9.2\ $GeV, (the experimental data [102]), and b) at $\sqrt{s}=13.7\
$GeV (points - the experimental data [102]).

Figure 23: The analyzing power $A_{N}$ of pp - scattering calculated: a) at
$\sqrt{s}=19.4\ $GeV (the experimental data [102]), and b) at $\sqrt{s}=23.4\
$GeV (points - the existing experimental data [102])

##  IX Polarization effects in proton-proton elastic scattering

In the Regge limit $t_{fix.}$ and $s\rightarrow\infty$ one can write the
Regge-pole contributions to the helicity amplitudes in the $s$-channel as

$\displaystyle\Phi^{B}_{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}}(s,t)\sim$
$\displaystyle\sum_{i}g^{i}_{\lambda_{1},\lambda_{2}}(t)g^{i}_{\lambda_{3},\lambda_{4}}(t)[\sqrt{|t|}]^{|\lambda_{1}-\lambda_{2}|+|\lambda_{3}-\lambda_{4}|}$
$\displaystyle(\frac{s}{s_{0}})^{\alpha_{i}}(1\pm e^{-i\pi\alpha_{i}}).$

The corresponding spin-correlation values presented in eq.(30).

Neglecting the $\Phi_{2}(s,t)-\Phi_{4}(s,t)$ contribution, the spin
correlation parameter $A_{N}(s,t)$ can be written taking into account the
phases of separate spin non-flip and spin-flip amplitudes as
$\varphi_{nf}(s,t),\varphi_{sf}(s,t)$ the analysis power is

$\displaystyle A_{N}(s,t)=$ $\displaystyle-\frac{4\pi}{s^{2}}[(F_{nf}(s,t)|\
|F_{sf}(s,t)|$ $\displaystyle
sin(\varphi_{nf}(s,t)-\varphi_{sf}(s,t))/\frac{d\sigma}{dt}.$

It is clearly seen that despite the large spin-flip amplitude, the analyzing
power can be near zero if the difference of the phases is zero in some region
of momentum transfer. The experimental data at some point of the momentum
transfer show the energy independence of the size of the spin correlation
parameter $A_{N}(s,t)$. Hence, the small value of the $A_{N}(s,t)$ at some $t$
(for example, very small $t$) does not serve as a proof that it will be small
in other regions of momentum transfer.

It is usually assumed that the imaginary and real parts of the spin-non-flip
amplitude have the exponential behavior with the same slope, and the imaginary
and real parts of the spin-flip amplitudes, without the kinematic factor
$\sqrt{|t|}$ [100], are proportional to the corresponding parts of the non-
flip amplitude. That is not so as regards the $t$ dependence shown in Ref.
[70], where $F^{fl}_{h}$ is multiplied by the special function dependent on
$t$. Moreover, one mostly takes the energy independence of the ratio of the
spin-flip parts to the spin-non-flip parts of the scattering amplitude. All
this is our theoretical uncertainty [87, 86].

In [88, 89] on the basis of generalization of the constituent-counting rules
of the perturbative QCD, the proton current matrix elements
$J^{\pm\delta\delta}_{p}$ for a full set of spin combinations corresponding to
the number of the spin-flipped quarks were calculated. This leads to part of
the spin-flip amplitude

$\displaystyle F_{h}^{sl}\sim\sqrt{-t}/(\frac{4}{9}m_{p}^{2})\
\sqrt{-t}/(\frac{4}{9}m_{p}^{2})\ \sqrt{-t}/(\frac{4}{9}m_{p}^{2}).$ (50)

Hence, such an amplitude gives large contributions at large momentum transfer.

Of course, at lower energies we need to take into account the energy
dependence parts of the spin-flip amplitudes. So the form of the spin-flip
amplitude is determined as

$\displaystyle
F_{sf1}(s,t)=ih_{sf1}q^{3}(1+q^{3}/\sqrt{\hat{s}})G_{em}^{2}e^{2tln{\hat{s}}}$
(51)

We take the second part of the spin-flip amplitude in the form

$\displaystyle
F_{sf2}(s,t)=i\sqrt{|t|}G_{em}^{2}(h_{5}+h_{6}(1+ih_{4})/ssc^{2})e^{2tln{\hat{s}}}$
(52)

This works in most part at low energies.

Our calculation for $A_{N}(t)$ is shown in Fig. 21 a,b at $\sqrt{s}=4.9$ GeV
and $\sqrt{s}=6.8$ GeV. For our high energy model it is a very small energy.
However, the description of the existing data is sufficiently good. At these
energies, the diffraction minimum is practically overfull by the real part of
the spin-non-flip amplitude and the contribution of the spin-flip-amplitude;
however, the $t$-dependence of the analysing power is very well reproduced in
this region of the momentum transfer. Note that the magnitude and the energy
dependence of this parameter depend on the energy behavior of the zeros of the
imaginary-part of the spin-flip amplitude and the real-part of the spin-
nonflip amplitude. Figure 22 shows $A_{N}(t)$ at $\sqrt{s}=9.2$ GeV and
$\sqrt{s}=13.7$ GeV. At these energies the diffraction minimum deepens and its
form affects the form of $A_{N}(t)$. At last, $A_{N}(t)$ is shown at large
energies $\sqrt{s}=19.4$ GeV and $\sqrt{s}=23.4$ GeV in Fig. 23. The
diffraction dip in the differential cross section has a sharp form and it
affects the sharp form of $A_{N}(t)$. The maximum negative values of $A_{N}$
coincide closely with the diffraction minimum.

We have found that the contribution of the spin-flip to the differential cross
sections is much less than the contribution of the spin-nonflip amplitude in
the examined region of momentum transfers from these figures; $A_{N}$ is
determined in the domain of the diffraction dip by the ratio

$\displaystyle A_{N}\sim Imf_{-}/Ref_{+}.$ (53)

The size of the analyzing power changes from $-45\%$ to $-50\%$ at
$\sqrt{s}=50$ GeV up to $-25\%$ at $\sqrt{s}=500$ GeV. These numbers give the
magnitude of the ratio Eq.(53) that does not strongly depend on the phase
between the spin-flip and spin-nonflip amplitudes. This picture implies that
the diffraction minimum is mostly filled by the real-part of the spin-nonflip
amplitude and that the imaginary-part of the spin-flip amplitude increases in
this domain as well.

##  X Conclusions

Practically for first time, a simultaneously research of proton-proton,
proton-antiproton and proton-neutron elastic scattering has been carried out
in a wide energy ((from $3.6$ GeV up to $13$ TeV) and momentum transfer region
(from $|t|=2.10^{-4}$ GeV2 up to $|t|=14$ GeV2. In the fitting procedure we
used only statistical errors. Systematic errors, which are mostly determined
by indefinite of a luminosity, were taken into account as additional
normalization coefficient. As a result, a wide range of possible forms of the
scattering amplitudes are pretty decreases. As a result, a simultaneously
description of the cross sections and spin correlation parameter of different
nucleon-nucleon reactions, including 90 sets of experimental data, with the
total number of data $N=4326$ gives very reasonably
$\sum_{i,j}\chi^{2}_{i,j}=4826$. The $pn$ case with $526$ experimental data,
where the basic parameters were fixed from $pp$ and $p\bar{p}$ scattering,
$\sum_{i,j}\chi^{2}_{i,j}=585$.

Our analysis is carried out by using a successful development of the HEGS
model which can be applied in the wide energy and momentum transfer regions.
The model of hadron interaction is based on the analyticity of the scattering
amplitude with taking into account the hadron structure, which is represented
by GPDs. Different origins of the non-linear behavior of the slope of the
scattering amplitude are analyzed. The possible contribution of a meson
threshold is compared with different forms of the approximations for a non-
linear slope at small momentum transfer.

The relative contributions of the possible different part of the scattering
amplitude were especially analyzed. It is remarkable that in the model the
main pomeron and odderon amplitudes have the same intercept. This leads after
eikonalization to the $\ln^{2}(s)$ energy dependence. In this sense, we have
some case of maximal Odderon. In the model, the odderon amplitude has a
special kinematic factor and does not give a visible contribution at zero
momentum transfer.

It was found that the new anomalous term with a large slope has the
complicated logarithmic energy dependence and has the cross even properties.
Hence, it is part of the pomeron amplitude and is also proportional to charge
distributions. Our analysis of the contribution of the so-called hard pomeron
with a large intercept does not show a visible contribution of this term. The
second additional term, which represents the additional oscillation properties
of the scattering amplitude at small momentum transfer with cross-odd
properties has an logarithmic energy dependence and is proportional to
gravitomagnetic form factor. Hence, it belongs to the odderon contribution in
the scattering amplitude.

This helps reduce decreased some tension between the TOTEM and ATLAS data. No
contribution is shown of hard-Pomeron in elastic hadron scattering. However,
the importance of Odderon’s contribution is shown. Also, a good description of
proton-neutron differential scattering with 526 experimental point including
the experimental data which reach extremely small momentum transfer $t=2\
10^{-4}$ GeV2 is obtained, on the basis the amplitudes obtained on $pp$ and
$p\bar{p}$ scattering. A good enough description of the polarization data was
also obtained, which reflects the true phases of the spin-non-flip and spin-
flip amplitudes, so the value of $A_{N}(s,t)\sim\varphi_{sp-n-
flip}-\varphi_{sp-flip}$.

Our work supports that GPDs reflect the basic properties of the hadron
structure and provide some bridge between many different reactions. The
determined new form of the momentum transfer dependence of GPDs allows one to
obtain different form factors, including Compton form factors, electromagnetic
form factors, transition form factor, and gravitational form factors. The
chosen form of the $t$-dependence of GPDs of the pion (the same as the
$t$-dependence of the nucleon) allow us to describe the electromagnetic and
GFF of pion and pion-nucleon scattering.

The impact of the different $t$ dependence of the real and imaginary parts of
the elastic scattering amplitude at small $t$ should be noted. The dispersion
relation shows that the real part has zero at small $t$ , approximately in the
domain $-t=0.1$; of course this depends on the energy. Hence, the contrary
behavior of the real part (growth at small $t$ \- so-called ”peripheral case”
of the phase of the scattering amplitude) which is examined in [32] has no
physical meaning.

Acknowledgement:  OVS would like to thank O. Teryaev and Yu. Uzikov for their
kind and helpful discussion. This research was carried out at the expense of
the grant of the Russian Science Foundation No. 23-22-00123,
https://rscf.ru/project/23-22-00123.

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